WO2025093253A1

WO2025093253A1 - High-frequency cascade readout

Info

Publication number: WO2025093253A1
Application number: PCT/EP2024/078611
Authority: WO
Inventors: Jacob Francis CHITTOCK-WOOD; Ross Cho Chun LEON; Michael Allan FOGARTY; Miguel Fernando GONZALEZ ZALBA
Original assignee: Quantum Motion Technologies Ltd
Current assignee: Quantum Motion Technologies Ltd
Priority date: 2023-10-30
Filing date: 2024-10-10
Publication date: 2025-05-08
Anticipated expiration: 2026-04-30
Also published as: TW202534580A

Abstract

A circuit for reading out the state of a qubit 100 having singlet and triplet spin states comprising: a double quantum dot 101 forming a qubit 100 comprising first and second quantum dots 105, 106. The circuit comprises a cascade quantum dot 102 capacitively coupled to the second quantum dot 106; a charge reservoir 103 tunnel coupled to the cascade quantum dot 102; and readout circuitry 104 coupled to the reservoir 103. When a charge carrier tunnels between the first and second quantum dots 105, 106, a charge carrier tunnels between the cascade quantum dot 102 and the charge reservoir 103. A frequency source connected to a gate electrode or the charge reservoir 103 is configured to apply an alternating potential 111 at a first frequency, h, thereby to cause cyclic tunnelling of a charge carrier only when the qubit is in the singlet state which can be detected by the resonator circuit 104.

Description

HIGH-FREQUENCY CASCADE READOUT

FIELD OF THE INVENTION

The present invention relates to a circuit for reading out the state of qubits. The device is suitable for performing measuring or reading out the state of a qubit at a distance from that qubit.

BACKGROUND

The measurement of spin qubits in semiconductor nanostructures is an extremely difficult task. Existing methods typically rely on the measurement of the charge of the spin-carrying particle rather than the spin itself. This approach to spin readout is known as spin-to-charge conversion and is ubiquitous in semiconductor-based quantum computing architectures.

The preferred mechanism for spin-to-charge conversion is that of Pauli spin blockade due to the possibility to utilize it at higher temperatures (approaching a few Kelvin) and low magnetic fields (B < 1 Tesla).

Pauli spin blockade requires a double quantum dot involving two quantum dots in which two spin-carrying particles exist (typically electrons or holes). The double quantum dot forms a qubit. The two spin-carrying particles may be in a singlet spin configuration or a triplet state configuration. The singlet and triplet spin configurations provide the two states of the qubit. T unnelling of a charge between the two quantum dots is possible when the spins are in the singlet spin configuration whereas tunnelling is suppressed when the two quantum dots are in the triplet configuration.

The tunnelling of charge carriers between the two quantum dots of the double quantum dot is typically detected using a charge measurement.

A charge measurement can be performed using dissipative charge sensors in close proximity to the double quantum dot, such as a single-electron transistor or a quantum point contact. These are sensitive sensors but occupy substantial on- chip real estate complicating the generation of highly connected qubit architecture. In the case of the single-electron transistor, an additional quantum dot and two charge reservoirs are required for readout.

Dispersive charge sensors, such as the single electron box, occupy less space than single-electron transistors or quantum point contacts. Therefore, dispersive charge sensors can be used to provide a more compact way of measuring charge because they only require one additional quantum dot and one charge reservoir, and a measurement of the alternating current between them.

The above techniques are used for measuring charge. Alternatively, the tunnelling of charge carriers between the two quantum dots can be detected using a charge polarisation measurement. Charge polarization measurements facilitate the detection of Pauli spin blockade, in comparison with charge measurements.

A charge polarisation measurement can be performed using dispersive readout. This technique involves directly connecting the double quantum dot system to an electrical resonator via one of the gates that already defines the double quantum dot. The absence or presence of charge polarisation results in a change in the frequency of the alternating potential detected by the resonator. This can be observed as a change in amplitude, or change in phase, in the alternating potential. The absence and presence of charge polarisation corresponds to the triplet and singlet states of the double quantum dot, respectively.

The dispersive readout method presents advantages in terms of minimal on-chip footprint because it does not require additional on-chip readout infrastructure as opposed to the case with charge sensors.

However, dispersive readout typically suffers from poor signal-to-noise ratio because the measured polarisation charge is a fraction of the full electronic charge, i.e. the charge of a single electron. It is effectively the image charge at the gate of the quantum dot that necessarily is less than one electron due to the separation between the quantum dot and the sensor. Furthermore, due to the fact that charge and charge polarisation measurements are governed by Coulomb electrostatic energies, the measurements typically need to be performed locally. This is achieved by placing a charge, or charge polarisation, sensor in close proximity to the target qubit. This presents problems with regards to measuring spin qubits in dense arrays since many sensors need to be deployed locally to sense all qubits.

In order to measure the charge at a distance, an electron cascade can be implemented in which a charge tunnelling event produces a chain reaction of charge tunnelling events that propagates the charge to the location of the charge readout sensor.

It is desirable to provide improved techniques for performing spin qubit measurements.

SUMMARY OF INVENTION

A first aspect of the invention provides a circuit for reading out the state of a qubit. The circuit comprises: a double quantum dot forming a qubit having a singlet spin state and a triplet spin state, wherein the double quantum dot comprises a first quantum dot and a second quantum dot, wherein the second quantum dot is tunnel coupled to the first quantum dot; a gate electrode configured to control the energy state of either the first or the second quantum dot; a cascade quantum dot capacitively coupled to the second quantum dot; a charge reservoir tunnel coupled to the cascade quantum dot; readout circuitry connected to the charge reservoir; and a frequency source connected to the gate electrode or the charge reservoir and configured to apply an alternating potential thereto. The cascade quantum dot is tuned to be close to a charge transition such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the cascade quantum dot and the charge reservoir. The frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa. The readout circuitry is configured to: detect a first alternating potential; measure one or more properties of the detected first alternating potential; and infer that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state.

In this way it is possible to infer the state of a qubit at a distance by connecting the double quantum dot to a charge reservoir and readout circuitry by way of a cascade quantum dot that is capacitively coupled in between. The tunnelling between the cascade quantum dot and the charge reservoir is due to the electrostatic coupling between systems, i.e. between the double quantum dot comprising the first and second quantum dots, and the cascade quantum dot. The circuit can therefore advantageously be used to perform a charge polarisation measurement at a distance with an improved signal-to-noise ratio. The readout circuitry can advantageously be used to infer the state of the qubit without requiring an external sensor at the nanostructure. Depending on a comparison between the one or more measured properties and expected values for singlet and triplet spin configurations respectively, the readout circuitry may either infer that the qubit is in the triplet spin state or the readout circuitry may infer that the qubit is in the singlet spin state.

The circuit can therefore facilitate detection of the state of a qubit using a readout sensor of the readout circuitry that may be some distance away from the qubit. A cascade Pauli spin blockade process, in which the tunnelling of charge carriers in a double dot system initiates a tunnelling process in a neighbouring system, advantageously provides improved signal-to-noise ratio. Advantageously, this means that a charge polarisation measurement can be performed at a distance to read out the state of the qubit.

Optionally, the detected first alternating potential may be detected in both real (in- phase) and imaginary (90 degrees out of phase) axes. The two signals may then be combined to deduce the change in amplitude or phase between the singlet and triplet spin states.

In this way, qubits in the circuit can be placed further from the readout sensor, which is advantageous in the context of a quantum computer where readout electronics take up significant space and require connection via classical control wiring.

Other techniques for transferring quantum information are known, including charge shuttling or performing SWAP operations. However, these techniques are subject to spin relaxation time and the fidelity of corresponding operations. Accordingly, a further advantage of the invention is that the fidelity of the readout measurement is improved because cascade Pauli spin blockade involves inducing successive tunnelling processes. In this way, the cascade Pauli spin blockade technique described herein beneficially enhances the charge detected similar to existing techniques involving latching, DC cascading, and spin-polarized singleelectron boxes whilst retaining the advantageous quantum non-demolition nature of in-situ dispersive readout methods. In this way, the system remains in an eigenstate of the system after readout as opposed to that information being destroyed or lost. Non-demolition readout therefore implies qubit initialisation can be done by measurement.

A further advantage stems from the cyclic nature of the tunnelling, which has the effect that a readout measurement is a quantum non-demolition measurement. The cyclic tunnelling typically occurs substantially synchronously. When the frequency source is connected to the gate electrode, the frequency source is configured to apply an alternating potential to the gate electrode; when the frequency source is connected to the charge reservoir, the frequency source is configured to apply an alternating potential to the charge reservoir. Optionally, the frequency source may be connected to both, such that the circuit can be flexibly configured. Typically, when an alternating potential is applied to the gate electrode, the tunnelling is initiated between the first and second quantum dot, which consequently induces tunnelling in the neighbouring tunnel-coupled system, i.e. the cascade quantum dot and the charge reservoir. Preferably, in this case, a transmission scattering coefficient is measured, i.e. S21. Alternatively, when an alternating potential is applied to the charge reservoir, the tunnelling is initiated between the cascade quantum dot and the charge reservoir, which consequently induces tunnelling in the neighbouring tunnel-coupled system comprising the first and second quantum dots. Preferably, in this case, a reflection scattering coefficient is measured, i.e. S11. In both cases, the tunnelling occurs continuously at a frequency matching the driving frequency, i.e. the frequency of the alternating potential, during the application of the alternating potential, and the tunnelling is substantially synchronous within all tunnel-coupled circuit elements capacitively coupled to neighbouring tunnel-coupled circuit elements.

Typically, the readout circuitry may comprise one or more resonant circuits having a pre-determined resonant frequency. Preferably, the frequency applied by the frequency source, used to drive the circuit, is a resonant frequency of a resonant circuit in the readout circuitry. In other examples, other off-chip or on-chip readout sensors may be employed as part of the readout circuitry. In addition to readout sensors, the readout circuitry preferably further comprises a processor to measure properties of the alternating potentials, and to infer the state of a qubit by checking the measured properties of the detected alternating potential against expected values for the singlet and triplet spin states respectively. It may be determined that the measured values substantially match the expected values when the measured and expected values agree within a pre-determined uncertainty value. Furthermore, the expected values may comprise a range of values, and it may be determined that the measured values substantially match the expected values when the measured values lie within the range of expected values. In each case, one or more measured properties may be compared with the corresponding one or more expected values.

Optionally, the expected values of the properties for the triplet spin state, and the expected values of the properties for the singlet spin state, are determined in a calibration step. Typically, a calibration step may involve determining expected values of the properties for the triplet spin state by recording the properties of an alternating potential detected by the readout circuitry when the double quantum dot is in the (1 ,1) charge configuration, i.e. with a charge carrier on each of the first and second quantum dots. The recorded properties from the (1 ,1) charge configuration may be determined to be the expected values of the properties for the triplet spin state. The calibration step may also involve determining expected values of the properties for the singlet spin state by recording the properties of an alternating potential detected by the readout circuitry when the double quantum dot is in the (0,2) charge configuration, i.e. with two charge carriers on the second quantum dot. The recorded properties from the (0,2) charge configuration may be determined to be the expected values of the properties for the singlet spin state.

Optionally, the expected values of the properties for the triplet spin state may comprise ranges of expected values. For example, if the measured frequency of the detected first alternating potential is greater than or equal to a first expected frequency value and less than or equal to a second expected frequency value larger than the first expected frequency value, then the readout circuitry may determine that the measured frequency matches the expected frequency value for the triplet spin state, thereby inferring that the qubit is in the triplet spin state. Similarly, the expected amplitude value may be between a first expected amplitude value and a second expected amplitude value larger than the first expected amplitude value; and the expected phase value may be between a first expected phase value and a second expected phase value larger than the first expected phase value.

Optionally, similarly, the expected values of the properties for the singlet spin state may comprise ranges of expected values. For example, if the measured frequency of the detected first alternating potential is greater than or equal to a third expected frequency value and less than or equal to a fourth expected frequency value larger than the third expected frequency value then the readout circuitry may determine that the measured frequency matches the expected frequency value for the singlet spin state, thereby inferring that the qubit is in the singlet spin state. Similarly, the expected amplitude value may be between a third expected amplitude value and a fourth expected amplitude value larger than the third expected amplitude value; and the expected phase value may be between a third expected phase value and a fourth expected phase value larger than the third expected phase value.

Preferably, the ranges of expected values of the properties for the triplet spin state do not overlap with the corresponding ranges of expected values of the properties for the singlet spin states.

Typically, the expected values may depend on system parameters such as material, temperature, surface factors, circuitry variability, and other such experimental factors deviating from theoretically determined values for the singlet and triplet spin states. Therefore, a confidence interval around the expected values may be determined, within which the readout circuitry can be used to infer that the measured one or more properties substantially match expected values for one of the singlet or the triplet spin states.

The circuit may further comprise one or more intermediate cascade double quantum dots each comprising a first intermediate cascade quantum dot and a second intermediate cascade quantum dot. The second intermediate cascade quantum dot is preferably tunnel coupled to the first intermediate cascade quantum dot; the second quantum dot is preferably capacitively coupled to the first intermediate cascade quantum dot in one of the one or more intermediate cascade double quantum dots; and the cascade quantum dot is preferably capacitively coupled to the second intermediate cascade quantum dot in one of the one or more intermediate cascade double quantum dots. Each intermediate cascade double quantum dot is preferably tuned to be close to a charge transition such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the first intermediate cascade quantum dot and the second intermediate cascade quantum dot of that intermediate cascade double quantum dot. The frequency source is preferably configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the first intermediate cascade quantum dot to the second intermediate cascade quantum dot of each of the one or more intermediate cascade double quantum dots sequentially, and vice-versa, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa.

Advantageously, the intermediate cascade double quantum dots increase the separation between the qubit and the readout sensor, with an improved signal-to- noise ratio at the readout sensor. The signal-to-noise ratio is improved because the qubit state information is transmitted using the cascade Pauli spin blockade process, in which the tunnelling of charge carriers in a double dot system initiates a tunnelling process in a neighbouring system. This amplifies the signal detected by the readout circuitry.

Using the cascade Pauli spin blockade process to trigger a cascade of cyclic quantum tunnelling events in a chain of double quantum dots advantageously results in a charge polarisation approaching a full electron charge e. In existing dispersive readout methods, the polarisation charge is ae, with a < 1, wherein a is the ratio between the difference between the gate capacitances to the first and second quantum dots, divided by the total capacitance of the measured quantum dot. For example, for the first quantum dot, a is the ratio between the gate capacitance to the first quantum dot minus the gate capacitance to the second quantum dot, divided by the total capacitance of the first quantum dot. Advantageously, the signal-to-noise ratio of the technique is enhanced by a factor F = 1 + (1 - e)²1 a², with e -> 0, wherein E is the ratio between the capacitance of the charge reservoir to the cascade quantum dot and the total capacitance of the cascade quantum dot. The capacitance of the charge reservoir to the cascade quantum dot can be engineered to tend to zero by increasing the size of the cascade quantum dot or increasing the separation between the cascade quantum dot and the charge reservoir. In all cases, the separation between the cascade quantum dot and the charge reservoir is within a quantum interaction distance whereby a charge carrier can tunnel between the charge reservoir and the cascade quantum dot. Accordingly, whereas the polarisation charge is significantly less than a full electron charge, ae < e, according to the invention the signal-to-noise ratio is increased by a factor greater than 1 , F > 1. Thus, the measured polarisation charge advantageously approaches the electron charge. The increased signal-to-noise ratio is in part due to the proximity of the final tunnelling process to the readout sensor and in part due to the measured charge carrier tunnelling directly involving the reservoir rather than merely being adjacent to the reservoir.

Any number of intermediate cascade double quantum dots may be provided to separate the qubit and the readout sensor by an arbitrary distance. When the circuit includes only one intermediate cascade double quantum dot, a first intermediate cascade double quantum dot, the second quantum dot is preferably capacitively coupled to the first intermediate cascade quantum dot of the first intermediate cascade double quantum dot; and the cascade quantum dot is preferably capacitively coupled to the second intermediate cascade quantum dot of the first intermediate cascade double quantum dot. In this way, one intermediate cascade double quantum dot is preferably both coupled to the double quantum dot forming a qubit and coupled to the cascade quantum dot.

When the circuit includes only two intermediate cascade double quantum dots, first and second intermediate cascade double quantum dots, the second quantum dot is preferably capacitively coupled to the first intermediate cascade quantum dot of the first intermediate cascade double quantum dot; and the cascade quantum dot is preferably capacitively coupled to the second intermediate cascade quantum dot of the second intermediate cascade double quantum dot. The second intermediate cascade quantum dot of the first intermediate cascade double quantum dot is preferably capacitively coupled to the first intermediate cascade quantum dot of the second intermediate cascade double quantum dot. In this way, one intermediate cascade double quantum dot is coupled to the double quantum dot forming a qubit, another intermediate cascade double quantum dot is coupled to the cascade quantum dot, and the first and second intermediate cascade double quantum dots are coupled to each other.

When the circuit includes three or more intermediate cascade double quantum dots, first, second, third, ... , n-th intermediate cascade double quantum dots, the second quantum dot is preferably capacitively coupled to the first intermediate cascade quantum dot of the first intermediate cascade double quantum dot; and the cascade quantum dot is preferably capacitively coupled to the second intermediate cascade quantum dot of the n-th intermediate cascade double quantum dot. The second intermediate cascade quantum dot of the k-th intermediate cascade double quantum dot is preferably capacitively coupled to the first intermediate cascade quantum dot of the (k+1)-th intermediate cascade double quantum dot, for 1 < k < n. In this way, the first intermediate cascade double quantum dot is coupled to the double quantum dot forming a qubit and the second intermediate cascade double quantum dot; the n-th intermediate cascade double quantum dot is coupled to the cascade quantum dot and the (n-1 )-th intermediate cascade double quantum dot; and the remaining intermediate cascade double quantum dots are each coupled to two neighbouring intermediate cascade double quantum dots.

In this way, the cascade Pauli spin blockade process proceeds sequentially from the double quantum dot forming the qubit, to each of the one or more intermediate cascade double quantum dots, to the cascade quantum dot and the reservoir. This advantageously improves the signal-to-noise ratio at the readout sensor which may advantageously be separated by an arbitrary distance from the qubit. The ability to separate the qubit and the readout sensor provides a further advantage that qubits in the circuit can be arranged in a dense array because space near the qubit does not need to be reserved for reservoirs and readout circuitry. Alternatively, the cascade Pauli spin blockade process may be initiated by driving the charge reservoir at the first frequency. In which case, the initial sequence of tunnelling events is reversed. However, once all the charge carriers are tunnelling (or not, depending on the spin state of the qubit) within the tunnel- coupled systems, the tunnelling will continue substantially synchronously for the duration of the application of the applied alternating potential.

Optionally, the circuit may further comprise: a second double quantum dot forming a second qubit having a singlet spin state and a triplet spin state, wherein the second double quantum dot comprises a second first quantum dot and a second second quantum dot, wherein the second second quantum dot is tunnel coupled to the second first quantum dot; a second gate electrode configured to control the energy state of either the second first or the second second quantum dot; and a second cascade quantum dot capacitively coupled to the second second quantum dot and tunnel coupled to the charge reservoir. The frequency source is preferably connected to the second gate electrode or the charge reservoir and is preferably configured to apply an alternating potential thereto. The frequency source is preferably further configured to apply an alternating potential at a second frequency thereby to cause cyclic tunnelling of a charge carrier from the second first quantum dot to the second second quantum dot and vice-versa only when the second qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the second cascade quantum dot to the reservoir, and vice- versa. The readout circuitry is preferably further configured to: detect a second alternating potential; measure one or more properties of the detected second alternating potential; and infer that the second qubit is in the triplet spin state when the one or more measured properties of the detected second alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the second qubit is in the singlet spin state when the one or more measured properties of the detected second alternating potential substantially match expected values of the properties for the singlet spin state.

Advantageously, the first and second qubits utilise a common charge reservoir and readout circuitry in this example. Further qubits may also share the charge reservoir and readout circuitry. Using this arrangement, the state of multiple qubits can be inferred using the same readout circuitry. This advantageously reduces the circuitry required to determine the state of a plurality of qubits. Fewer reservoirs are required, and less readout circuitry is required. Reducing the circuitry requirements in this way beneficially enables a denser qubit arrangement.

Optionally, when the readout circuitry comprises resonant circuits, the readout circuitry may comprise n resonant circuits connected in parallel to measure the state of n qubits respectively. Each resonant circuit preferably has a different resonant frequency. Advantageously, use of different resonant frequencies enables the signals to be distinguished even if combined. Typically, the output of the n resonant circuits is combined using a capacitive coupling.

Furthermore, in comparison with other techniques for transferring quantum information such as charge shuttling or SWAP operations, the circuit can advantageously be used to substantially simultaneously readout the state of a plurality of qubits.

Optionally, alternatively or additionally, the circuit may further comprise: a third double quantum dot forming a third qubit having a singlet spin state and a triplet spin state, wherein the third double quantum dot comprises a third first quantum dot and a third second quantum dot, wherein the third second quantum dot is tunnel coupled to the third first quantum dot; and a third gate electrode configured to control the energy state of either the third first or the third second quantum dot. The cascade quantum dot is preferably further capacitively coupled to the third second quantum dot. The frequency source is preferably connected to the third gate electrode or the charge reservoir and is preferably configured to apply an alternating potential thereto. The frequency source is preferably further configured to apply an alternating potential at a third frequency thereby to cause cyclic tunnelling of a charge carrier from the third first quantum dot to the third second quantum dot and vice-versa only when the third qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa. The readout circuitry is preferably further configured to: detect a third alternating potential; measure one or more properties of the detected third alternating potential; and infer that the third qubit is in the triplet spin state when the one or more measured properties of the detected third alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the third qubit is in the singlet spin state when the one or more measured properties of the detected third alternating potential substantially match expected values of the properties for the singlet spin state.

Advantageously, the first and third qubits utilise a common charge reservoir, readout circuitry, and cascade quantum dot. Further qubits may also share the charge reservoir, readout circuitry and cascade quantum dot. In some cases, a plurality of qubits may share a charge reservoir and readout circuitry. In some cases, a plurality of qubits may share a charge reservoir, readout circuitry and a cascade quantum dot. In some cases, a first plurality of qubits may share a charge reservoir and readout circuitry; and a second plurality of qubits may share a charge reservoir, readout circuitry and a cascade quantum dot.

Sharing a charge reservoir, readout circuitry, and a cascade quantum dot between a plurality of qubits advantageously reduces the circuitry required to determine the state of the plurality of qubits. Fewer reservoirs, less readout circuitry, and fewer cascade quantum dots are required. Reducing the circuitry requirements in this way beneficially enables a denser arrangement of qubits. Specifically, sharing cascade quantum dots between qubits advantageously enables the area occupied by the reservoir to be reduced: the reservoir requires fewer cascade quantum dots to be arranged adjacent to the reservoir around its circumference to couple the same number of qubits to the reservoir.

Optionally, one or more of the intermediate cascade double quantum dots are common to the double quantum dot and the second and/or third double quantum dots. In comparison with other techniques for transferring quantum information such as charge shuttling or SWAP operations, the circuit can advantageously be used to substantially simultaneously readout the state of a plurality of qubits. The intermediate cascade double quantum dots can be used to transfer quantum information relating to the first, second and/or third qubits concurrently.

Optionally, one or more of the intermediate cascade double quantum dots are common to the double quantum dot and the second double quantum dot. Optionally, one or more of the intermediate cascade double quantum dots are common to the double quantum dot and the third double quantum dot. Optionally, one or more of the intermediate cascade double quantum dots are common to the second double quantum dot and the third double quantum dot. Optionally, one or more of the intermediate cascade double quantum dots are common to the double quantum dot, the second double quantum dot, and the third double quantum dot.

Advantageously, wherein one or more of the double quantum dots forming qubits in the circuit share one or more intermediate cascade double quantum dots, the area occupied by the circuit can be reduced. By using common intermediate cascade double quantum dots, the routes between each qubit and the reservoir partially overlap, which advantageously saves space. For example, the qubits may be arranged in a nested architecture, advantageously enabling a dense qubit array.

The first, second and third frequencies may be the same. In this case, the state of the first, second and third qubits can be inferred sequentially. However preferably, the second and/or third frequencies are different from the first frequency.

Optionally, the second frequency is different from the first frequency and the third frequency is different from the first frequency. Typically, the first, second and third frequencies are different from each other.

Optionally, for a circuit comprising m double quantum dots forming m qubits, the frequency source may be configured to apply alternating potentials at f different frequencies, wherein 1 < f < m. Alternatively, the circuit may comprise a plurality of frequency sources configured to apply alternating potentials at one or more different frequencies.

Advantageously, applying alternating potentials having different frequencies to different qubits (wherein the frequencies are applied to the respective gate electrodes of the first quantum dots of the double quantum dots forming the respective qubits) enables a frequency division multiplexing readout technique. This means that the state of a plurality of qubits can advantageously be simultaneously readout, because the different frequencies can be distinguished by the readout circuitry. The different frequencies are typically distinguished by utilising different resonator circuits, each resonator circuit having a different resonant frequency. The frequency source is preferably configured to drive the circuit at one or more frequencies corresponding to resonant frequencies of the plurality of resonator circuits. Typically, one or more frequencies are mutually orthogonal.

Therefore, advantageously the readout circuitry can be employed for reading the state of multiple qubits at substantially the same time. This can be achieved by applying a different alternating potential to different qubits, i.e. an alternating potential having a different frequency. Thus, the comparison between the properties of the detected alternating potential with determined expected values of the properties for the different spin configurations of the qubit can be used to infer the state of each qubit. By using several different frequencies it may be possible to use frequency multiplexing to infer the state of different qubits. The determined expected values may be determined by measuring the circuit in a known charge configuration corresponding to the singlet and triplet spin state configurations respectively and recording the measured properties as reference values used to subsequently infer a spin state of a to-be-measured qubit. Typically a confidence interval may be applied such that measured values are considered to match the reference values (the expected values) when the measured values of the properties of the alternating potential lie within a range of values.

A second aspect of the invention provides a circuit for reading out the state of a qubit. The circuit comprises: a double quantum dot comprising a first quantum dot having a first spin state and a second spin state and a second quantum dot having the first spin state and the second spin state, wherein the second quantum dot is tunnel coupled to the first quantum dot; wherein the first quantum dot forms a data qubit and the second quantum dot forms an ancillary qubit; or the first quantum dot forms an ancillary qubit and the second quantum dot forms a data qubit; a gate electrode configured to control the energy state of either the first or the second quantum dot; control circuitry connected to the ancillary qubit, the control circuitry configured to initialise the ancillary qubit in the first spin state; a cascade quantum dot capacitively coupled to the second quantum dot; a charge reservoir tunnel coupled to the cascade quantum dot; readout circuitry connected to the charge reservoir; and a frequency source connected to the gate electrode or the charge reservoir and configured to apply an alternating potential thereto. The cascade quantum dot is tuned to be close to a charge transition such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the cascade quantum dot and the charge reservoir. The frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the data qubit is in the second spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa. The readout circuitry is configured to: detect a first alternating potential; measure one or more properties of the detected first alternating potential; and infer that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; and infer that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state.

Similarly to the first aspect, it is possible to infer the state of a qubit at a distance by connecting the qubit to a charge reservoir and readout circuitry by way of a cascade quantum dot that is capacitively coupled in between. The tunnelling between the cascade quantum dot and the charge reservoir is due to the electrostatic coupling between systems. The circuit can therefore advantageously be used to perform a charge polarisation measurement at a distance with an improved signal-to-noise ratio. The readout circuitry can advantageously be used to infer the state of the qubit without requiring an external sensor at the nanostructure. Depending on a comparison between the one or more measured properties and expected values for spin states which are parallel and anti-parallel respectively to the ancillary qubit spin state, the readout circuitry may either infer that the data qubit is in the first spin state or the readout circuitry may infer that the data qubit is in the second spin state.

The advantageous features of the first aspect also apply to the second aspect. In the first aspect, the circuit is suitable for reading out the state of a singlet-triplet qubit in the double quantum dot; in the second aspect, the circuit is suitable for reading out the state of a single spin qubit in one quantum dot of the double quantum dot. Therefore, in the first aspect, the measured state is the singlet spin state or the triplet spin state, whereas in the second aspect the measured state is a first or second spin state corresponding to spin up and down. The cascade process involving substantially synchronous cyclic tunnelling in capacitively coupled neighbouring double quantum dot systems, is utilised in both the first and second aspects of the invention.

The expected values of the properties for the first spin state correspond to expected values of the properties of the detected first alternating potential when both the data qubit and ancillary qubit have the same spin state. Typically, the method may include a calibration step comprising determining expected values of the properties for the first spin state by recording the properties of an alternating potential detected by the readout circuitry when the double quantum dot is in the (1 ,1) charge configuration, i.e. with a charge carrier on each ofthe first and second quantum dots. The recorded properties from the (1 ,1) charge configuration may be determined to be the expected values of the properties for the first spin state, i.e. when the data qubit and the ancillary qubit have the same, parallel, spin state.

The expected values of the properties for the second spin state correspond to expected values of the properties of the detected first alternating potential when the data qubit and ancillary qubit have opposite spin states. Typically, the method may include a calibration step comprising determining expected values of the properties for the second spin state by recording the properties of an alternating potential detected by the readout circuitry when the double quantum dot is in the (0,2) charge configuration, i.e. with two charge carriers on the second quantum dot. The recorded properties from the (0,2) charge configuration may be determined to be the expected values of the properties for the second spin state, i.e. when the data qubit and the ancillary qubit have opposite, anti-parallel, spin states relative to each other.

A third aspect of the invention provides a method of reading out the state of a qubit, wherein the qubit is a double quantum dot qubit comprising first and second tunnel-coupled quantum dots, the qubit having a singlet spin state and a triplet spin state, wherein a cascade quantum dot is capacitively coupled to the double quantum dot qubit, tunnel coupled to a charge reservoir, and tuned to be close to a charge transition, and wherein a frequency source is connected to the charge reservoir or a gate electrode configured to control the energy state of either the first or the second quantum dot, the frequency source being configured to apply an alternating potential thereto. The method comprises: applying, using the frequency source, an alternating potential at a first frequency to the gate electrode or the charge reservoir; and detecting, using readout circuitry coupled to the charge reservoir, a first alternating potential; measuring, using the readout circuitry, one or more properties of the detected first alternating potential; and inferring that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; or inferring that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state. The double quantum dot qubit is configured such that when an alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa is caused only when the qubit is in the singlet spin state, and also cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir is caused, and vice-versa.

Advantageously, this method can be used to infer the state of a qubit at a distance. The state of the qubit can be readout using a charge polarisation measurement technique. A cascade Pauli spin blockade process, in which the tunnelling of charge carriers in a double dot system initiates a tunnelling process in a neighbouring system, advantageously provides improved signal-to-noise ratio. In this way, the method can be used to infer the state of qubits which are positioned further from the readout sensor, which is advantageous in the context of a quantum computer where readout electronics take up significant space and require connection via classical control wiring.

A fourth aspect of the invention provides a method of reading out the state of a data qubit, wherein the data qubit is a quantum dot having a first spin state and a second spin state, the data qubit is tunnel coupled to another quantum dot having the first spin state and the second spin state, the another quantum dot forming an ancilla qubit, wherein the data qubit and ancilla qubit form a double quantum dot, wherein a cascade quantum dot is capacitively coupled to the double quantum dot, tunnel coupled to a charge reservoir, and tuned to be close to a charge transition, and wherein a frequency source is connected to the charge reservoir or a gate electrode configured to control the energy state of either the quantum dot or the another quantum dot, the frequency source being configured to apply an alternating potential thereto. The method comprises: initialising, using control circuitry connected to the ancillary qubit, the ancillary qubit in the first spin state; applying, using the frequency source, an alternating potential at a first frequency to the gate electrode or the charge reservoir; and detecting, using readout circuitry connected to the charge reservoir, a first alternating potential; measuring, using the readout circuitry, one or more properties of the detected first alternating potential; and inferring that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; or inferring that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state; wherein the double quantum dot is configured such that when an alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the quantum dot to the another quantum dot and vice-versa is caused only when the data qubit is in the second spin state, and also cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir is caused, and vice-versa. Advantageously, this method can be used to infer the state of a qubit at a distance. The state of the qubit can be readout using a charge polarisation measurement technique. A cascade Pauli spin blockade process, in which the tunnelling of charge carriers in a double dot system initiates a tunnelling process in a neighbouring system, advantageously provides improved signal-to-noise ratio.

In this way, the method can be used to infer the state of qubits which are positioned further from the readout sensor in the readout circuitry, which is advantageous in the context of a quantum computer where readout electronics take up significant space and require connection via classical control wiring.

The third and fourth aspects of the invention correspond to the first and second aspects respectively and have corresponding advantageous features.

Another aspect of the invention provides a circuit for reading out the state of a qubit. The circuit comprises: a double quantum dot forming a qubit having a singlet spin state and a triplet spin state, wherein the double quantum dot comprises a first quantum dot and a second quantum dot, wherein the second quantum dot is tunnel coupled to the first quantum dot; a gate electrode configured to control the energy state of either the first or the second quantum dot; a cascade quantum dot capacitively coupled to the second quantum dot; a charge reservoir tunnel coupled to the cascade quantum dot; readout circuitry connected to the charge reservoir; and a frequency source connected to the gate electrode or the charge reservoir and configured to apply an alternating potential thereto. The cascade quantum dot is arranged such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the cascade quantum dot and the charge reservoir. The frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa. The readout circuitry is configured to: detect a first alternating potential; measure one or more properties of the detected first alternating potential; and infer that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state.

A further aspect of the invention provides a circuit for reading out the state of a qubit. The circuit comprises: a double quantum dot comprising a first quantum dot having a first spin state and a second spin state and a second quantum dot having the first spin state and the second spin state, wherein the second quantum dot is tunnel coupled to the first quantum dot; wherein the first quantum dot forms a data qubit and the second quantum dot forms an ancillary qubit; or the first quantum dot forms an ancillary qubit and the second quantum dot forms a data qubit; a gate electrode configured to control the energy state of either the first or the second quantum dot; control circuitry connected to the ancillary qubit, the control circuitry configured to initialise the ancillary qubit in the first spin state; a cascade quantum dot capacitively coupled to the second quantum dot; a charge reservoir tunnel coupled to the cascade quantum dot; readout circuitry connected to the charge reservoir; and a frequency source connected to the gate electrode or the charge reservoir and configured to apply an alternating potential thereto. The cascade quantum dot is arranged such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the cascade quantum dot and the charge reservoir. The frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the data qubit is in the second spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa. The readout circuitry is configured to: detect a first alternating potential; measure one or more properties of the detected first alternating potential; and infer that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; and infer that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state.

Another aspect of the invention provides a method of reading out the state of a qubit, wherein the qubit is a double quantum dot qubit comprising first and second tunnel-coupled quantum dots, the qubit having a singlet spin state and a triplet spin state, wherein a cascade quantum dot is capacitively coupled to the double quantum dot qubit, and tunnel coupled to a charge reservoir, and wherein a frequency source is connected to the charge reservoir or a gate electrode configured to control the energy state of either the first or the second quantum dot, the frequency source being configured to apply an alternating potential thereto. The method comprises: applying, using the frequency source, an alternating potential at a first frequency to the gate electrode or the charge reservoir; and detecting, using readout circuitry coupled to the charge reservoir, a first alternating potential; measuring, using the readout circuitry, one or more properties of the detected first alternating potential; and inferring that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; or inferring that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state. The double quantum dot qubit is configured such that when an alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa is caused only when the qubit is in the singlet spin state, and also cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir is caused, and vice-versa.

A further aspect of the invention provides a method of reading out the state of a data qubit, wherein the data qubit is a quantum dot having a first spin state and a second spin state, the data qubit is tunnel coupled to another quantum dot having the first spin state and the second spin state, the another quantum dot forming an ancilla qubit, wherein the data qubit and ancilla qubit form a double quantum dot, wherein a cascade quantum dot is capacitively coupled to the double quantum dot, and tunnel coupled to a charge reservoir, and wherein a frequency source is connected to the charge reservoir or a gate electrode configured to control the energy state of either the quantum dot or the another quantum dot, the frequency source being configured to apply an alternating potential thereto. The method comprises: initialising, using control circuitry connected to the ancillary qubit, the ancillary qubit in the first spin state; applying, using the frequency source, an alternating potential at a first frequency to the gate electrode or the charge reservoir; and detecting, using readout circuitry connected to the charge reservoir, a first alternating potential; measuring, using the readout circuitry, one or more properties of the detected first alternating potential; and inferring that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; or inferring that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state; wherein the double quantum dot is configured such that when an alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the quantum dot to the another quantum dot and vice-versa is caused only when the data qubit is in the second spin state, and also cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir is caused, and vice-versa.

BRIEF DESCRIPTION OF DRAWINGS

Embodiments of the invention will now be described with reference to the accompanying drawings in which:

Figure 1 is a schematic illustration of a circuit;

Figure 2A is a schematic illustration of a circuit including a plurality of intermediate cascade double quantum dots;

Figure 2B is a schematic illustration of a circuit including a plurality of intermediate cascade double quantum dots;

Figure 3 is a schematic illustration of a circuit involving frequency multiplexing; Figure 4 is a schematic illustration of a circuit involving frequency multiplexing including shared intermediate cascade double quantum dots;

Figures 5A, 5B and 5C are schematic illustrations of a circuit;

Figures 6A, 6B and 6C are schematic illustrations of measurements in different layers of a circuit;

Figures 7A and 7B are schematic illustrations demonstrating resilience to errors;

Figure 8 is a schematic illustration of a circuit;

Figure 9 is a flow chart of a method of reading out the state of a qubit;

Figure 10 is a flow chart of a method of reading out the state of a qubit; and Figure 11 is a schematic illustration of singlet and triplet spin state frequency responses.

DETAILED DESCRIPTION

The figures schematically illustrate circuits and methods for reading out the state of one or more qubits in a quantum device. The qubits are either double quantum dot spin qubits or single quantum dot spin qubits, each typically implemented in semiconductor nanostructures. The readout mechanism utilises spin-to-charge conversion in the form of Pauli spin blockade. Pauli spin blockade can be utilized both at temperatures above millikelvin temperatures, approaching a few Kelvin and at low magnetic fields, with a magnetic field strength below about 1 Tesla. For Pauli spin blockade to be implemented, two quantum dots are necessary in which two spin-carrying particles exist (typically electrons or holes). For a double quantum dot qubit, tunnelling of a spin-carrying particle, or charge carrier, between quantum dots separated by a tunnel barrier is possible when the spins are in a singlet spin configuration whereas tunnelling is suppressed when they are in a triplet configuration. In a singlet state, the double quantum dot has spin 0, and in a triplet state, the double quantum dot has spin 1 . Similarly, for a single quantum dot qubit, tunnelling of a spin-carrying particle between quantum dots separated by a tunnel barrier is possible when the spins are opposite (i.e. antiparallel), whereas tunnelling is suppressed when the spins are the same (i.e. parallel).

Figure 1 schematically illustrates a circuit suitable for reading out the state of a qubit. The circuit comprises a double quantum dot 101 , a cascade quantum dot 102, a charge reservoir 103, and a resonator circuit 104. The resonator circuit forms part of the readout circuitry used to infer the state of the qubit. The readout circuitry typically also includes a processor for processing the received signals. Figure 1 illustrates a ladder diagram for the double quantum dot 101 , the cascade quantum dot 102, and the reservoir 103. The ladder diagram illustrates the relative alignment of the electrochemical potentials of the quantum dots in the circuit prior to performing a readout measurement of the state of the qubit 100. According to the ladder diagram, an electron can tunnel from a higher electrochemical potential to a lower electrochemical potential separated by a tunnel barrier. The ladder diagram indicates only a selection of the electrochemical potential levels near the Fermi level, and typically there are higher, unoccupied, levels above the Fermi level and lower, occupied, levels below the Fermi level.

The electrochemical potentials can be tuned, i.e. raised or lowered, by applying a potential to respective gate electrodes (not shown). In this way, the respective gate electrodes can be used to control the energy states of the quantum dots.

The double quantum dot 101 is formed from two quantum dots: a first quantum dot 105 and a second quantum dot 106. The first and second quantum dots 105, 106 are separated by a first tunnel barrier 107. Charge carriers, i.e. electrons or holes, can pass through the first tunnel barrier 107 from the first quantum dot 105 to the second quantum dot 106, and also from the second quantum dot 106 to the first quantum dot 105, in a quantum tunnelling process. The double quantum dot 101 is defined by a first electrostatic barrier 109 and a second electrostatic barrier 110. The first and second electrostatic barriers 109, 110 strongly suppress quantum tunnelling, thereby confining charge carriers within the double quantum dot 101. The second electrostatic barrier 110 separates the double quantum dot 101 and the cascade quantum dot 102, thereby capacitively coupling the double quantum dot 101 and the cascade quantum dot 102.

The cascade quantum dot 102 and the reservoir 103 are separated by a second tunnel barrier 108. Therefore, charge carriers can pass through the second tunnel barrier 108 from the cascade quantum dot 102 to the reservoir 103, and also from the reservoir 103 to the cascade quantum dot 102, in a quantum tunnelling process. The reservoir 103 is typically an electron reservoir.

The electrostatic barriers and tunnel barriers illustrated in Figure 1 are provided by applying a bias potential to a gate electrode. The height of the barrier is determined by the magnitude of the bias potentials, thereby defining a tunnel barrier for smaller bias potentials or an electrostatic barrier for larger bias potentials.

In this way, the first quantum dot 105, second quantum dot 106, cascade quantum dot 102, reservoir 103, first and second tunnel barriers 107, 108 and first and second electrostatic barriers 109, 110 are all controlled using corresponding gate electrodes (not shown). The occupation of the quantum dots 105, 106, 102 and the height of the barriers 107-110 can therefore be modified by controlling the potential applied to the associated gate electrode.

The first and second quantum dots 105, 106 and the cascade quantum dot 102 have discrete energy levels as shown in the ladder diagram in Figure 1. This is due to the confinement of electrons in the quasi-zero-dimensional structures. In contrast, the energy levels of the reservoir 103 are continuous, not discretised.

In Figure 1 , the first and second quantum dots 105, 106 are each occupied with a single electron. In this configuration, the qubit 100 may be in a singlet state or a triplet state. In a singlet state, the spin of the double quantum dot is 0. In a triplet state, the spin of the double quantum dot is 1. In other examples, the first and second quantum dots 105, 106 may be occupied by more than a single electron, however typically the circuit is operated in an isolated regime. This means only one charge transition is probed. In other examples, the charge carriers may be holes. The description in relation to electrons here applies in a corresponding manner when the charge carriers are holes.

The plunger gate electrode of the first quantum dot 105 controls the energy state of the first quantum dot 105, which causes the electrochemical potential levels to be raised or lowered according to the sign of the potential. Typically, applying a positive bias to the plunger gate electrode lowers the electrochemical potential level and applying a negative bias to the plunger gate electrode raises the electrochemical potential level. Therefore, in this case, when a negative potential is applied to the plunger gate electrode of the first quantum dot 105, the electrochemical potential level illustrated in the first quantum dot 105 is raised above the electrochemical potential level in the second quantum dot 106 corresponding to occupation of the second quantum dot 106 with two opposing spin states.

This means that the electrochemical potentials in the second quantum dot 106 are tuned such that, when a negative potential is applied to the plunger gate electrode of the first quantum dot 105, due to Pauli spin blockade, the electron in the first quantum dot 105 can only tunnel to the second quantum dot 106 when the qubit 100 is in the singlet state. This tunnelling process is indicated in Figure 1 at 1a. The amplitude of the potential applied to the plunger gate electrode is such that if the qubit 100 is in the triplet state, the electron on the first quantum dot 105 will not tunnel to the second quantum dot 106 when the potential is applied because of Coulomb repulsion between the charge carrier spins.

The cascade quantum dot 102 is tuned to be close to a charge transition using a plunger gate electrode for the cascade quantum dot. This means that, due to the electrostatic barrier 110 providing capacitive coupling between the cascade quantum dot 102 and the second quantum dot 106, a change in charge carrier occupation of the second quantum dot 106 induces a change in charge carrier occupation of the cascade quantum dot 102. This occurs because, when the tunnelling of an electron as indicated at 1a occurs, the cascade quantum dot 102 enters a higher energy state, as indicated at 1 b in Figure 1 . The cascade quantum dot 102 is further tuned such that, from this higher energy state, the electron tunnels from the cascade quantum dot 102 through the second tunnel barrier 108 to the reservoir 103, as indicated at 1c in Figure 1.

In this example, a frequency source is configured to apply an alternating potential 111 at a first frequency, h, to the plunger gate electrode of the first quantum dot 105. The first alternating potential typically has a sinusoidal form, V = 7_in cos(< >₀t), with < >₀ = 2nf₀. The frequency source may be any commercially available frequency source capable of applying one or more frequencies to a plurality of gate electrodes. The alternating potential is used to drive the circuit. In this example, the alternating potential is applied to the plunger gate electrode of the first quantum dot. In other examples, the alternating potential may be applied to the plunger gate electrode of the second quantum dot, or to the charge reservoir.

Application of an alternating potential alternately raises and lowers the electrochemical potential level of the first quantum dot 105. When the electrochemical potential is raised, the charge carrier movement is as described above and as shown by arrows at 1a, 1 b and 1c. When the electrochemical potential is lowered, the charge carrier movement is reversed. This is shown by arrows at 2a, 2b and 2c. Firstly, at 2a, an electron tunnels from the second quantum dot 106 to the first quantum dot 105 through the first tunnel barrier 107. This change in occupation of the second quantum dot 106 induces a change in the neighbouring cascade quantum dot 102 because the energy state lowers as indicated at 2b. Finally, at 2c, an electron tunnels from the reservoir 103 to the cascade quantum dot 102 because the electrochemical potential is below the Fermi level of the reservoir 103.

In this way, the alternating potential applied to the gate electrode of the first quantum dot 105 causes cyclic tunnelling of a charge carrier from the first quantum dot 105 to the second quantum dot 106 (1a) and vice-versa (2a) only when the qubit is in the singlet state. Cyclic tunnelling involves the tunnelling of charge carriers back and forth, here between the first and second quantum dots 105, 106. The tunnelling is dependent on the spin state of the qubit 100. As described above, the cascade quantum dot 102 is tuned such that this tunnelling also results in cyclic tunnelling of a charge carrier from the cascade quantum dot 102 to the reservoir 103 (1c), and vice-versa (2c). This is achieved by tuning the cascade quantum dot 102 such that the electrochemical potential without a charge carrier in the neighbouring second quantum dot 106 is below the Fermi level of the reservoir 103, and the electrochemical potential with a charge carrier in the neighbouring second quantum dot 106 is above the Fermi level of the reservoir 103.

Figure 1 illustrates a resonator circuit 104 coupled to the reservoir 103. The resonator circuit 104 is an LC resonator typically comprising a capacitor and an inductor connected in parallel or series to form a resonant circuit having a resonant frequency used for a radio-frequency (RF) measurement. That is to say that the resonator circuit 104 has RF readout capability. The resonator circuit 104 provides a readout sensor which can be used to infer the state of the qubit 100. In this example, the qubit 100 is a double quantum dot qubit and therefore the qubit 100 has two states: (1) singlet state or (2) triplet state. In other examples, either the first quantum dot or the second quantum dot may form a single spin qubit to be measured: a data qubit. The circuit is operated in the same manner, the difference being that the quantum dot which is not being measured forms an ancilla qubit, the spin state of which is prepared in advance of measuring the state of the data qubit.

The resonator circuit 104 is configured to detect a first alternating potential 112 in the vicinity of the first frequency, h using a homodyne detection method typically used in RF measurements. The first frequency, h, is the same frequency that is applied by the frequency source to the plunger gate electrode for the first quantum dot 105. The RF output detected by the readout circuitry 104 has the form V = V_out cos(c ₀t + cf> The properties of the output will change depending on whether the singlet-triplet qubit is in a singlet state or a triplet state. When the qubit is in a singlet spin state, the RF output is V = V _ut cos(c ₀t + cf>^s When the qubit is in a triplet spin state, the RF output is V =

differs from V _ut and cf>^T differs from <p^s . Accordingly, the properties of the detected first alternating potential 112 are different depending on the spin state of the qubit. Thus, depending on the measured properties of the detected first alternating potential, such as frequency, amplitude and/or phase, it can be deduced that the qubit 100 is in the triplet or singlet state. The expected RF output values are calibrated by recording the properties of an alternating potential detected by the readout circuit in the (1 ,1) and (0,2) charge configurations to determine the expected values for the properties for the triplet and singlet spin states respectively.

In other examples involving single spin data qubits, the properties of the RF output will change in a similar manner described above depending on whether the spin state of the data qubit is anti-parallel, or parallel, to that of the ancillary qubit. The RF outputs for the anti-parallel and parallel spin states approximately correspond to the RF outputs for the singlet spin state and triplet spin state as described above due to the presence and absence of cyclic tunnelling. When the spin state of the data qubit is opposite to the spin state of the ancillary qubit, the RF output is V = 7₀“ _t cos(c ₀t + cf>^a When the spin state of the data qubit is the same as the spin state of the ancillary qubit, the RF output is V =

V_o ^a _ut differs from V^b _ut and <p^a differs from <f>^b . Accordingly, the properties of the detected first alternating potential are different depending on the spin state of the qubit. Thus, depending on the measured properties of the detected first alternating potential, such as frequency, amplitude and/or phase, it can be deduced that the data qubit is in the first or second spin state, using the expected values of the RF output and the known spin state of the ancillary qubit.

The expected RF output values are calibrated by recording the properties of an alternating potential detected by the readout circuit in the (1 ,1) and (0,2) charge configurations to determine the expected values for the properties for the parallel and anti-parallel data-ancillary qubit spin configurations respectively.

The circuit pictured In Figure 1 therefore operates to perform a charge polarisation measurement to infer the state of the qubit 100.

Figures 2A and 2B schematically illustrate one dimensional chains each linking the qubit 200 to be read out, and the reservoir 203, which is coupled to the readout sensor 204 in the form of a resonator circuit. Similarly to Figure 1 , Figures 2A and 2B illustrate the ladder diagram for the components of the circuit. It is noted that the figures are schematic representations and further electrochemical potential levels may typically be present. In order to simplify the illustration, only the electrochemical potential levels relevant for the cascade Pauli spin blockade process are shown. In practice, the quantum dots may be occupied by any number of electrons. In Figures 2A and 2B, the double quantum dot 201 forming the qubit 200 comprises a first quantum dot 205 and a second quantum dot 206, separated by a tunnel barrier 207. The double quantum dot 201 is defined by electrostatic barriers 209, 210 which confine charge carriers within the double quantum dot 201 .

Again, similarly to Figure 1 , Figures 2A and 2B include a cascade quantum dot 202 separated from a reservoir 203 by a tunnel barrier 208. The reservoir 203 is coupled to a resonator circuit 204 comprising an inductor and a capacitor as in Figure 1.

In contrast to Figure 1 , the embodiment illustrated in Figures 2A and 2B also includes intermediate cascade double quantum dots 220, 230 arranged between the qubit 200 and the cascade quantum dot 202. In existing techniques, charge polarisation measurements are performed locally because the electrostatic energy detected is small. This means that a readout sensor, i.e. a charge polarisation sensor, needs to be placed in close proximity to the target qubit, the state of which is to be measured. However, as discussed, this is incompatible with measuring the state of a plurality of qubits arranged in dense arrays.

Each intermediate cascade double quantum dot 220, 230 comprises a first intermediate cascade quantum dot 221 , 231 and a second intermediate cascade quantum dot 222, 232 separated by a tunnel barrier 223, 233. Adjacent, neighbouring, intermediate cascade double quantum dots are separated by electrostatic barriers 218, 224, 234 thereby forming a capacitive coupling between neighbouring intermediate cascade double quantum dots.

Therefore, Figures 2A and 2B illustrate a one-dimensional chain of double quantum dots in which tunnelling of charge carriers is possible between the two quantum dots within each double quantum dot, and there is capacitive coupling between the other nearest neighbours, i.e. there is capacitive coupling between neighbouring quantum dots of neighbouring double quantum dots. In addition, spin dependent tunnelling events are suppressed between neighbouring double quantum dots.

A first intermediate cascade double quantum dot 220, defined by left and right electrostatic barriers 210, 224, comprises a first first intermediate cascade quantum dot 221 tunnel coupled to a first second intermediate cascade quantum dot 222 by way of a tunnel barrier 223. The first first intermediate cascade quantum dot 221 is capacitively coupled to the second quantum dot 206 by way of the left electrostatic barrier 210. The right electrostatic barrier 224 provides a capacitive coupling between the first second intermediate cascade quantum dot 222 and a second intermediate cascade double quantum dot 230. In another example including only one intermediate cascade double quantum dot, the first second intermediate cascade quantum dot neighbours the cascade quantum dot and is thereby directly capacitively coupled to the cascade quantum dot.

The second intermediate cascade double quantum dot 230, defined by left and right electrostatic barriers 224, 234, comprises a second first intermediate cascade quantum dot 231 tunnel coupled to a second second intermediate cascade quantum dot 232 by way of a tunnel barrier 233. The left electrostatic barrier 224 capacitively couples the second first intermediate cascade quantum dot 231 to the first second intermediate cascade quantum dot 222. The right electrostatic barrier 234 capacitively couples the second second intermediate cascade quantum dot 232 to a further intermediate cascade double quantum dot (not shown). In another example including only two intermediate cascade double quantum dots, the second second intermediate cascade quantum dot neighbours the cascade quantum dot and is thereby directly capacitively coupled to the cascade quantum dot.

Further intermediate cascade double quantum dots are indicated between the second intermediate cascade double quantum dot 230 and the cascade quantum dot 202. Afinal intermediate cascade double quantum dot (not shown) is arranged between another intermediate cascade double quantum dot (either the second intermediate cascade double quantum dot 230 or another, not shown, intermediate cascade double quantum dot) and the cascade quantum dot 202. In this way, the final first intermediate cascade quantum dot is capacitively coupled to the another second intermediate cascade quantum dot; and the final second intermediate cascade quantum dot is capacitively coupled to the cascade quantum dot. The final second intermediate cascade quantum dot is separated from the cascade quantum dot by an electrostatic barrier 218.

In examples including three or more intermediate cascade double quantum dots, the circuit includes a first intermediate cascade double quantum dot arranged as described above, a final intermediate cascade double quantum dot arranged as described above, and one or more second intermediate cascade double quantum dots arranged as described above. The one dimensional chain is therefore defined by the sequential arrangement of the qubit 200, the first (and optionally second, third, ... n-th) intermediate cascade double quantum dot(s) 220, 230, and the cascade quantum dot 202. The cascade quantum dot 202 is directly coupled to the reservoir 203 with a tunnel barrier 208.

In Figures 2A and 2B, a frequency source is used to apply an alternating potential 211 at a first frequency, h, to the gate electrode of the first quantum dot 205. In other examples the frequency source can be used to apply the alternating potential to the gate electrode of the second quantum dot, or to the charge reservoir. The resonator circuit 204 is configured to detect a first alternating potential 212, and measure the properties of the detected alternating potential to infer the state of the qubit 200. The properties include frequency, amplitude and/or phase of the alternating potentials.

Each intermediate cascade double quantum dot 220, 230 is tuned to be close to a charge transition. In this way, a change in charge carrier occupation of the second quantum dot 206 induces a change in charge carrier occupation of the first intermediate cascade double quantum dot 220, which induces a change in charge carrier occupation of the second intermediate cascade double quantum dot 230 and so on until a change in charge carrier occupation of the cascade quantum dot 202 is induced which is detected by the resonator circuit 204.

When the qubit 200 is in a triplet state in Figure 2A, no tunnelling occurs. Theoretically the RF output detected by the resonator circuit 204 could be the same as the RF input in this case. However, due to imperfections in the circuitry, the detected alternating potential 212 is typically different. Therefore, in order to determine the spin state of the qubit, one or more properties of the detected alternating potential 212 are compared with expected values of the properties for the triplet and singlet spin states. When the qubit 200 is in a triplet state, it is determined by the readout circuitry that the one or more measured values substantially match the corresponding expected values of the properties for the triplet spin state. In this way, it can be inferred that the qubit 200 is in the triplet state.

When the qubit 200 is in a singlet state, the cascade Pauli spin blockade illustrated in Figure 2A proceeds as follows. An electron tunnels through the tunnel barrier 207 from the first quantum dot 205 to the second quantum dot 206 (A⁰). Due to the capacitive coupling between the double quantum dot 201 and the first intermediate cascade double quantum dot 220, the first first intermediate cascade quantum dot 221 enters a higher energy state (A ). An electron tunnels through the tunnel barrier 223 from the first first intermediate cascade quantum dot 221 to the first second intermediate cascade quantum dot 222 (A^). Due to the capacitive coupling between the first and second intermediate cascade double quantum dots 220, 230, the second first intermediate cascade quantum dot 231 enters a higher energy state (Aj). An electron tunnels through the tunnel barrier 233 from the second first intermediate cascade quantum dot 231 to the second second intermediate cascade quantum dot 232 (A^).

Due to the capacitive coupling between neighbouring intermediate cascade double quantum dots, the cascade Pauli spin blockade process proceeds by the k-th first intermediate cascade quantum dot entering a higher energy state (A ), and an electron tunnelling through a tunnel barrier from the k-th first intermediate cascade quantum dot to the k-th second intermediate cascade quantum dot (A ).

The cascade process continues until the cascade quantum dot 202 is reached. At this point, the process is similar. Due to the capacitive coupling between the final intermediate cascade double quantum dot and the cascade quantum dot 202, the cascade quantum dot enters a higher energy state (A"). An electron tunnels from the cascade quantum dot 202 through a tunnel barrier 208 to the reservoir 203.

As described in relation to Figure 1 , the application of an alternating potential by the frequency source alternately raises and lowers the electrochemical potential level of the first quantum dot 205. When the electrochemical potential is raised, the charge carrier movement is as described above and as shown by arrows at

When the electrochemical potential is lowered, the charge carrier movement is reversed. This is shown in Figure 2A by arrows at B°, Bl, Bl Bl, Bl, ... Bl Bl ... B B . At B° an electron tunnels from the second quantum dot 206 to the first quantum dot 205 through the tunnel barrier 207. This change in charge carrier occupation of the second quantum dot 206 causes the energy state of the first first intermediate cascade quantum dot 221 to drop at B , enabling an electron to tunnel from the first second intermediate cascade quantum dot 222 to the first first intermediate cascade quantum dot 221 at Bl Corresponding charge carrier movement occurs at the second intermediate cascade double quantum dot at B , B and at all further intermediate cascade double quantum dots which are not illustrated at Bl, Bl Finally, the energy state of the cascade quantum dot 202 is caused to drop at Bl thus enabling an electron to tunnel from the reservoir 203 to the cascade quantum dot 202 at Bl

In this way, the alternating potential applied to the gate electrode of the first quantum dot 205 results in a cascade effect initiating synchronised cyclic tunnelling in a chain of coupled circuit elements when the qubit is in the singlet state. The cascade involves a quantum tunnelling process in each double quantum dot system. The application of the alternating potential results in a cyclic tunnelling process (A, B) in each double quantum dot system. The cascade of charge carriers is triggered by a first quantum tunnelling process in the double quantum dot which induces further quantum tunnelling processes in each of the intermediate cascade double quantum dots. Finally, a charge carrier tunnels from the cascade quantum dot into the reservoir which is coupled to the resonator circuit which is configured to detect a first alternating potential 212, measure the properties of the detected first alternating potential (i.e. the frequency, amplitude and/or phase), and infer that the double quantum dot qubit 200 is in the singlet state if the properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state. In this way, the state of the qubit 200 can be determined using a charge polarisation measurement even though the qubit may be separated from the cascade quantum dot, and thus the reservoir and the readout apparatus, by any number of intermediate cascade double quantum dots.

Figure 2B illustrates a similar cascade process to that illustrated in Figure 2A. In Figure 2A, the cascade is initiated by a charge carrier tunnelling from the first quantum dot 205 to the second quantum dot 206. In Figure 2B, the cascade is initiated by a charge carrier tunnelling from the second quantum dot 206 to the first quantum dot 205.

When the qubit 200 is in a triplet state in Figure 2B, no tunnelling occurs. In this way, the resonator circuit 204 detects a first alternating potential 212 having properties which substantially match expected values of the properties for the triplet spin state, and it can be inferred that the qubit 200 is in the triplet state.

When the qubit 200 is in a singlet state, the cascade Pauli spin blockade illustrated in Figure 2B proceeds as follows. An electron tunnels through the tunnel barrier 207 from the second quantum dot 206 to the first quantum dot 205 (A⁰). Due to the capacitive coupling between the double quantum dot 201 and the first intermediate cascade double quantum dot 220, the first first intermediate cascade quantum dot 221 enters a lower energy state (A ). An electron tunnels through the tunnel barrier 223 from the first second intermediate cascade quantum dot 222 to the first first intermediate cascade quantum dot 221 (A^). Due to the capacitive coupling between the first and second intermediate cascade double quantum dots 220, 230, the second first intermediate cascade quantum dot 231 enters a lower energy state (Aj). An electron tunnels through the tunnel barrier 233 from the second second intermediate cascade quantum dot 232 to the second first intermediate cascade quantum dot 231 (A^).

Due to the capacitive coupling between neighbouring intermediate cascade double quantum dots, the cascade Pauli spin blockade process proceeds by the k-th first intermediate cascade quantum dot entering a lower energy state (A ), and an electron tunnelling through a tunnel barrier from the k-th second intermediate cascade quantum dot to the k-th first intermediate cascade quantum dot A^k ₂).

The cascade process continues until the cascade quantum dot 202 is reached. At this point, the process is similar. Due to the capacitive coupling between the final intermediate cascade double quantum dot and the cascade quantum dot 202, the cascade quantum dot enters a lower energy state (A"). An electron tunnels through a tunnel barrier 208 from the reservoir 203 to the cascade quantum dot 202.

As described in relation to Figure 1 , the application of an alternating potential by the frequency source alternately raises and lowers the electrochemical potential level of the first quantum dot 205. When the electrochemical potential is lowered, the charge carrier movement is as described above and as shown by arrows at

When the electrochemical potential is raised, the charge carrier movement is reversed. This is shown in Figure 2B by arrows at B°,Bl,BlB ,B ... Bl,B%

At B° an electron tunnels from the first quantum dot 205 to the second quantum dot 206 through the tunnel barrier 207. This change in charge carrier occupation of the second quantum dot 206 causes the energy state of the first first intermediate cascade quantum dot 221 to rise at B , enabling an electron to tunnel from the first first intermediate cascade quantum dot 221 to the first second intermediate cascade quantum dot 222 at Bl Corresponding charge carrier movement occurs at the second intermediate cascade double quantum dot at B , B and at all further intermediate cascade double quantum dots which are not illustrated at B ,B . Finally, the energy state of the cascade quantum dot 202 is caused to rise at B , thus enabling an electron to tunnel from the cascade quantum dot 202 to the reservoir 203 at B .

In this way, the alternating potential applied to the gate electrode of the first quantum dot 205 results in a cascade effect when the qubit is in the singlet state. The cascade involves a quantum tunnelling process in each double quantum dot system. The application of the alternating potential results in a cyclic tunnelling process in each double quantum dot system. The cascade of charge carriers is triggered by a first quantum tunnelling process in the double quantum dot which initiates further quantum tunnelling processes in each of the intermediate cascade double quantum dots. Finally, a charge carrier tunnels from the cascade quantum dot into the reservoir which is coupled to the resonator circuit which is configured to detect a first alternating potential 212 having properties that substantially match expected values of the properties for the singlet spin state. In this way, the state of the qubit 200 can be determined using a charge polarisation measurement even though the qubit may be separated from the cascade quantum dot, and thus the reservoir and the readout apparatus, by any number of intermediate cascade double quantum dots.

The examples in Figures 2A and 2B have been described using electrons as the charge carriers. However, it would be understood that in different examples the same process can be implemented using holes as the charge carriers. Similarly, although a double quantum dot qubit measurement has been described in relation to Figures 2A and 2B, the same circuit can be used to measure a single spin qubit if the other qubit in the double quantum dot is prepared with a known spin state such as spin down.

Figure 3 schematically illustrates a circuit comprising first and second qubits 301 , 311 formed from two double quantum dots. Each qubit has a singlet state and a triplet state. The circuit includes a shared charge reservoir 303, shared by the first and second qubits 301 , 311 . In this way, the readout sensor 304 coupled to the reservoir is configured to measure the state of both the first and second qubits 301 , 311. The readout sensor in this example includes two resonator circuits connected in parallel, each comprising an inductor and a capacitor in the form of an LC resonator. The first resonator circuit has a first resonant frequency and the second resonator circuit has a second resonator frequency. The output of the two resonator circuits can be combined to produce a single output single comprising multiple frequency components.

In Figure 3, each of the first and second qubits 301 , 311 are capacitively coupled to respective first intermediate cascade double quantum dots 321 , 331 which are each further capacitively coupled to respective second intermediate cascade double quantum dots 322, 332. One or more further intermediate cascade double quantum dots are arranged between each second intermediate cascade double quantum dot and respective cascade quantum dots 302, 312. The total number of intermediate cascade double quantum dots separating the qubits 301 , 311 from the cascade quantum dots 302, 312 may be the same or different.

Each intermediate cascade double quantum dot is separated from neighbouring quantum dots by an electrostatic barrier 307, 308, 309, 310, 317, 318, 319, 320 forming a capacitive coupling between neighbouring circuit elements. Respective final intermediate cascade double quantum dots are capacitively coupled to respective cascade quantum dots 302, 312 which are tunnel coupled to the same reservoir 303. In this case, the circuit elements between the first qubit 301 and the first cascade quantum dot 302 are electrically separated from the circuit elements between the second qubit 302 and the second cascade quantum dot 312. In some cases, two or more reservoirs may be provided. In each case, a readout sensor is provided for each reservoir. In this way, the readout sensor coupled to each reservoir is configured to infer the state of all the qubits coupled to that reservoir, which may be one or more.

Each intermediate cascade double quantum dot 321 , 322, 331 , 332 comprises first and second intermediate cascade quantum dots which are separated by a tunnel barrier (barrier not shown in Figure 3). The double quantum dots forming the first and second qubits 301 , 311 each comprise a first quantum dot 305, 315 and a second quantum dot 306, 316 separated by a tunnel barrier (barrier not shown in Figure 3).

As described above, when an alternating potential is applied by a frequency source to the gate electrode of the first quantum dot 305, 315 of a qubit 301 , 311 , this triggers a cascade of charge carrier tunnelling.

In Figure 3, an alternating potential 341 at a first frequency, h, is applied to the gate electrode of the first quantum dot 305 of the first qubit 301. Similarly, an alternating potential 342 at a second frequency, f₂, is applied to the gate electrode of the first quantum dot 315 of the second qubit 311.

When the first qubit 301 is in a triplet state, no tunnelling occurs. In this way, a first alternating potential 351 detected by the first resonator circuit of the readout circuitry 304, has similar properties to the applied alternating potential 341. The expected values of the properties of the detected alternating potential for the triplet state will typically differ to the applied alternating potential due to non-ideal features of the circuit. Therefore, whilst theoretically the detected alternating potential may match the applied alternating potential when a qubit is in the triplet state, in practice the circuit typically absorbs or reflects the voltage due to nonideal features of the circuit such as mis-matched impedances or environmental factors. The expected RF output for when the first qubit is in a triplet state can be determined by monitoring the RF output signal when the double quantum dot is in a (1 ,1) charge configuration. In this way, expected values of the properties of the detected first alternating potential can be determined, and it can be inferred, by the readout circuitry, that the first qubit 301 is in the triplet state when the measured one or more properties of the detected first alternating potential substantially match the corresponding expected values. The expected values might be a range of expected values, and the readout circuitry may determine that the measured properties substantially match when they fall within the expected range of values.

When the second qubit 311 is in a triplet state, no tunnelling occurs. In this way, a second alternating potential 352 detected by the second resonator circuit of the readout circuitry 304, has similar properties to the applied alternating potential 342, differing slightly due to experimental factors similar to the first qubit. As for the first qubit, it can be inferred that the second qubit 311 is in the triplet state when one or more measured properties of the detected second alternating potential substantially match the known, expected values of those properties for the triplet state.

When the first qubit 301 is in a singlet state, a cascade of quantum tunnelling processes is caused by the application of an alternating potential 341 to the gate electrode of the first quantum dot 305 of the first qubit 301 as described in relation to Figure 2A or Figure 2B. Consequently, the readout circuitry 304 detects a first alternating potential 351 having modified properties relative to the applied alternating potential 341 due to the tunnelling. Specifically, the properties of the detected first alternating potential when the first qubit 301 is in a singlet state substantially match the properties of an alternating potential detected by the readout circuitry when the double quantum dot is in the (0,2) charge configuration. In this way, it can be inferred that the first qubit 301 is in the singlet state by determining whether one or more of the measured properties of the detected first alternating potential substantially match expected values, where the expected values correspond to the values of the properties of an alternating potential detected by the readout circuitry when the double quantum dot is in the (0,2) charge configuration.

When the second qubit 311 is in a singlet state, a cascade of quantum tunnelling processes is caused by the application of an alternating potential 342 to the gate electrode of the first quantum dot 315 of the second qubit 311 as described in relation to Figure 2Aor Figure 2B. Consequently, the readout circuitry 304 detects a second alternating potential 352 having properties that substantially match expected values of the properties for the singlet state, and thus it can be inferred that the second qubit 311 is in the singlet state.

When a single reservoir and connected resonator circuits are used to measure a plurality of qubits, the state of the qubits can be determined using multiplexing methods. In some examples, time-division multiplexing can be used. In time- division multiplexing, the plurality of qubits sharing a single reservoir and resonator circuit can be measured sequentially. In time-division multiplexing, the first frequency h is applied to the gate electrode of the first quantum dot 305 of the first qubit 301 at a first time, and the second frequency f₂ is applied to the gate electrode of the second quantum dot 315 of the second qubit 311 at a second time later than the first time. In this case, the first and second frequencies h, f₂, may the same or may be different. The measurements can be distinguished at the resonator circuit by separation in time.

In preferred examples, frequency-division multiplexing can be used. In frequencydivision multiplexing, the plurality of qubits sharing a single reservoir and connected resonator circuits can be measured simultaneously. Alternating potentials at first and second frequencies h, f₂ are applied, typically substantially simultaneously, to the first quantum dots 305, 315 of the first and second qubits 301 , 311 respectively. In this case, the first and second frequencies h, f₂, are different from each other and correspond to the resonant frequencies of the first and second resonant circuits respectively. The measurements can therefore be distinguished at the resonator circuits by separation in frequency. Therefore, for frequency-division multiplexing, the readout circuitry 304 is configured to detect a plurality of different frequencies using a plurality of different resonant circuits.

In multiplexing techniques, the readout circuitry, including the resonator circuit 304, typically further includes a mapping relationship between each qubit and the corresponding time and/or frequency. In this way, when a particular time and/or frequency is detected by the resonator circuit, the readout circuitry determines the corresponding qubit according to the mapping relationship. Preferably the mapping relationship is a one-to-one relationship.

Figure 3 illustrates two qubits 301 , 311. In a circuit, any number of qubits may share a common reservoir and resonator circuit. For time-division multiplexing, there is no limit on the number of qubits sharing a reservoir, and a larger number of qubits per reservoir has space-saving advantages. However, measuring the state of more qubits using a time-division multiplexing technique requires more time to perform, which impacts the efficiency. For frequency-division multiplexing, increasing the number of qubits per reservoir has space-saving advantages without any loss of efficiency, because the states of each of the plurality of qubits sharing the reservoir can be inferred substantially simultaneously. The total number of qubits per reservoir is only limited by the bandwidth capability and the spectral resolution.

Typically, the frequencies applied by the frequency source are between from 20 kilohertz (kHz) to 300 gigahertz (GHz), typically between from 1 megahertz (MHz) to 10 GHz, with a spectral resolution of between from 1 to 50 MHz. This spectral resolution avoids overlap between adjacent frequencies. The frequency source may be any commercially available unit capable of outputting an alternating current at one or more selected frequencies. Preferably, the frequency source is capable of outputting a plurality of selected frequencies to facilitate frequencydivision multiplexing.

Figure 4 schematic illustrates a two-dimensional qubit array. Within the two- dimensional array, the neighbouring quantum dots are arranged within a quantum interaction distance such that barriers between neighbouring quantum dots can be tuned to facilitate, or suppress, tunnelling between neighbouring quantum dots. The qubit array comprises quantum dots 426, 427 which are configured to perform as a data qubit or an ancilla qubit. When used as a data qubit, the quantum dots 426, 427 store quantum information relevant to the quantum computation. The state of the data qubits is measured to record the outcome of a quantum computation. Therefore, the circuit is for reading out the state of data qubits. When used as an ancilla qubit, the quantum dots 426, 427 indirectly relay quantum information. For example, intermediate cascade quantum dots comprise ancilla qubits.

In Figure 4, a first subset of quantum dots 427 can be addressed by a frequency source. Addressing a quantum dot in the first subset of quantum dots 427 involves applying, using the frequency source, an alternating potential to the plunger electrode of that quantum dot in the first subset of quantum dots 427. The same frequency source may be used to address each quantum dot in the first subset of quantum dots 427. Alternatively, a plurality of frequency sources may be provided, each configured to address one or more quantum dots in the first subset of quantum dots 427.

In this example, the first subset of quantum dots 427 forms approximately half of the total number of quantum dots 426, 427 in the array. In this example, quantum dots 427 which can be addressed by a frequency source alternate with quantum dots 426 which cannot be addressed by a frequency source. In this way, vertically or horizontally neighbouring quantum dots can form a double quantum dot with one addressable quantum dot 427. It is not necessary to be able to address every single quantum dot in the array and therefore this arrangement reduces the control circuitry required to connect the quantum dots and the frequency source, whilst maximising flexibility in routing quantum information through the circuit from data qubits to readout circuitry. One quantum dot (surrounded by quantum dots in the first subset of quantum dots 427) is replaced by a reservoir 424 in this example. In other examples, a plurality of quantum dots may be replaced by one or more reservoirs.

The state of a plurality of data qubits can be measured using readout circuitry 425. The readout circuitry 425 is coupled to the reservoir 424. In this example, the wiring connecting the reservoir 424 and the readout circuitry 425 is out of plane. This means no accommodation needs to be made within the quantum dot array for the wiring, thus facilitating a dense array of quantum dots. Use of readout circuitry 425 in this way results in minimal on-chip footprint because no additional infrastructure is required to read out the state of qubits. The readout circuitry 425 comprises a plurality of readout sensors such as LC resonators.

Figure 4 schematically illustrates two paths 40, 41 for quantum information. A first path 40 connects a first qubit 401 to the reservoir 424. A second path 41 connects a second qubit 411 to the same reservoir 424. The first and second qubits 401 , 411 are data qubits. The first and second qubits 401 , 411 comprise first and second tunnel coupled quantum dots forming a double quantum dot qubit with two states: singlet and triplet, as described above. In this example, the first and second paths 40, 41 overlap, as described below. In the first path 40, the first qubit 401 is capacitively coupled to a first first intermediate cascade double quantum dot 402; the first first intermediate cascade double quantum dot 402 is capacitively coupled to a first second intermediate cascade double quantum dot 403; the first second intermediate cascade double quantum dot 403 is capacitively coupled to a first third intermediate cascade double quantum dot 404; the first third intermediate cascade double quantum dot 404 is capacitively coupled to a first shared intermediate cascade double quantum dot 421 ; the first shared intermediate cascade double quantum dot 421 is capacitively coupled to a second shared intermediate cascade double quantum dot 422; the second shared intermediate cascade double quantum dot 422 is capacitively coupled to a shared cascade quantum dot 423; and the shared cascade quantum dot 423 is tunnel coupled to the reservoir 424.

In the second path 41 , the second qubit 411 is capacitively coupled to a second first intermediate cascade double quantum dot 412; the second first intermediate cascade double quantum dot 412 is capacitively coupled to the first shared intermediate cascade double quantum dot 421 ; the first shared intermediate cascade double quantum dot 421 is capacitively coupled to the second shared intermediate cascade double quantum dot 422; the second shared intermediate cascade double quantum dot 422 is capacitively coupled to the shared cascade quantum dot 423; and the shared cascade quantum dot 423 is tunnel coupled to the reservoir 424.

As can be seen, the first and second shared intermediate cascade double quantum dots 421 , 422 and the cascade quantum dot 423 are present in both first and second paths 40, 41. Sharing portions of the cascade chain in this way increases the robustness of the circuit against faulty nanostructures.

In other examples, other path shapes and lengths can be implemented. The shapes and lengths of different paths within the same circuit can be different or the same. The number of overlapping circuit elements varies depending on the circuit requirements such as the relative locations of the data qubits to be measured and the reservoir. In other examples, some intermediate cascade double quantum dots may be shared by two paths without the cascade quantum dot and reservoir being shared between those two paths.

As described in relation to Figure 3, the state of each of a plurality of qubits can be measured using frequency-division multiplexing or time-division multiplexing. In this example, an alternating potential 441 at a first frequency, h, is applied to the plunger gate of the first quantum dot of the first qubit 401 and an alternating potential 442 at a second frequency, f₂, different from the first frequency h, is applied to the plunger gate of the first quantum dot of the second qubit 411. Each plunger gate is configured to control the energy state of the respective first quantum dot by raising and lowering the electrochemical potential levels according to the potential applied to the plunger gate.

The readout circuitry 425 is therefore configured to detect first and second alternating potentials 451 , 452, measure one or more properties of the alternating potentials, and compare one or more of the measured properties of the detected first and second alternating potentials with expected values of the properties of the alternating potentials for singlet and triplet states. The expected values differ depending on the frequency of the applied alternating potential as well as the spin state of the qubit. Therefore the state of the first and second qubits 401 , 411 can be measured substantially simultaneously using frequency-division multiplexing. Any shared circuit elements, i.e. the first and second shared intermediate cascade double quantum dots 421 , 422 and the cascade quantum dot 423 in this example, can relay the quantum information for the first and second qubits 401 , 411 simultaneously by using different frequencies to address the first and second qubits 401 , 411.

Figures 5A-5C schematically illustrate a possible circuit architecture for reading out the state of a plurality of qubits, including exemplary paths for the transmission of quantum information and a frequency multiplexing example using this circuit architecture. This circuit architecture is to be defined using gate-defined silicon metal oxide semiconductor (Si-MOS) quantum dots in a two-dimensional array. Within the two-dimensional array, the neighbouring quantum dots are arranged within a quantum interaction distance such that barriers between neighbouring quantum dots can be tuned to facilitate, or suppress, tunnelling between neighbouring quantum dots. Each quantum dot is defined by a plurality of gate electrodes which can be individually controlled. Therefore, by applying voltages to the gate electrodes, the potential landscape can be defined as shown schematically in Figures 5A-5C. For each quantum dot, a plunger gate for controlling the electrochemical potential of that quantum dot is provided along with barrier gates for controlling the tunnel couplings between neighbouring quantum dots, or between a quantum dot and a reservoir. The potentials applied to each gate can be controlled in accordance with a capacitance matrix. In this way the potential landscape can be flexibly defined to include the desired paths, couplings, and barriers.

The circuit architecture forms a unit cell which can be repeated a plurality of times across a chip of a quantum device. Each unit cell comprises one charge reservoir 505 at the centre of the array of quantum dots. The reservoir is an n-doped electron reservoir. Readout circuitry 506 is coupled to the charge reservoir 505 out of plane. In this example, the readout circuitry comprises a plurality of LC resonators (i.e. resonator circuits or tank circuits).

In the exemplary unit cell shown in Figures 5A-5C, there are 40 quantum dots 501 , 502, 503, 504 surrounding one reservoir 505. Of the 40 quantum dots, 16 quantum dots 502, 504 are coupled to a frequency source. This means that the frequency source can apply an alternating potential at a selected frequency to these 16 quantum dots 502, 504. The quantum dots are arranged in concentric layers around the reservoir 505, each successive layer surrounding the previous layer. The arrangement enables vertically or horizontally neighbouring quantum dots to form a double quantum dot, as described in relation to Figure 4. The substantially central positioning of the reservoir 505 within the unit cell enables a denser arrangement of qubits because fewer reservoirs are required to address the same number of qubits. This also enables the length of the one-dimensional chains to be reduced.

Figure 5B illustrates exemplary paths for the quantum information using double headed arrows. To create each path, vertically or horizontally neighbouring quantum dots are able to form a double quantum dot, transmitting the quantum information relating to the state of the data qubit to the charge reservoir using the cascade Pauli spin blockade process. Other paths involving vertically or horizontally neighbouring quantum dots are also readily defined by modifying the gate potentials of the quantum dots in the unit cell.

An initial, zeroth, layer L0 arranged next to the reservoir 505 comprises four ancilla quantum dots 504 each coupled to the reservoir 505. The quantum dots 504 in the zeroth layer are coupled to a frequency source.

The quantum dots 503, 502, 501 in the first, second and third layers L1 , L2, L3 can be used to form data qubits. Each data qubit may be a single spin data qubit tunnel coupled to neighbouring qubit which forms an ancilla qubit (also referred to as an ancillary qubit), or may be a double quantum dot qubit. In both cases, a double quantum dot is formed and one of the two quantum dots forming the double quantum dot is coupled to a frequency source. This means that 36 of the 41 elements can be used to form data qubits: the space occupied by quantum dots which can be used to form data qubits is 87.8% of the footprint. This enables a dense arrangement of data qubits.

The quantum dots 503, 502 in the first and second layers L1 , L2 are also utilised as ancilla qubits when measuring the state of data qubits in layers further from the charge reservoir 505 as is described in relation to Figures 6B and 6C.

In other examples, additional layers of quantum dots may be present in the unit cell. In each case, the outermost layer of quantum dots can only be used to form data qubits. The intermediate layers of quantum dots can be used as data or ancilla qubits depending on which qubit is to be measured. When the reservoir occupies the space of one quantum dot in the array, the number of quantum dots in a layer, N_d, is N_d = 4(d + 1), where d is the layer number.

The number of different RF frequencies, N_M, required to address the different quantum dots in the array depends on the number of quantum dots in the highest layer with RF connected, i.e. N_M = 4 x max(d, even), where d is the layer number. In this case, the highest layer of quantum dots coupled to the frequency source is the second layer, L2, which includes twelve quantum dots. Correspondingly, in this example twelve different resonator circuits are required, having twelve different resonant frequencies. Optionally, a resonator circuit used to detect an alternating potential applied to a quantum dot in the second layer, L2, may also be used to detect an alternating potential applied to a quantum dot in the zeroth layer, L0. This reduces the total number of resonator circuits required, reducing the readout circuitry requirements.

Figure 5C depicts the use of two different RF frequencies applied to quantum dots in the second layer, L2, to measure the data qubits in the third layer, L3. An alternating potential 541 at a first frequency, f , is applied to a first ancilla qubit in the second layer, L2, the first ancilla qubit neighbouring a first data qubit in the third layer, L3; and an alternating potential 542 at a second frequency, f₂, which is different from the first frequency, is applied to a second ancilla qubit in the second layer, L2, the second ancilla qubit neighbouring a second data qubit in the third layer, L3. The tunnelling of charge carriers between the first and second data qubits and the charge reservoir is indicated by the double headed arrows. The readout circuitry 506 is configured to detect a first alternating potential 551 using a first LC resonator having a resonant frequency substantially matching the first frequency, and detect a second alternating potential 552 using a second LC resonator having a resonant frequency substantially matching the second frequency, fa The first and second LC resonators (not shown) are connected in parallel in the readout circuitry. The output of the resonant circuits can be combined to form a signal comprising the first and second alternating potentials.

In the arrangement shown in Figures 5A-5C, there are more quantum dots in the third layer, L3, than in the second layer, L2. Therefore, in order to measure the state of all the qubits in the third layer, L3, two measurement steps are required. In a first measurement step, the state of up to twelve different data qubits in the third layer, L3, can be measured by applying an alternating potential having a different frequency to each of the twelve quantum dots in the second layer, L2. In a second measurement step, the state of the remaining data qubits in the third layer, L3, can be measured by applying an alternating potential having a different frequency to a corresponding number of neighbouring quantum dots in the second layer, L2.

In Figure 5B, paths for the first measurement step for each layer are indicated using a solid double headed arrow, and paths for the second measurement step for each layer are indicated using a dashed double headed arrow.

The unit cells can be positioned adjacent to one another such that an outermost layer in a unit cell is adjacent to outermost layers of a plurality of additional unit cells.

In other examples, other arrangements of circuit elements are possible. In each case, there is capacitive coupling between nearest neighbours and there is RF readout capability. A tunnel barrier can selectively be formed by modifying the potential applied to the barrier electrode separating neighbouring quantum dots to reduce the barrier height, thereby enabling tunnel coupling between those neighbouring quantum dots. In this way, a path between each to-be-measured data qubit and the charge reservoir can be defined. To define a path, the bias potential applied to every other barrier gate along the to-be-defined path is reduced to lower the barrier height to form double quantum dots. Then, the bias potential applied to the plunger gates is modified such that the quantum dots are close to a charge transition. Barriers between quantum dots in the to-be-defined path and quantum dots which are not in the to-be-defined path are tuned such that the charge carrier occupation of quantum dots outside the to-be-defined path remains unchanged. In this way, the relationship between the plunger gates of the quantum dots in the array, and all other circuit elements, can be characterised as a capacitance matrix to determine the path(s) for quantum information. Furthermore, based on the values in this matrix, a set of potentials can be applied to gate electrodes of quantum dots next to, and not forming part of, the path. This means that any influence on other circuit elements which do not form part of the path is reduced. Figures 6A-6C schematically illustrate measurements of data qubits in each of the three layers L1-L3 depicted in the unit cell of Figures 5A-5C, using the cascade process. In Figures 6A-6C the one-dimensional chains are illustrated along a line, however in the circuit each coupling between neighbouring quantum dots may be formed vertically or horizontally in the quantum dot array.

Figure 6A illustrates a measurement involving the first layer, L1 , of the unit cell. This measurement is similar to existing charge polarisation measurements. The measurement circuit for reading out the spin state involves a first-layer quantum dot 603, a zeroth-layer quantum dot 604, and the charge reservoir 605. A frequency source is connected to a gate electrode of the zeroth-layer quantum dot 604.

In Figure 6A, the first-layer quantum dot 603 is tunnel coupled to the zeroth-layer quantum dot 604, and the zeroth-layer quantum dot is capacitively coupled to the charge reservoir 605. The frequency source is configured to apply an alternating potential 641 at a first frequency, f , to the zeroth-layer quantum dot 604. Depending on the spin states of the two quantum dots 603, 604, this can result in cyclic tunnelling of a charge carrier from the zeroth-layer quantum dot 604 to the first-layer quantum dot 603 and vice-versa. When cyclic tunnelling occurs, an image charge ae (or mirror charge, or mirror image charge) is detected by readout circuitry connected to the charge reservoir 605. The readout circuitry is not shown in Figures 6A-6C.

A first measurement type using the circuit shown in Figure 6A is a single spin qubit measurement. In this example, the first-layer quantum dot 603 forms a single quantum dot data qubit and the zeroth-layer quantum dot 604 forms an ancilla qubit. The read out spin state of the first-layer quantum dot 603 can be either spin up or spin down. Prior to the readout measurement, the zeroth-layer quantum dot 604 is initialised in a known spin state, typically spin down. The readout circuitry is used to infer when the data qubit is in a spin up state or a spin down state. This can be inferred because when the spin state of the data qubit is anti-parallel to, i.e. opposite to, the spin state of the ancilla qubit, the cyclic tunnelling described above will occur; and when the spin state of the data qubit is parallel to, i.e. the same as, the spin state of the ancilla qubit, the cyclic tunnelling process described above will not occur. Therefore, due to the presence or absence of tunnelling, the expected values of the properties for the first and second spin states will differ and this can be used to infer the state of the qubit.

When the cyclic tunnelling process occurs in response to an alternating potential applied by a frequency source to the zeroth-layer quantum dot 604, the properties of the alternating potential detected by the readout circuitry will substantially match expected values for an anti-parallel spin configuration; and when the cyclic tunnelling process does not occur, the properties of the detected alternating potential will substantially match expected values for a parallel spin configuration. Therefore, if the ancilla qubit is initialised in a spin down state, the readout circuitry will infer that the data qubit is in a spin down state if the measured properties of the detected alternating potential substantially match the expected values for the spin down state. Alternatively, the readout circuitry will infer that the data qubit is in a spin up state if the measured properties of the detected alternating potential substantially match the expected values for the spin up state.

A second measurement type using the circuit shown in Figure 6A is a double quantum dot qubit measurement. In this example, the first-layer quantum dot 603 forms a first quantum dot of the double quantum dot, and the zeroth-layer quantum dot 604 forms a second quantum dot of the double quantum dot. The read out state of the double quantum dot can be either a singlet spin configuration or a triplet spin configuration. Prior to the readout measurement, the zeroth-layer quantum dot 604 can optionally be initialised in a random spin state.

The readout circuitry is used to infer when the double quantum dot qubit is in a singlet spin state or triplet spin state. This can be inferred because when the qubit is the singlet spin state, the cyclic tunnelling described above will occur; and when the qubit is in the triplet spin state, the cyclic tunnelling process described above will not occur.

Similarly to the first measurement type, when the cyclic tunnelling process occurs in response to an alternating potential applied by a frequency source to the zeroth- layer quantum dot 604, the properties of the alternating potential detected by the readout circuitry will substantially match expected values of the properties for the singlet spin state; and when the cyclic tunnelling process does not occur, the properties of the detected alternating potential will substantially match expected values of the properties for the triplet spin state. Therefore, the readout circuitry will infer that the qubit is in a triplet spin state or a singlet spin state according to the expected values of the properties for the triplet and singlet spin states respectively. The expected values are therefore used as reference values against which the detected alternating potential can be compared to infer the spin state of the qubit.

Figure 6B illustrates a measurement involving the second layer, L2, of the unit cell shown in Figures 5A-5C. The circuit arrangement is similar to that shown in Figure 1 . In Figure 6B, the second-layer quantum dot 602 is tunnel coupled to the first-layer quantum dot 603, forming a double quantum dot, and the zeroth-layer quantum dot 604 is tunnel coupled to the charge reservoir 605. The first-layer quantum dot 603 is capacitively coupled to the zeroth-layer quantum dot 604. A frequency source is connected to a gate electrode of the second-layer quantum dot 602. A frequency source is also connected to a gate electrode of the zeroth- layer quantum dot 604, although this is not used in this example.

The frequency source is configured to apply an alternating potential 641 at a first frequency, f , to the second-layer quantum dot 602. Depending on the spin states of the first- and second-layer quantum dots 602, 603, this can result in cyclic tunnelling of a charge carrier back-and-forth between the first- and second-layer quantum dots 602, 603. The cyclic tunnelling of a charge carrier between the firstand second-layer quantum dots 602, 603 induces a tunnelling process back-and- forth between the zeroth-layer quantum dot 604 and the charge reservoir 605. In this way, when the cascade cyclic tunnelling process occurs, a charge carrier tunnels into the charge reservoir. In this way, a full electron charge e is detected by readout circuitry connected to the charge reservoir 605, thereby amplifying the signal detected by the readout circuitry and increasing the signal-to-noise ratio relative to the measurements described in relation to Figure 6A. Similarly to Figure 6A, the circuit shown in Figure 6B can be used for first and second measurement types depending on the initial state of the first-layer quantum dot 603. For the first measurement type, if the first-layer quantum dot 603 is initialised with a known spin state, a single quantum dot qubit measurement can be performed to determine the spin state of the second-layer quantum dot 602, i.e. spin up or spin down. As described above, cyclic tunnelling only occurs when the spins are anti-parallel due to Pauli spin blockade and Coulomb repulsion, and therefore the qubit spin state can be inferred by using the readout circuitry to determine whether or not the cyclic tunnelling is happening in response to the applied alternating potential. The detected alternating potential will have different properties depending on whether the cyclic tunnelling is occurring or not, and therefore measuring the properties of the detected alternating potential and performing a comparison against a reference value allows the spin state of the qubit to be measured.

For the second measurement type, if the first-layer quantum dot 603 may be initialised with a random spin state, the first- and second-layer quantum dots 602, 603 will form a double quantum dot, the state of which can be determined to be a singlet spin state or a triplet spin state as described above.

Figure 6C illustrates a measurement involving the third layer, L3, of the unit cell shown in Figures 5A-5C. In Figure 6C, the third-layer quantum dot 601 is tunnel coupled to the second-layer quantum dot 602; and the first-layer quantum dot 603 is tunnel coupled to the zeroth-layer quantum dot 604. In this way, the second- and third-layer quantum dots 601 , 602 form a first double quantum dot; and the zeroth- and first-layer quantum dots 603, 604 form a second double quantum dot. The first double quantum dot is capacitively coupled to the second double quantum dot; and the second double quantum dot is capacitively coupled to the charge reservoir 605. Specifically, the first- and second-layer quantum dots 602, 603 are capacitively coupled; and the zeroth-layer quantum dot 604 is capacitively coupled to the charge reservoir 605. A frequency source is connected to gate electrodes of the second- and zeroth-layer quantum dots 602, 604. In this example, the same frequency source is connected to both quantum dots 602, 604, although this is not required.

The frequency source is configured to apply an alternating potential 641 at a first frequency, f , to the second-layer quantum dot 602. Depending on the spin states of the second- and third-layer quantum dots 601 , 602, this can result in cyclic tunnelling of a charge carrier back-and-forth between the second- and third-layer quantum dots 601 , 602. The cyclic tunnelling of a charge carrier between the second and third-layer quantum dots 601 , 602 induces a tunnelling process back- and-forth between the zeroth- and first-layer quantum dots 603, 604. In this way, when the cascade cyclic tunnelling process occurs, a charge carrier tunnels adjacent to the charge reservoir, and an image charge ae is detected by readout circuitry connected to the charge reservoir 605. In this case, the cascade process delivers the quantum information relating to the third layer, L3, of quantum dots closer to the reservoir 605, thereby increasing the signal-to-noise ratio relative to a circuit omitting the zeroth- and first-layer quantum dots 603, 604. The signal-to- noise ratio is increased because the tunnelling of charge carriers is closer to the reservoir.

Similarly to Figures 6A and 6B, the circuit shown in Figure 6C can be used for first and second measurement types depending on the initial state of the second-layer quantum dot 602. For the first measurement type, the second-layer quantum dot 602 is initialised with a known spin state, such that a single quantum dot qubit measurement can be performed to determine the spin state of the third-layer quantum dot 601 , i.e. spin up or spin down. As described above, cyclic tunnelling only occurs when the spins in the neighbouring second- and third-layer quantum dots are anti-parallel due to Pauli spin blockade and Coulomb repulsion, and therefore the single quantum dot qubit spin state can be inferred by using the readout circuitry to determine whether or not the cyclic tunnelling is happening in response to the applied alternating potential.

For the second measurement type, the first-layer quantum dot 603 is optionally initialised with a random spin state, and the second- and third-layer quantum dots 601 , 602 form a double quantum dot, the state of which can be determined to be a singlet spin state or a triplet spin state as described above.

In the circuit arrangements illustrated in Figures 6A-6C, the couplings between neighbouring quantum dots are engineered by modifying the electrostatic barriers between neighbouring quantum dots. Furthermore, in these circuit arrangements, the alternating potential used to stimulate cyclic tunnelling is applied to a gate electrode of a quantum dot. In this way, the S21 transmission scattering component is measured by the readout circuitry. In other examples, the alternating potential used to stimulate cyclic tunnelling may be applied to the charge reservoir. In those examples, the S11 reflection scattering component is measured by the readout circuitry.

Figures 7A and 7B schematically illustrate the resilience of the circuit architecture shown in Figures 5A-5C to faults. The same principles apply to different circuit architectures involving an array of quantum dots. As in Figures 5A-5C, the unit cell comprises zeroth-layer quantum dots 704, first-layer quantum dots 703, second-layer quantum dots 702 and first-layer quantum dots 701 . The quantum dots 701 , 702, 703, 704 are arranged concentrically around a charge reservoir 705 which, in this example, occupies the same space as one quantum dot in the array. Readout circuitry 706 is connected to the charge reservoir 705.

In Figure 7A, a first first-layer quantum dot 7031 has a fault and is therefore deemed to be a “dead” qubit, which can neither be measured nor be used in a path between another qubit to be measured and the charge reservoir. The dashed double-headed arrows indicate a path between a third-layer quantum dot 701 and the charge reservoir 705 involving the first first-layer quantum dot 7031 which may have been used had the first first-layer quantum dot 7031 not been a dead qubit. However, as the path can be flexibly determined and couplings can be formed between vertically and horizontally neighbouring quantum dots in the array, it is straightforward to re-route the path for transmitting quantum information to a second first-layer quantum dot 7032 which is not dead. The solid double-headed arrows indicate an alternative path which can be used to transmit quantum information from the second- and third-layer quantum dots 701 , 702 to the charge reservoir 705 without involving the dead qubit 7031. Of course, any one of a plurality of paths involving different combinations of vertical and/or horizontal couplings may be defined according to experimental requirements. Typically a short, and optionally the shortest possible, path will be selected.

Figure 7B illustrates two adjacent unit cells: a first unit cell 710 and a second unit cell 720. Each unit cell 710, 720 has zeroth-, first-, second- and third-layer quantum dots 701-704 arranged around respective charge reservoirs 715, 725. In this example, the first reservoir 715 in the first unit cell 710 is functional and the second reservoir 725 in the second unit cell 720 is not functional, or “dead”. In this example, the quantum dots 701-704 in the second unit cell 720 cannot be readout by readout circuitry connected to the second reservoir 725. However, as indicated by the double-headed arrows, the state of a qubit can be read out by forming a path to a charge reservoir in a different unit cell. Figure 7B indicates a path between a first-layer quantum dot 703 in the second unit cell 720 and the first charge reservoir 715 in the first unit cell 710. Typically a short, and optionally the shortest possible, path will be selected. Therefore quantum information from other to-be-measured qubits in the second unit cell 720 may be routed to charge reservoirs in other adjacent unit cells. Typically the circuit architecture includes a plurality of unit cells with their outer layers neighbouring as shown in Figure 7B.

Figure 8 is a schematic illustration of a circuit. The circuit in Figure 8 includes a first quantum dot 801 , a second quantum dot 802, a third quantum dot 803, a charge reservoir 804, a resonator circuit 805 and an RF output 806. Each of the first, second and third quantum dots 801 , 802, 803 are capacitively coupled to respective first, second and third gate electrodes 811 , 812, 813. An RF input signal 821 is applied to the first gate electrode 811 of the first quantum dot 801 , thereby alternately raising and lowering the energy levels within the first quantum dot 801 as described above, initiating a tunnelling process. In other examples, the RF input signal can be applied to the second gate electrode 812 of the second quantum dot 802. The first and second quantum dots 801 , 802 are tunnel coupled and therefore the application of an RF input signal 821 forces a charge carrier to tunnel from the first quantum dot 801 to the second quantum dot 802 or from the second quantum dot 802 to the first quantum dot 801 depending on the relative energy states of the first and second quantum dots 801 , 802. The third quantum dot 803 is capacitively coupled to the second quantum dot 802 and tunnel coupled to the charge reservoir 803. Therefore, when cyclic tunnelling occurs between the first and second quantum dots 801 , 802, the capacitive coupling between the second and third quantum dots 802, 803 creates synchronous cyclic tunnelling between the third quantum dot 803 ang the charge reservoir 804. The transmitted signal is picked up at the output of the resonant circuit, here a resonator circuit 805 also referred to as an LC resonator.

The resonator circuit 805 connected to the charge reservoir 804 comprises an inductor 815 and a capacitor 825. The resonator circuit 805 has a resonant frequency, ₀. To measure the state of a qubit, a frequency source connected to the first gate electrode 811 is configured to apply an alternating potential at the resonant frequency of the resonator circuit 805. Typically, the readout mechanism involves homodyne detection, in which the applied alternating potential applied to the gate electrode 811 is simultaneously applied to the resonator circuit 805. In this way, if no tunnelling occurs between the tunnel coupled quantum dots, then the RF output 806 is substantially the same as the RF input 821. Conversely, if tunnelling does occur, then the RF output 806 differs from the RF input 821. Typically, the frequency of the RF output 806 when tunnelling occurs is shifted negatively. In the frequency domain this is seen as a straightforward negative frequency shift. In the time domain, measuring at a particular frequency (usually the resonant frequency, ₀), this means that the amplitude and/or phase of the RF output 806 is lower when the cascade cyclic tunnelling process occurs.

The arrangement depicted in Figure 8 can be used to measure (a) the spin state of the first quantum dot 801 , (b) the spin state of the second quantum dot 802, or (c) the spin state of the double quantum dot comprising the first and second quantum dots 801 , 802. To measure the spin state of the first quantum dot 801 , the second quantum dot 802 is initialised with a known spin state (for example, spin down). To measure the spin state of the second quantum dot 802, the first quantum dot 801 is initialised with a known spin state. When measuring the spin state of the double quantum dot, optionally a quantum dot may be initialised with a random spin state, or simply measured without an initialisation step.

In other examples, additional qubits (not shown) can be measured simultaneously using frequency multiplexing as described above. In those examples, additional resonator circuits arranged in parallel to the illustrated resonator circuit 805 would be present, each having a different resonant frequency. The resonant frequency is determined by the inductance, L, and capacitance, C, or the inductor and capacitor respectively. Specifically, the resonant frequency is f_res =

The output of each resonant circuit can be combined to provide a single RF output signal with multiple frequency components.

In the example illustrated in Figure 8, a transmission scattering parameter, S₂₁, is measured. In alternative examples, the RF input may be applied to the charge reservoir. In those examples, a reflection scattering parameter, S^, is measured. The remaining details of such a set up are as described above.

Figure 9 is a flow chart of a method of reading out the state of a qubit. Step S91 involves applying, using a frequency source, an alternating potential at a first frequency to a gate electrode of a first quantum dot, a gate electrode of a second quantum dot, or a charge reservoir. The first and second quantum dots form a double quantum dot forming the to-be-measured qubit.

The gate electrode is typically the plunger gate electrode used to raise and lower the electrochemical potential levels within the associated quantum dot. For a plurality of qubits, the frequency source can further be used to apply further alternating potentials at a corresponding plurality of frequencies to gate electrodes of quantum dots of the double quantum dot forming each qubit or the charge reservoir.

The double quantum dot qubit is configured such that when an alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa is caused only when the qubit is in the singlet state, and consequently cyclic tunnelling of a charge carrier from the cascade quantum dot to the reservoir is caused, and vice- versa.

Therefore, in order to infer the state of the qubit, the method further comprises in step S92 detecting, using readout circuitry connected to the charge reservoir, a first alternating potential. The detected first alternating potential is detected by a resonator circuit having the first frequency as its resonant frequency.

Step S93 involves measuring, using the readout circuitry, one or more properties of the detected first alternating potential. The one or more properties include frequency, amplitude and phase. For example, in the frequency domain optionally only the frequency may be measured, or in the time domain optionally only the amplitude may be measured. Of course, two or more properties can be measured.

Step S94 involves inferring that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; or inferring that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state. In other words, if the frequency, amplitude, and/or phase of the detected alternating potential substantially match the expected values, or lie within a range of expected values, for the triplet spin state or for the singlet spin state, it can be inferred that the double quantum dot qubit is in the triplet spin or singlet spin configuration respectively. Typically, a singlet spin configuration is reflected by a negative frequency shift relative to the triplet spin configuration which results in a reduced amplitude and phase at the resonant frequency.

The method of Figure 9 may be applied to an array of qubits. An example readout method for the unit cell shown in Figures 5A-5C is described. A first group of qubits comprises double quantum dot qubits formed from four of the eight first- layer quantum dots 503 and the zeroth-layer quantum dots 504. The state of the first group of qubits is measured by applying alternating potentials at first, second, third and fourth frequencies to the four zeroth-layer quantum dots 504 respectively (see step S91 of Figure 9 and Figure 6A). Steps S92-S94 are performed in parallel for each of the four double quantum dot qubits, using four resonant circuits connected in parallel to detect the four alternating potentials respectively. The zeroth-layer quantum dots 504 are subsequently loaded with a random spin. Then, a second group of qubits comprising double quantum dot qubits formed from the remaining four of the eight first-layer quantum dots 503 tunnel coupled to the zeroth-layer quantum dots 504. The measurement of the state of each qubit in the second group of qubits proceeds in the same manner as for the first group. Subsequently, the first-layer quantum dots 503 are initialised with a random spin.

A third group of qubits comprises eight double quantum dot qubits formed from eight of the twelve second-layer quantum dots 502 respectively tunnel coupled to one of the first-layer quantum dots 503. The state of each qubit in the third group of qubits is determined in parallel using the above-described frequency multiplexing technique (see steps S91-S94 of Figure 9 and Figure 6B). Therefore, eight alternating potentials each having a different frequency are applied to respective second-layer quantum dots 502. A fourth group of qubits comprises double quantum dot qubits formed from the remaining four of the twelve second- layer quantum dots 502 tunnel coupled with four of the first-layer quantum dots 503 respectively. Prior to measuring the fourth group of qubits, at least the first- layer quantum dots 503 forming part of the to-be-measured double quantum dot qubits are prepared in a random spin state. Subsequently, the second-layer quantum dots 502 are initialised with a random spin.

A fifth group of qubits comprises twelve double quantum dot qubits formed from twelve of the sixteen third-layer quantum dots 501 respectively tunnel coupled to one of the second-layer quantum dots 502. The state of each qubit in the fifth group of qubits is inferred in parallel using above-described frequency multiplexing technique (see steps S91-S94 of Figure 9 and Figure 6C), applying the alternating potential to the second-layer quantum dot 602. A sixth group of qubits comprises double quantum dot qubits formed from the remaining four of the sixteen third- layer quantum dots 501 tunnel coupled with four of the second-layer quantum dots 502 respectively. Prior to measuring the state of each of the qubits in the sixth group of qubits, at least the second-layer quantum dots 502 forming part of the to- be-measured double quantum dot qubits are prepared in a random spin state.

In each case, the maximum number of qubits in each group is determined by the total number of (functioning) qubits in the inwardly-neighbouring layer. The number of readout cycles, i.e. the number of groups of qubits to be measured separately in time, scales logarithmically with the number of qubits in the unit cell.

Figure 10 is a flow chart of a method of reading out the state of a single spin data qubit. The data qubit being measured is tunnel coupled to an ancillary qubit (otherwise referred to as an ancilla qubit). Therefore the data and ancillary qubits form a double quantum dot. Each of the data and ancilla qubits have first and second spin states. For example, the first spin state may be spin down and the second spin state may be spin up. Alternatively, the first spin state may be spin up and the second spin state may be spin down. The first and second spin states are anti-parallel.

Step S100 involves initialising, using control circuitry connected to the ancillary qubit, the ancillary qubit in the first spin state. The spin state of the ancillary qubit can be prepared using any well-known initialising techniques, for example by tuning the energy levels to energetically exclude the spin up state. In this way, the spin state will necessarily be spin down.

Steps S101 , S102 and S103 are similar to the corresponding steps S91-S93 shown in Figure 9. Step S101 involves applying, using a frequency source, an alternating potential at a first frequency to a gate electrode of the data qubit or the ancillary qubit, or a charge reservoir. As in Figure 9, the frequency source can further be used to apply a plurality of alternating potentials to measure a corresponding plurality of data qubits.

The double quantum dot is configured such that when the alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier between the data and ancillary qubits is caused only when the data qubit is in the second spin state, i.e. in a spin state which is antiparallel to the spin state of the ancillary qubit. Cyclic tunnelling of a charge carrier between the cascade quantum dot and the charge reservoir is also caused when the data qubit is in the second spin state.

Step S102 involves detecting, using readout circuitry connected to the charge reservoir, a first alternating potential. The detected first alternating potential is detected by a resonator circuit having the first frequency as its resonant frequency.

Step S103 involves measuring, using the readout circuitry, one or more properties, such as frequency, amplitude and/or phase, of the detected first alternating potential.

Step S104 involves inferring that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; or inferring that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state. The expected values of the properties for the first and second spin states are different because the cascade cyclic tunnelling occurs when the data qubit is in the anti-parallel spin state relative to the ancillary qubit, and does not occur when the data is in the parallel spin state relative to the ancillary qubit.

The method of Figure 10 may be applied to an array of qubits. An example readout method for the unit cell shown in Figures 5A-5C is described. Firstly, the zeroth- layer quantum dots 504 are initialised in a first, known, spin state (step S100). A first group of qubits comprises single spin qubits formed from four of the eight first- layer quantum dots 503. The zeroth-layer quantum dots 504 form ancillary qubits. The state of the first group of qubits is measured by applying alternating potentials at first, second, third and fourth frequencies to the four zeroth-layer quantum dots 504 respectively (see step S101 of Figure 10 and Figure 6A). Steps S102-S104 are performed in parallel for each ofthe four single spin qubits, using four resonant circuits in parallel to detect the four alternating potentials. The zeroth-layer quantum dots 504 are subsequently initialised, again, with the first spin state. Then, the state of each of a second group of qubits comprising single spin qubits formed from the remaining four of the eight first-layer quantum dots 503 is measured in the same manner as for the first group. Subsequently, the first-layer quantum dots 503 are prepared in the first spin state (S100).

A third group of qubits comprises eight single spin qubits formed from eight of the twelve second-layer quantum dots 502. The state of each qubit in the third group of qubits is determined in parallel using the above-described frequency multiplexing technique (see steps S101-S104 of Figure 10 and Figure 6B). After this measurement, the first-layer quantum dots 503 are again prepared in the first spin state (S100). A fourth group of qubits comprises single spin qubits formed from the remaining four of the twelve second-layer quantum dots 502. Once the second-layer quantum dots 502 have been measured, they are each loaded with a charge carrier having a known spin state, here the first spin state (S100).

A fifth group of qubits comprises twelve single spin qubits formed from twelve of the sixteen third-layer quantum dots 501. The state of each qubit in the fifth group of qubits is determined in parallel using above-described frequency multiplexing technique (see steps S101-S104 of Figure 10 and Figure 6C). After this measurement, the second-layer quantum dots 502 are again prepared in the first spin state (S100). A sixth group of qubits comprises single spin qubits formed from the remaining four of the sixteen third-layer quantum dots 501 .

Figure 11 shows an example of possible S21 scattering parameter measurements in the frequency domain for two double quantum dot qubits. Detected alternating potentials for the singlet and triplet states for both first and second qubits are illustrated in Figure 11.

The readout circuitry used for the readout measurement comprises two resonant circuits having first and second resonant frequencies ₀ and f respectively. A frequency source is configured to apply alternating potentials having first and second frequencies ₀, f respectively to the first and second qubits respectively. The graph schematically illustrates detected first and second alternating potentials corresponding to the first and second qubits respectively.

If the first qubit is in the triplet spin state, T, the frequency, amplitude and phase of the detected first alternating potential substantially match the expected values of the properties of the alternating potential for the triplet spin state. The properties comprise one or more of frequency, amplitude and phase. A schematic illustration of an example measurement is shown by the first qubit triplet spin state response 11 . If the first qubit is in the singlet spin state, S, the frequency shifts negatively due to the change in impedance of the circuit. This is shown schematically by the first qubit singlet spin state response 12. At the first frequency ₀ which is the resonant frequency of just the first resonant circuit, the negative frequency shift results in a reduced amplitude and/or reduced phase of the detected first alternating potential at the first frequency ₀.

The second frequency f is higher than the first frequency ₀ in Figure 11. The second frequency f is the resonant frequency of just the second resonant circuit. Figure 11 shows the second qubit triplet spin state response 21 and the second qubit singlet spin state response 22. Similarly to the first qubit, when the second qubit is in the triplet spin state, T, the properties of the detected second alternating potential substantially match expected values of the properties of the alternating potential for the triplet spin state; and when the second qubit is in the singlet spin state, S, the properties of the detected second alternating potential substantially match expected values of the properties of the alternating potential for the singlet spin state. For the singlet spin state, the frequency shifts negatively relative to the triplet spin state due to the change in impedance of the circuit. In Figure 11 , the triplet spin state responses 11 , 21 are substantially aligned with the respective resonant frequencies. However, in practice, these may also be shifted in frequency due to the non-ideal circuit features.

Figure 11 shows the S21 response, however the S11 response is similar. Figure 11 describes the singlet and triplet states measured when the qubit is a double quantum dot qubit. For examples involving the measurement of single quantum dot qubits, the properties of a detected alternating potential for a data qubit having a spin state parallel to the known spin of the ancillary qubit will be substantially matched to expected values of the properties for the parallel spin state (as shown in the triplet spin state responses 11 , 21 in Figure 11); and the properties of a detected alternating potential for a data qubit having a spin state anti-parallel to the known spin of the ancillary qubit will be substantially matched to expected values of the properties for the anti-parallel spin state (as shown in the singlet spin state responses 12, 22 in Figure 11). Typically, the anti-parallel spin state response is lower in frequency relative to the parallel spin state response.

The examples described above differ in the arrangement of the circuit elements including the position of the data qubits relative to the reservoir (and therefore the number of intermediate circuit elements required), and the arrangement of intermediate circuit elements. In each case, a charge polarisation measurement, driven by a high-frequency alternating potential applied by a frequency source, is detected at a distance using a cascade process in which the quantum information is successively passed on from one circuit element to the next. The charge carriers may be electrons or holes; for examples involving a plurality of qubits, the qubits may be driven by the same or different frequencies; the qubits may be single quantum dot or double quantum dot qubits; paths for different qubits may be partially shared or separate; one or more frequency sources, reservoirs and resonator circuits may be provided for a plurality of qubits.

As will be appreciated, a radio-frequency charge carrier cascade methodology for the readout of charge polarisation in semiconductor spin qubits at an arbitrary distance from the qubits is provided by a circuit enabling a cascade of cyclic tunnelling events. The circuit is driven by a high-frequency sinusoidal excitation which causes cyclic tunnelling of charge carriers when a double quantum dot qubit is in a singlet state or when a single quantum dot qubit has an anti-parallel spin relative to its neighbouring ancillary qubit, which in each case subsequently induces a tunnelling event in a neighbouring, capacitively coupled, quantum dot. The cascade terminates with a charge carrier tunnelling to and from a reservoir, which can be measured using readout circuitry comprising resonant circuits. The cascade can be used to transmit quantum information within a quantum dot array over an arbitrary distance.

Claims

1 . A circuit for reading out the state of a qubit, the circuit comprising: a double quantum dot forming a qubit having a singlet spin state and a triplet spin state, wherein the double quantum dot comprises a first quantum dot and a second quantum dot, wherein the second quantum dot is tunnel coupled to the first quantum dot; a gate electrode configured to control the energy state of either the first or the second quantum dot; a cascade quantum dot capacitively coupled to the second quantum dot; a charge reservoir tunnel coupled to the cascade quantum dot; readout circuitry connected to the charge reservoir; and a frequency source connected to the gate electrode or the charge reservoir and configured to apply an alternating potential thereto; wherein the cascade quantum dot is tuned to be close to a charge transition such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the cascade quantum dot and the charge reservoir; wherein the frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa; and wherein the readout circuitry is configured to: detect a first alternating potential; measure one or more properties of the detected first alternating potential; and infer that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state.

2. The circuit of claim 1 , further comprising one or more intermediate cascade double quantum dots each comprising a first intermediate cascade quantum dot and a second intermediate cascade quantum dot, wherein: the second intermediate cascade quantum dot is tunnel coupled to the first intermediate cascade quantum dot; the second quantum dot is capacitively coupled to the first intermediate cascade quantum dot in one of the one or more intermediate cascade double quantum dots; and the cascade quantum dot is capacitively coupled to the second intermediate cascade quantum dot in one of the one or more intermediate cascade double quantum dots; wherein each intermediate cascade double quantum dot is tuned to be close to a charge transition such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the first intermediate cascade quantum dot and the second intermediate cascade quantum dot of that intermediate cascade double quantum dot; and wherein the frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the first intermediate cascade quantum dot to the second intermediate cascade quantum dot of each of the one or more intermediate cascade double quantum dots sequentially, and vice-versa, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice- versa.

3. The circuit of claim 1 or claim 2, further comprising: a second double quantum dot forming a second qubit having a singlet spin state and a triplet spin state, wherein the second double quantum dot comprises a second first quantum dot and a second second quantum dot, wherein the second second quantum dot is tunnel coupled to the second first quantum dot; a second gate electrode configured to control the energy state of either the second first or the second second quantum dot; and a second cascade quantum dot capacitively coupled to the second second quantum dot and tunnel coupled to the charge reservoir; wherein the frequency source is connected to the second gate electrode or the charge reservoir and is configured to apply an alternating potential thereto; wherein the frequency source is further configured to apply an alternating potential at a second frequency thereby to cause cyclic tunnelling of a charge carrier from the second first quantum dot to the second second quantum dot and vice-versa only when the second qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the second cascade quantum dot to the charge reservoir, and vice-versa; and wherein the readout circuitry is further configured to: detect a second alternating potential; measure one or more properties of the detected second alternating potential; and infer that the second qubit is in the triplet spin state when the one or more measured properties of the detected second alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the second qubit is in the singlet spin state when the one or more measured properties of the detected second alternating potential substantially match expected values of the properties for the singlet spin state.

4. The circuit of claim 1 or claim 2, further comprising: a third double quantum dot forming a third qubit having a singlet spin state and a triplet spin state, wherein the third double quantum dot comprises a third first quantum dot and a third second quantum dot, wherein the third second quantum dot is tunnel coupled to the third first quantum dot; and a third gate electrode configured to control the energy state of either the third first or third second quantum dot; wherein the cascade quantum dot is further capacitively coupled to the third second quantum dot; wherein the frequency source is connected to the third gate electrode or the charge reservoir and is configured to apply an alternating potential thereto; wherein the frequency source is further configured to apply an alternating potential at a third frequency thereby to cause cyclic tunnelling of a charge carrier from the third first quantum dot to the third second quantum dot and vice-versa only when the third qubit is in the singlet spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice-versa; and wherein the readout circuitry is further configured to: detect a third alternating potential; measure one or more properties of the detected third alternating potential; and infer that the third qubit is in the triplet spin state when the one or more measured properties of the detected third alternating potential substantially match expected values of the properties for the triplet spin state; and infer that the third qubit is in the singlet spin state when the one or more measured properties of the detected third alternating potential substantially match expected values of the properties for the singlet spin state.

5. The circuit of claim 3 or claim 4, when dependent on claim 2, wherein one or more of the intermediate cascade double quantum dots are common to the double quantum dot and the second and/or third double quantum dots.

6. The circuit of any of claims 3 to 5, wherein the second and/or third frequencies are different from the first frequency.

7. A circuit for reading out the state of a qubit, the circuit comprising: a double quantum dot comprising a first quantum dot having a first spin state and a second spin state and a second quantum dot having the first spin state and the second spin state, wherein the second quantum dot is tunnel coupled to the first quantum dot; wherein the first quantum dot forms a data qubit and the second quantum dot forms an ancillary qubit; or the first quantum dot forms an ancillary qubit and the second quantum dot forms a data qubit; a gate electrode configured to control the energy state of either the first or the second quantum dot; control circuitry connected to the ancillary qubit, the control circuitry configured to initialise the ancillary qubit in the first spin state; a cascade quantum dot capacitively coupled to the second quantum dot; a charge reservoir tunnel coupled to the cascade quantum dot; readout circuitry connected to the charge reservoir; and a frequency source connected to the gate electrode or the charge reservoir and configured to apply an alternating potential thereto; wherein the cascade quantum dot is tuned to be close to a charge transition such that when a charge carrier tunnels between the first quantum dot and the second quantum dot, a charge carrier tunnels between the cascade quantum dot and the charge reservoir; wherein the frequency source is configured to apply an alternating potential at a first frequency thereby to cause cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa only when the data qubit is in the second spin state, and to also cause cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir, and vice- versa; and wherein the readout circuitry is configured to: detect a first alternating potential; measure one or more properties of the detected first alternating potential; and infer that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; and infer that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state.

8. A method of reading out the state of a qubit, wherein the qubit is a double quantum dot qubit comprising first and second tunnel-coupled quantum dots, the qubit having a singlet spin state and a triplet spin state, wherein a cascade quantum dot is capacitively coupled to the double quantum dot qubit, tunnel coupled to a charge reservoir, and tuned to be close to a charge transition, and wherein a frequency source is connected to the charge reservoir or a gate electrode configured to control the energy state of either the first or the second quantum dot, the frequency source being configured to apply an alternating potential thereto, wherein the method comprises: applying, using the frequency source, an alternating potential at a first frequency to the gate electrode or the charge reservoir; and detecting, using readout circuitry connected to the charge reservoir, a first alternating potential; measuring, using the readout circuitry, one or more properties of the detected first alternating potential; and inferring that the qubit is in the triplet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the triplet spin state; or inferring that the qubit is in the singlet spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the singlet spin state; wherein the double quantum dot qubit is configured such that when an alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the first quantum dot to the second quantum dot and vice-versa is caused only when the qubit is in the singlet spin state, and also cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir is caused, and vice-versa.

9. A method of reading out the state of a data qubit, wherein the data qubit is a quantum dot having a first spin state and a second spin state, the data qubit is tunnel coupled to another quantum dot having the first spin state and the second spin state, the another quantum dot forming an ancilla qubit, wherein the data qubit and ancilla qubit form a double quantum dot, wherein a cascade quantum dot is capacitively coupled to the double quantum dot, tunnel coupled to a charge reservoir, and tuned to be close to a charge transition, and wherein a frequency source is connected to the charge reservoir or a gate electrode configured to control the energy state of either the quantum dot or the another quantum dot, the frequency source being configured to apply an alternating potential thereto, wherein the method comprises: initialising, using control circuitry connected to the ancillary qubit, the ancillary qubit in the first spin state; applying, using the frequency source, an alternating potential at a first frequency to the gate electrode or the charge reservoir; and detecting, using readout circuitry connected to the charge reservoir, a first alternating potential; measuring, using the readout circuitry, one or more properties of the detected first alternating potential; and inferring that the data qubit is in the first spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the first spin state; or inferring that the data qubit is in the second spin state when the one or more measured properties of the detected first alternating potential substantially match expected values of the properties for the second spin state; wherein the double quantum dot is configured such that when the alternating potential at the first frequency is applied, cyclic tunnelling of a charge carrier from the quantum dot to the another quantum dot and vice-versa is caused only when the data qubit is in the second spin state, and also cyclic tunnelling of a charge carrier from the cascade quantum dot to the charge reservoir is caused, and vice-versa.