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WO2025074301A2 - Procédés et systèmes de reconstruction d'état quantique à l'aide d'une tomographie d'états quantiques d'ombres neuronales - Google Patents

Procédés et systèmes de reconstruction d'état quantique à l'aide d'une tomographie d'états quantiques d'ombres neuronales Download PDF

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WO2025074301A2
WO2025074301A2 PCT/IB2024/059692 IB2024059692W WO2025074301A2 WO 2025074301 A2 WO2025074301 A2 WO 2025074301A2 IB 2024059692 W IB2024059692 W IB 2024059692W WO 2025074301 A2 WO2025074301 A2 WO 2025074301A2
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quantum
state
classical
shadow
loss function
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WO2025074301A3 (fr
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Bohdan KULCHYTSKYY
Shunji Matsuura
Wirawat KOKAEW
Pooya Ronagh
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1QB Information Technologies Inc
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1QB Information Technologies Inc
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/60—Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms

Definitions

  • quantum data may need to be stored in a classical data structure as a proxy for access to information about a given quantum state.
  • this data structure is the list of the measurements performed (e.g., a tomographically complete set of measurements such as those used in the classical shadow technique in Huang et al., “Predicting Many Properties of a Quantum System from Very Few Measurements,” Nature Physics 16, 1050–1057, 2020, which is incorporated herein by reference in its entirety).
  • a raw dataset already includes the information that has been collected from a quantum state. This raises the question as to what the role of a neural quantum state may be when it cannot learn more about the physics of the state than what it is provided in the training data.
  • (d) comprises Monte Carlo (MC) sampling from said classical shadow state.
  • said snapshot state comprises a stabilizer state.
  • (c) and (d) comprise evaluating said overlap between said stabilizer state and said ansatz state; wherein said overlap is evaluated based at least in part on samples from said stabilizer state.
  • a validation dataset is used to prevent overfitting in training.
  • a training dataset is separated into mini- batches to calculate gradients.
  • said loss function comprises cross- entropy that is estimated using an empirical data distribution.
  • the method may comprise: (a) obtaining one or more quantum measurements; (b) constructing a classical shadow state from a classical shadow technique based at least in part on said one or more quantum measurements; (c) constructing a loss function of a machine learning (ML) model representative of a parameterized quantum state based at least in part on said classical shadow state, wherein said loss function comprises an overlap between a snapshot state representative of said one or more quantum measurements and an ansatz state provided by said ML model; (d) calculating a gradient of said loss function based at least in part on samples from said classical shadow state; and (e) training said ML model, wherein said training comprises using said gradient of said loss function to update one or more parameters of said ML model.
  • ML machine learning
  • the present disclosure provides a hybrid system for a quantum state reconstruction.
  • the system may comprise (a) a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system; and (b) a high- performance computing platform operatively coupled to said quantum computer, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.
  • said high-performance computing platform is operatively coupled to said quantum computer using a digital computer.
  • the system further comprises a cloud-based computing platform.
  • said hybrid system provides computational services on the cloud.
  • said hybrid system is configured to operate in a distributed computing environment.
  • the present disclosure provides a system for a quantum state reconstruction.
  • the system may comprise a high-performance computing platform operatively coupled to a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.
  • the system may comprise (a) a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, wherein said quantum computer is operatively coupled to a high-performance computing platform, said high- performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.
  • said high-performance computing platform is operatively coupled to said quantum computer using a digital computer.
  • the system further comprises a cloud-based computing platform.
  • said hybrid system provides computational services on the cloud.
  • said hybrid system is configured to operate in a distributed computing environment.
  • Another aspect of the present disclosure provides a system comprising one or more computer processors and computer memory coupled thereto.
  • the computer memory comprises machine executable code that, upon execution by the one or more computer processors, implements any of the methods above or elsewhere herein.
  • FIG.1 is a schematic of an example hybrid computing system for a quantum state reconstruction, in accordance with some embodiments.
  • FIG.2 is a flowchart of an example method for a quantum state reconstruction, in accordance with some embodiments.
  • FIG.3A is a diagram of an example Clifford measurement framework: randomly selecting a unitary U from the n-qubit Clifford group.
  • FIG. 3B is a diagram of an example Pauli measurement framework: randomly selecting a unitary U by taking an n-qubit tensor product of single-qubit unitaries independently sampled from the single-qubit Clifford group, in accordance with some embodiments.
  • FIGS.6A-6E are a group of graphs of a performance comparison of reconstructed six-qubit GHZ mixed states using classical shadows and neural quantum states collected from 5000 random Pauli measurements, in accordance with some embodiments.
  • FIG.7A and FIG.7B are a pair of graphs depicting the discrepancy in angle for the Monte Carlo-estimated gradients compared to the exact gradient throughout the training process, in accordance with some embodiments.
  • FIG.8A and FIG.8B are a pair of graphs of the Kullback–Leibler (KL) divergence which compares the proxy for a typically unknown data distribution with empirical and shadow-weight distributions, both accessible from measurement data, in accordance with some embodiments.
  • KL Kullback–Leibler
  • the term “e.g.” explains that “instructions” are an example of “data” that the computer may send over the Internet, and also explains that “a data structure” is an example of “data” that the computer may send over the Internet.
  • both “instructions” and “a data structure” are merely examples of “data,” and other things besides “instructions” and “a data structure” can be “data.”
  • the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values the term “at least,” “greater than,” or “greater than or equal to” applies to each of the numerical values in that series of numerical values. For example, greater than or equal to 1, 2, or 3 is equivalent to greater than or equal to 1, greater than or equal to 2, or greater than or equal to 3.
  • a quantum processor or quantum computer may comprise one or more quantum gate arrays, one-way quantum computers, topological quantum computers, superconductor-based quantum computers, trapped ion quantum computers, trapped atom quantum computers, optical lattices, quantum dot computers, spin-based quantum computers, spatial-based quantum computers, Loss–DiVincenzo quantum computers, nuclear magnetic resonance (NMR) based quantum computers, solution-state NMR quantum computers, solid-state NMR quantum computers, solid-state NMR Kane quantum computers, electrons-on-helium quantum computers, cavity-quantum- electrodynamics based quantum computers, molecular magnet quantum computers, fullerene-based quantum computers, linear optical quantum computers, diamond-based quantum computers, nitrogen-vacancy (NV) diamond-based quantum computers, Bose– Einstein condensate-based quantum computers, transistor-based quantum computers, and rare-earth-metal-ion-doped inorganic crystal based quantum
  • a quantum device with a limitation of a two-dimensional structure of a quantum chip or a limitation on how many neighboring qubits each qubit is connected to may benefit from methods and systems disclosed herein.
  • suitable quantum computers may include, by way of non-limiting examples, including the associated references, each of which is incorporated by reference herein in its entirety: superconducting quantum computers (qubits implemented as small superconducting circuits—Josephson junctions) (Clarke et al., “Superconducting quantum bits”, Nature 453, no. 7198, pp.
  • spatial-based quantum dot computers (qubits implemented as electron positions in a double quantum dot) (Fedichkin et al., “Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot”, arXiv:quant-ph/0006097, 2000); coupled quantum wires (qubits implemented as pairs of quantum wires coupled by quantum point contact) (Bertoni et al., “Quantum logic gates based on coherent electron transport in quantum wires”, Physical Review Letters 84, no.25, p.5912, 2000); nuclear magnetic resonance quantum computers (qubits implemented as nuclear spins and probed by radio waves) (Cory et al., “Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing”, arXiv: quant-ph/9709001, 1997); solid-state NMR Kane quantum computers (qubits implemented as the nuclear spin states of phosphorus donors in silicon) (Kane, “A silicon-based nuclear spin quantum computer”, Nature 393, no.
  • Quantum computer hardware based on rare- earth-ion-doped inorganic crystals”, Optics Communications 201, no. 1–3, pp. 71–77, 2002; metal-like carbon nanospheres based quantum computers (qubits implemented as electron spins in conducting carbon nanospheres) (Náfrádi et al., “Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres”, arXiv:cond- mat/1611.07690, 2016); topological quantum computers (qubits implemented as non- Abelian anyons) (Nayak et al., “Non-Abelian Anyons and Topological Quantum Computation,” arXiv:0707.1889, 2007); photonic continuous-variable quantum computing hardware (quantum variables represented by the quadrature operators of the quantum harmonic oscillators in a quantum optical mode) (Arrazola et al., “Quantum circuits with many photons on a programmable nanophotonic chip,” Nature 591, pp.
  • quantum computing hardware based on bosonic codes error-protected qubits are formed by embedding a finite-dimensional code space within the infinite- dimensional Fock space associated with a bosonic quantum field mode; examples include the Gottesman–Kitaev–Preskill (GKP) code, cat codes, and binomial codes, respectively) (Gottesman et al., “Encoding a qubit in an oscillator,” Physical Review A 64, 012310, 2001; Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes,” PRX Quantum 3, 010329, 2022; Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode,” Physical Review X 6, 031006, 2016); quantum hardware based on coherent network computing (operating by sampling low-energy eigenstates of an Ising Hamiltonian by encoding the spins in a network of optical parametri
  • a GPU may be a specialized electronic circuit optimized for high throughput, which can perform the same set of operations in parallel on many data blocks at a time.
  • a matrix multiplication device may be a TPU.
  • a TPU may be a type of ASIC developed for low bit precision processing by Google® Inc.
  • a matrix multiplication device may be an FPGA.
  • An FPGA may be an integrated circuit chip that comprises configurable logic blocks and programmable interconnects. It can be programmed after manufacturing to execute custom algorithms.
  • a matrix multiplication device may be an ASIC.
  • An ASIC may be an integrated circuit chip that is customized to run a specific algorithm. In some case, an ASIC cannot be programmed after manufacturing.
  • a matrix multiplication device may be a TSP.
  • the classical computer is connected to the Internet such that it accesses the World Wide Web.
  • the classical computer is connected to one or more computer servers, which can enable distributed computing, such as a cloud computing infrastructure.
  • the classical computer is connected to an intranet and/or extranet or an intranet and/or extranet that is in communication with the Internet.
  • the classical computer is connected to a data storage device.
  • the network is a telecommunication and/or data network.
  • the network is a peer- to-peer network, which may enable devices coupled to the computer system to behave as a client or a server.
  • the operating system may be, for example, software, including programs and data, which manages the device’s hardware and provides services for execution of applications.
  • Suitable server operating systems include, by way of non- limiting examples, FreeBSD, OpenBSD, NetBSD®, Linux®, Apple® Mac OS X Server®, Oracle® Solaris®, Windows Server®, and Novell® NetWare®.
  • Suitable personal computer operating systems may include, by way of non-limiting examples, Microsoft® Windows®, Apple® Mac OS X®, Apple® macOS®, UNIX®, and UNIX- like operating systems such as GNU/Linux®.
  • the operating system is provided by cloud computing.
  • Suitable mobile smart phone operating systems may include, by way of non-limiting examples, Nokia® Symbian® OS, Apple® iOS®, Research In Motion® BlackBerry OS®, Google® Android®, Microsoft® Windows Phone® OS, Microsoft® Windows Mobile® OS, Linux®, and Palm® WebOS®.
  • Suitable media streaming device operating systems may include, by way of non-limiting examples, Apple TV®, Roku®, Boxee®, Google TV®, Google Chromecast®, Amazon Fire®, and Samsung® HomeSync®.
  • Suitable video game console operating systems may include, by way of non-limiting examples, Sony® PS3®, Sony® PS4®, Microsoft® Xbox 360®, Microsoft® Xbox One®, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®.
  • the non-volatile memory comprises ferroelectric random-access memory (FRAM). In some cases, the non-volatile memory comprises phase-change random-access memory (PRAM). In some cases, the non-volatile memory comprises resistive random-access memory (RRAM).
  • the device comprises a storage device including, by way of non-limiting examples, CD-ROMs, DVDs, flash memory devices, magnetic disk drives, magnetic tapes drives, optical disk drives, and cloud computing-based storage. In some cases, the storage and/or memory device comprises a combination of devices such as those disclosed herein. [0054]
  • the classical computer includes a display to send visual information to a user. In some cases, the display is a cathode ray tube (CRT).
  • CTR cathode ray tube
  • the input device is a pointing device including, by way of non-limiting examples, a mouse, trackball, track pad, joystick, game controller, or stylus.
  • the input device is a touch screen or a multi-touch screen.
  • the input device is a microphone to capture voice or other sound input.
  • the input device is a video camera or other sensor to capture motion or visual input.
  • the input device is a Kinect®, Leap Motion®, or the like.
  • the input device is a combination of devices such as those disclosed herein.
  • a classical shadow may be created by repeatedly performing the following procedure: at every iteration ⁇ , a random unitary transformation ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ is applied, after which all the qubits in the computational basis are measured.
  • the pairs ( ⁇ ⁇ , ⁇ ⁇ ) may be recorded in classical memory.
  • ⁇ ⁇ ⁇ be the associated density matrix and stabilizer state, respectively.
  • the mapping M: ⁇ ⁇ ⁇ [ ⁇ ⁇ ] is invertible if the measurement protocol is tomographically complete.
  • the classical shadow ⁇ ⁇ suffices to predict ⁇ arbitrary linear and nonlinear l og ( ⁇ ) max ⁇ ⁇ ⁇ 2 functions of the state ⁇ up to an additive error ⁇ if it is of size ⁇ ⁇ ⁇ shadow
  • the shadow norm may depend on the ensemble from which the unitaries ⁇ ⁇ are drawn for creating the classical shadow and has efficient bounds. In some cases, a shadow norm may have efficient bounds if the ensemble satisfies a unitary 3-design property.
  • FIG. 3A illustrates a diagram of an example Clifford measurement framework: randomly selecting a unitary U from the n- qubit Clifford group.
  • FIG. 3B illustrates a diagram of an example Pauli measurement framework: randomly selecting a unitary U by taking an n-qubit tensor product of single- qubit unitaries independently sampled from the single-qubit Clifford group.
  • the Clifford measurement may comprise randomly selecting a unitary transformation U from the n-qubit Clifford group.
  • the Pauli measurement may comprise randomly selecting a unitary transformation U by taking an n-qubit tensor product of single-qubit unitaries independently sampled from the single-qubit Clifford group.
  • the Clifford measurement may use Clifford circuits generated by CNOT, Hadamard, or phase gates.
  • the Pauli measurement may use single-qubit Clifford gates, which may be decomposed into tensor products.
  • the classical snapshot may be written as where the external index ⁇ refers to the measurement index while the internal index ⁇ refers to the qubit index.
  • infidelity may offer a better optimization landscape due to its metric properties and bounded range.
  • the infidelity may not be constrained within the physically valid [0, 1] range.
  • infidelity cannot guarantee bounded errors in approximating many quantities of interest from the trained model.
  • the loss may not generalize to ansatz for mixed states.
  • the NSQST approach may face many practical challenges in its training protocol. The trainability of the ansatz may use an ad hoc initialization step, without which the infidelity-based optimization stalls.
  • Methods and systems disclosed herein may improve upon NSQST and may provide an alternative approach to improving the predictive power of Pauli ensembles (e.g., the shallowest possible shadows) by using a neural network (or other parameterized ansatz). Whereas the raw classical shadow is powerful at predicting many observables efficiently, it may not always be sample efficient due to the unphysical nature of the state it provides.
  • neural quantum states may allow for natively imposing physicality constraints on a reconstructed state.
  • Another advantage of the unitary design property of the Clifford ensemble is that it may allow for the efficient prediction of nonlinear functions of a quantum state. This may be particularly useful in estimating entropy-based quantities such as the Rényi entropies, since they involve two copies of a state, and thus quadratic functions of it. Neural networks may also improve the predictive power of such nonlinear quantities when combined with a purification technique.
  • Neural Shadow Quantum State Tomography is a technique that may be used for harnessing the capabilities of artificial neural networks to effectively extract essential information about quantum states from experimental measurements. This approach may achieve its goal by training a neural network to recognize underlying patterns within measurement data. NNQST may approximate a neural network wave function that closely matches some class of target states.
  • FIG. 1 there is shown a schematic of an example hybrid computing system for a quantum state reconstruction.
  • the system may comprise a quantum computer 126 and a high-performance computing (HPC) platform 100.
  • the system may comprise a digital computer 114.
  • the quantum computer 126 and the HPC platform 100 may be operatively coupled using the digital computer 114.
  • each node corresponds to a classical computing instance that can perform a specific individual task, with the digital computer 114 combining the results from the HPC nodes.
  • the HPC platform 100 may be configured to at least (i) obtain one or more quantum measurements from the quantum computer 126; (ii) construct a classical shadow state using a classical shadow technique based at least in part on the one or more quantum measurements; (iii) use the classical shadow state to construct a loss function of an ML model representative of a parameterized quantum state; (iv) use samples from the classical shadow state to estimate the gradient of the loss function; and (v) use the gradient of the loss function to train the ML model by updating its parameters.
  • the quantum state may comprise a pure quantum state or a mixed quantum state.
  • the mixed quantum state may correspond to a state prepared via a noisy quantum circuit.
  • the mixed quantum state may correspond to a subsystem embedded into a larger pure quantum state.
  • the mixed quantum state may comprise an ensemble of pure quantum states prepared via repeated execution of a quantum circuit containing measurement gates.
  • one or more quantum measurements are obtained.
  • the one or more quantum measurements may comprise a tomographically complete set of quantum measurements that permit a complete reconstruction of the measured quantum state.
  • the gradient of the loss function is estimated using samples from the classical shadow state.
  • the classical shadow state may comprise the stabilizer state, the snapshot state, or the sum of the snapshot or stabilizer states.
  • the estimation may comprise Monte Carlo (MC) sampling from the classical shadow state.
  • the estimation may comprise MC sampling of stabilizer states from the classical shadow state.
  • the estimation may comprise MC sampling from the stabilizer state.
  • the estimation may comprise MC sampling of computational basis measurements from the stabilizer state.
  • the gradient of the loss function may be estimated using a hybrid computing system comprising an HPC platform, such as the hybrid computing system disclosed herein with respect to FIG.1.
  • this sum may be exponentially large in the computational basis with respect to the number of qubits and may be intractable. However, it may be estimated using Monte Carlo sampling in at least two ways.
  • the second estimator may keep the training process more analogous to supervised learning.
  • the stable statistics collected from ⁇ may provide a steady target during training.
  • ⁇ ⁇ may take the simpler form
  • FIG.7A and FIG.7B there are shown a pair of graphs depicting the discrepancy in angle for the Monte Carlo-estimated gradients compared to the exact gradient throughout the training process.
  • the dataset consists of 1000 Clifford measurements performed on a three-qubit GHZ state.
  • NS neural state-based sampling
  • SS stabilizer-based sampling
  • FIG. 7A The gradient errors are evaluated over (FIG. 7A) mini-batches and (FIG. 7B) a full batch.
  • the error bars correspond to the standard deviation in the error over all mini-batches.
  • the graphs depict the error in the gradient direction as quantified by the angle between the gradient evaluated exactly and the gradient estimated with estimators using NS and SS.
  • the errors are evaluated on samples from mini-batches and a full batch, as shown in FIG.7A and FIG.7B, respectively.
  • the Kullback–Leibler (KL) divergence may be considered.
  • the loss function of the ML model representative of the parameterized quantum state may comprise the KL divergence between the distribution corresponding to the one or more quantum measurements in a random basis and the distribution corresponding to the ansatz state.
  • the loss function of the ML model representative of the parameterized quantum state may comprise the KL divergence between the distribution corresponding to the ansatz state and the distribution corresponding to the one or more quantum measurements in a random basis.
  • the loss function L ECE ( ⁇ ) may be viewed in light of the empirical distribution is an indicator function. Since ⁇ emp data is an unbiased estimator for ⁇ , the loss function LECE ( ⁇ ) may be rewritten as [0110] The new summation runs over distinct elements of the training data without repetition. This reformulation helps alternative unbiased estimators to be considered for the data distribution. An approach based on the classical shadow state ⁇ ⁇ may be considered.
  • the normalized shadow weights may now replace the empirical averaging in the cross-entropy approximation, justifying the new loss function which is referred to as the shadow-based cross-entropy (SCE) loss function.
  • SCE shadow-based cross-entropy
  • the new loss function for L SCE ( ⁇ ) may leverage the classical shadow state to inject further signals useful for the training dynamics.
  • the loss function comprising the cross-entropy may be derived using the estimator .
  • the ML model is trained by updating its parameters using the gradient of the loss function.
  • the training may comprise pretraining using synthetic data.
  • a fixed training dataset may be used for the ML model’s training.
  • a validation dataset may be used to prevent overfitting in training.
  • FIG. 8A and FIG. 8B there is shown a pair of graphs of the Kullback–Leibler (KL) divergence which compares the proxy for a typically unknown data distribution with empirical and shadow-weight distributions, both accessible from measurement data.
  • the KL divergence is examined as a function of (FIG.8A) the system size for a dataset of 1000 shadows and (FIG.8B) the number of shadows for a six-qubit system.
  • the measurement type is specified by the marker type: square for Clifford measurements and circle for Pauli measurements.
  • the target distribution type is specified by the line styles: dashed and solid curves for the empirical distribution and shadow weights, respectively.
  • the KL divergence KL ⁇ data target emp ⁇ was considered between this typically unknown distribution and ⁇ target , which denotes either of the two empirically accessible distributions used for training: ⁇ emp is based on the equation ⁇ emp whereas ⁇ sh is specified by the equation ⁇ sh Specifically, the dependence was studied on the size of the dataset relative to the size of the system by scaling the number of qubits given a fixed dataset, as disclosed herein with respect to FIG.8A, and scaling the number of measurements in the dataset for a fixed number of qubits, as disclosed herein with respect to FIG.8B.
  • FIG.8A and FIG.8B indicate that this benefit can be reclaimed by increasing the number of measurements performed.
  • a crucial factor in interpreting the plots disclosed herein with respect to FIG.8A and FIG.8B may be in considering the difference in the configuration space dimensions of ⁇ e d m at p a associated with Pauli and Clifford measurements.
  • Pauli measurements can reconstruct 4 ⁇ distinct stabilizers
  • Clifford measurements can reconstruct 2 0.5+ ⁇ (1) ⁇ 2 distinct stabilizers
  • n 6 qubits
  • the dataset sizes considered in FIG. 8B were compared to the dimension of the stabilizer space.
  • the dataset ranges from 5% to 500% of the total set size of distinct stabilizers.
  • the maximum number of measurements is less than 0.001% of the corresponding stabilizer set size.
  • FIG. 7A shows that both estimators struggle most with capturing the correct gradient direction during the early stages of training, where the neural quantum state is likely the most different from the target state.
  • the stabilizer-based estimation significantly improves, while the neural state estimation remains relatively high and inconsistent.
  • the stabilizer-based estimator consistently provides estimates with much lower variance, which is essential for training stability. Additionally, the estimator proposed herein is consistently more accurate than the neural-based estimator.
  • the stochastic error due to mini-batching is expected to be cancelled out when averaged over the mini-batches and thus can be lowered using smaller learning rates, albeit at the cost of slower training dynamics.
  • FIG.7B shows whether this expected behaviour holds for the given estimators.
  • the stabilizer-based estimator functions as an unbiased estimator, reducing gradient errors by averaging over mini-batches.
  • the neural state-based estimator exhibits a systematic bias throughout the entire training duration, resulting in an extreme accumulation of errors from the true gradient. To eliminate this finite-sample bias, a significant increase in the samples generated by the neural state may be necessary.
  • the estimators may be employed in evaluating the overlap between the neural network (ansatz state) and stabilizer states.
  • the superscripted abbreviations “NS” (for neural state-based samples) and “SS” (for stabilizer-based samples) appear with the corresponding loss function L to distinguish between the two estimators.
  • FIG. 4A-FIG. 4F there are shown a group of graphs of the infidelity and infidelity loss, along with the cross-entropy loss, between a reconstructed state and the actual state for four methods in reconstructing both six- and eight-qubit GHZ states. The results are averaged over five training trials on the same dataset trained over 50 epochs. Gradient estimation is based on 500 Monte Carlo samples.
  • Each estimate of the overlap between a neural quantum state and a stabilizer is based on 500 newly generated samples.
  • the final states for each loss function and trial are represented by markers. Square and triangle markers represent the samples for the stabilizer- and neural state- based infidelity loss functions, respectively. Circle and “x” markers represent the samples for the empirical and shadow-based entropy loss functions, respectively.
  • the upper row displays the infidelity loss (left y-axis) and the cross-entropy loss (right y-axis) for the loss functions, while the lower row shows the associated infidelity with respect to the true state as a measure of the generalization power of the neural quantum state.
  • the cross- entropy loss is normalized to be below one in order to facilitate a visual comparison with the infidelity loss.
  • FIG. 4A – FIG. 4F show the trainability and the generalization power of these training methods via the optimization progress curves of loss functions evaluated on the training set (upper row) and infidelity with respect to the true states (lower row), over 50 epochs.
  • FIG.4A and FIG.4B show the superiority of cross- entropy-based training over the infidelity-based approach. The latter struggles to achieve significant progress in approximating the underlying quantum state.
  • the benefits of the cross-entropy-based loss function become even more pronounced when considering computational costs. Both loss functions rely on computing the overlap between the neural quantum state and stabilizers.
  • each Pauli measurement leads to a proliferation of stabilizers that grows super-exponentially with the number of qubits.
  • Using a Monte Carlo approach to address this prohibitive scaling of the infidelity-based approach through sampling of stabilizers from a classical shadow state leads to unstable training.
  • this prohibitive scaling is entirely absent for both cross-entropy-based loss functions, as each Pauli measurement contributes a single stabilizer.
  • stabilizer-based sampling improves in-training performance across all loss functions (the results for neural state-based cross-entropy loss functions have been omitted for visual clarity).
  • these improvements during training translate to enhanced generalization only for cross-entropy loss functions.
  • the target quantum state to be reconstructed may be a mixed quantum state generated by noisy circuits, wherein the reconstruction is from Pauli or Clifford measurements.
  • This approach allows physical constraints to be imposed on the ansatz.
  • Pauli Observables Although classical shadows based on Pauli measurements predict local observables with high accuracy, their predictive accuracy may worsen in the case of many-body observables. The benefits of the neural network approach were demonstrated over classical shadows in predicting Pauli observables. To do so, a set of 5000 random Pauli strings was generated by sampling each single-qubit Pauli operator from a uniform distribution. [0134] Now referring to FIG. 5A and FIG.
  • FIG. 5B there are shown a pair of graphs of the predictive capabilities based on neural quantum states, classical shadows, and the simplex projection of the latter, for noisy observables of a mixed GHZ state prepared using a series of noisy gates with noise levels p ranging from 0.0 to 0.5.
  • the dataset used for reconstructions contains 5000 random Pauli measurements.
  • FIG. 5A shows absolute error, ⁇ , in predicting random Pauli observables with standard deviations represented by error bars.
  • FIG.5A shows accuracy, defined as the average of the absolute error over the standard deviation, ⁇ ⁇ / ⁇ ⁇ .
  • the results are differentiated by marker style, with neural quantum states shown using squares, classical shadows using circles, and the simplex projections of classical shadows using triangles.
  • FIG.6E there are shown a group of graphs of a performance comparison of reconstructed six-qubit GHZ mixed states using classical shadows and neural quantum states collected from 5000 random Pauli measurements.
  • FIG.6A shows theoretical values of purity for the ideal state (shown using a solid curve), along with the reconstructed states from neural quantum states (shown using square markers), classical shadows (shown using circles), and the simplex projections of classical shadows (shown using triangles).
  • FIG.6B shows trace distance with respect to the ideal state.
  • the grey-scale bar as well as the height of each data point indicates the magnitude of each component of the density matrix.

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Abstract

La présente divulgation concerne des procédés et des systèmes pour une reconstruction d'états quantiques. Les procédés peuvent consister à : obtenir une ou plusieurs mesures quantiques ; utiliser la ou les mesures quantiques pour construire un état d'ombre classique à l'aide d'une technique d'ombre classique ; utiliser l'état d'ombre classique pour construire une fonction de perte d'un modèle d'apprentissage automatique (ML) représentatif d'un état quantique paramétré, la fonction de perte comprenant un chevauchement entre un état d'instantané représentatif de ladite mesure ou desdites mesures quantiques et un état d'ansatz fourni par le modèle ML ; utiliser des échantillons de l'état d'ombre classique pour estimer le gradient de la fonction de perte ; et utiliser le gradient de la fonction de perte pour entraîner le modèle ML. Les systèmes peuvent comprendre un système hybride comprenant un calculateur quantique comprenant une puce quantique comprenant une pluralité de bits quantiques et un système de commande, et une plateforme informatique à hautes performances.
PCT/IB2024/059692 2023-10-03 2024-10-03 Procédés et systèmes de reconstruction d'état quantique à l'aide d'une tomographie d'états quantiques d'ombres neuronales Pending WO2025074301A2 (fr)

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