WO2023232175A1 - Method for determining ground state energies of a physical system by means of a quantum computer - Google Patents

Abstract

The invention relates to a method for determining ground state energies of a physical system by means of a quantum computer, wherein a test state for the physical system is prepared on the quantum computer. Aside from the two most common methods to determine the ground state energy on the quantum computer, that is to say the variational approach and quantum phase estimation (QPE), the method according to the invention proposes determining ground state energies of a physical system by means of a quantum computer by determining at least one expected value of an energy with regard to the test state and at least one expected value of an observable, wherein the at least one determined expected value of the observable is used to determine the ground state energies of the test state by using the at least one determined expected value of the observable and the at least one expected value of the energy with regard to the test state to generate a functional for the energy on the basis of the expected values of the observable for the physical system and determining the ground state energy by applying an energy minimization to the functional.

Description

METHOD FOR DETERMINING BASIC STATE ENERGIES

OF A PHYSICAL SYSTEM

USING A QUANTUM COMPUTER

The invention relates to a method for determining ground state energies of a physical system using a quantum computer.

A typical use case for quantum computers is determining the ground state energy of the Hamiltonian of a quantum mechanical system. The two most common methods for estimating ground state energy on quantum computers are variational approaches and quantum phase estimation (QPE).

In a Variational Quantum Eigensolver (VQE), different test states are generated, depending on classical parameters. The energy of the test states is measured and minimized using the classical parameters. The test state with the lowest energy provides an approximate ground state. The accuracy of the variational methods depends heavily on the approach chosen for the test states that are prepared on the quantum computer.

Widely used approaches range from very general test states with many free classical parameters in uCCSD (Romero, J. et al. Strategies for quantum computing molecular energies using the unitary coupled cluster approach. Quantum Science and Technology 4, 014008, October 2018) to very problem-specific approaches with few free parameters in the Variational Hamiltonian approach (VHA) (Wecker, D., Hastings, MB & Troyer, M. Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (October 4, 2015)).

wobei 9 der Vektor der klassischen Parameter ist, die U^di) unitäre Transformationen sind, die auf dem Quantencomputer durchgeführt werden und von 6 abhängen und

ein Anfangszustand ist, der auf dem Quantencomputer effizient initialisiert werden kann. The preparation of the test states can be generally written as,

where 9 is the vector of classical parameters, which U^di) are unitary transformations performed on the quantum computer and depend on 6 and

is an initial state that can be efficiently initialized on the quantum computer.

Minimizing over 6 gives the approximate ground state energy,

When determining the optimal classical parameters 6 _opt, the following problems can arise, among others:

1. The application of the unitary transformations U^di) on NISQ quantum computers is not error-free, where NISQ stands for Noisy-Intermediate-Scale Quantum and refers to logic gate-based quantum computers.

2. Dissipation processes and other errors can change the available space of test states, so that a good approximation of the ground state is not found. An example is the loss of electrons during simulation of a particle number-preserving electron system on a quantum computer with dissipation.

3. As the number of classical variables increases, the classical optimization of 9 becomes increasingly difficult. It is from McClean et al. (McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nature Communications 9, 4812. issn: 2041-1723, 2018) known that the Potential landscape E(ö) with determination of the ground state energy by functional construction based on measurements of a quantum computer of increasing dimension of 6 becomes flat and the probability of finding a non-vanishing gradient becomes exponentially small.

4. Regardless of the size of 0, statistical fluctuations in the measured energy must be expected on NISQ quantum computers. The energy measurement is done by repeated projective measurements of the qubits and the finite number of measurements leads to limited readout accuracy.

5. Due to the statistical fluctuations, the application of gradient-based optimization methods, such as the “conjugate gradient” (CG) method or quasi-Newton methods such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, is extremely difficult and must be done less performant, gradient-free optimization methods, such as the “Constrained optimization by linear approximation” (COBYLA) method, can be used.

6. Even when gradient-based optimizers are used, the results are highly dependent on the chosen starting values of the classic parameters, 0 _O , and experience shows that ten or more repetitions are required of the optimization process, starting with different starting values of the classic parameters in order to reliably obtain a good result.

Some of the problems mentioned above, especially those mentioned under points 4 to 6, can in principle be counteracted by a larger number of measurements.

However, on NISQ quantum computers, especially on hardware with slow gates and measurements such as ion traps, the applicability of the variational approaches is significantly limited by the large total number of unitary operations and measurements required.

A clear summary of the points mentioned above can be found in the review article by McArdle et al. (Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin & Xiao Yuan. Quantum computational chemistry. arXiv: 1808.10402 [quant-ph], arXiv: 1808. 10402. http://arxiv.org/abs/1808.10402 (2018 ) (August 30, 2018)).

With the help of Quantum Phase Estimation (QPE) better results can be achieved than with variational approaches, but the resource requirement is significantly greater. In order to apply QPE to a quantum mechanical system with m base states, n qubits are required in addition to m system qubits, in which the state of the quantum mechanical system is stored. The anzilla qubits are brought into a superposition state using Hadamard gates and for each anzilla qubit a controlled time evolution is applied to the m qubits,

In the end, a quantum Fourier transform is applied to the anzilla qubits. With a probability that depends on the initial state of the m system qubits, after measuring the anzilla qubits one obtains an stood in which the m system qubits contain the ground state. In this case, measuring the ancestral qubits results in a binary representation of the phase Eoto, from which the ground state energy Eo is obtained.

On classical computers, once the quantum mechanical system reaches a certain size, the state vector can no longer be stored (“exponential wall”). Approximation methods are necessary to calculate the ground state energy. One of the most important approximation methods in general and by far the most important method in quantum chemistry is density functional theory (DFT).

In the context of DFT, based on the Hohenberg-Kohn theorem (Hohenberg, P. & Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 136, B864-B871 (3B Nov. 1964)), for example, the ground state energy is determined by minimization of an energy functional. Formally, the DFT is a Legendre-transformed description of the quantum mechanical problem, in which a change from the local potentials to the local densities is made as a parameter of the system. The energy is no longer viewed as a functional of the local potentials, but as a functional of the local densities. In contrast to conventional variational methods in quantum mechanics, the variational parameters are not external quantities that characterize a class of wave functions, for example, but rather expected values of the quantum mechanical problem itself, the electron density in the case of DFT. Since not all contributions to the energy functional are explicitly known, they must be approximated.

Approximations of explicitly unknown energy contributions are primarily obtained from the following two methods:

1. Calculation using an effective, non-interacting quantum system (Kohn-Sham construction (Kohn, W. & Sham, LJ Self-Consistent Equa- tions Including Exchange and Correlation Effects. Phys. Rev. 140, A1133- Al 138 (4A Nov. 1965))), e.g. for kinetic energy.

2. Calculation from explicit approximation formulas of variable complexity, e.g. for the so-called exchange-correlation energy.

Furthermore, DFT can be generalized to, among other things, reduce the contributions to be approximated in the energy functional. In addition to the electron density, spin magnetization, electron and spin current, kinetic energy density, stress tensor, double occupancy, etc. can be considered as additional variational parameters.

In the following, the term functional theories is used to describe methods that describe physical systems through the extremization of a functional that depends on intrinsic system quantities or combinations of intrinsic and external quantities. This includes DFT and the generalizations mentioned above as well as time-dependent density functional theory (Time-Dependent DFT, TD-DFT), DFT at finite temperature (Finite-Temperature / Thermal-DFT, FT-DFT) and approaches from many-body theory, e.g., many-body perturbation theory (MBPT) , Dynamical Mean Field Theory (DMFT) and embedding methods such as Density Matrix Embedding Theory (DMET).

Furthermore, the term “functional theories” is used to describe inverse mean-field methods in which the optimal values for the parameters are measured directly using the many-body wave functions, especially on the quantum computer, without carrying out an explicit minimization of the functional.

The object of the invention is, among other things, the determination of the ground state energy of a physical system using a quantum computer using a particularly small number of measurements on the quantum computer.

This task is solved using a method according to current claim 1. Advantageous embodiments of the method according to the invention can be found in the dependent claims.

According to the invention, it is a method for determining ground state energies of a physical system using a quantum computer. The method is characterized in that a test state is prepared for the physical system on the quantum computer, at least one expected value is determined together with commutating partial amounts of the at least one expected value of the test state and at least one expected value of an observable, using the at least one specific expected value of the observable the ground state energies of the test states are determined by generating a functional of the expected value of the test state as a function of the expected values of the observables for the physical system from the at least one specific expected value of the observables and the at least one expected value of the test state and by applying an energy minimization to the functional the ground state energy is determined.

In other words: In the invention proposed here, the energy and expected values of (local) observables obtained in the context of a quantum simulation are related to one another in the sense of a functional theory. This means that a functional is constructed from the measurement data, the minimization of which provides a new approximate value for the ground state energy. In contrast to previous methods, the external variational parameters are replaced by system-internal expected values. Examples of such observables are listed and described in more detail below. To determine the ground state energy, the constructed functional is minimized. The selection of the expected values represents an adjustment screw that allows the invention to be adapted to the specific problem. Essentially, the selection of observables influences the complexity of the functional to be approximated and the so-called N-representability conditions, which guarantee that the expected value of an observable can be realized by a quantum state \ip).

The use of known N-representability conditions to reduce the required number of measurements has been proposed in particular in the international patent application PCT/US2017/067095: Known N-representability conditions are used to reduce the number of measurements. In this invention presented here, a different, complementary strategy is pursued, which differs in that the functional connection of the ground state energy and a selection of observables is constructed from the measurements of variational quantum states. This means that work is carried out under the assumption that the synthesized quantum states fulfill the N-representability conditions by construction and thereby make it possible to span the space of physical expectation values. In particular, the strategy proposed in the international patent application PCT/US2017/067095 and the method described here are not mutually exclusive, but can even be combined.

According to the method according to the invention, a starting state I 'init) is first initialized on a qubit register of the quantum computer. To determine the ground state energy of a quantum mechanical system, one ideally chooses a state that already has a large overlap with the real ground state, but can be initialized efficiently with a few gates on the quantum computer. An example is the ground state of an interacting electron system described within the framework of a so-called mean field approximation, which can be obtained from functional theories. The Ground states of square mean-field Hamiltonians can be efficiently produced on the quantum computer (see, for example, Jiang, Z., Sung, KJ, Kechedzhi, K., Smelyanskiy, VN & Boixo, S. Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions. Phys. Rev. Applied 9, 044036 (Apr. 2018).

After preparing the test state by applying a sequence of unitary transformations to the selected ground state, various expected values of various observables are measured with respect to the test state. For the common variational methods it is sufficient to measure the expected value of the energy.

The invention presented has several advantages compared to usual variational approaches, which result from the combination of variational quantum computer algorithms with methods from functional theory.

Variational approaches such as VQE are limited to the Fock space of possible test states that can be reached by the unitary transformations Ui ß ) from the starting state \ipintt). By evaluating the functional, the presented invention is not strictly limited to the Fock space of the test states and can in principle also find the ground state energy if the ground state does not correspond to any of the possible test states. However, since states close to the ground state must be sampled for a precise determination of the ground state energy, this only applies to ground states with small deviations from possible test states. For example, extrapolation is not possible if the true ground state and the possible test states have completely different symmetries.

The possibility of finding minima outside the Fock space of the test states also helps with the fault tolerance of the presented invention. Errors in NISQ quantum computers during the preparation of the test states can cause it It can happen that the Fock space of the test states moves away from the true ground state.

so führt in der Jor- dan-Wigner-Repräsentation die Depolarisation der Elektronenorbitale auf dem Quantencomputer dazu, dass sich die Zahl der Anregungen ändert und die Teilchenzahl in den Testzuständen nicht erhalten ist. Ein üblicher variationeller Ansatz kann in diesem Fall nicht die gesuchte Grundzustandsenergie finden, da kein Zustand mit richtiger Teilchenzahl gefunden werden kann. Im Vergleich dazu ist es mit der vorgestellten Erfindung möglich, zur Grundzustandsenergie bei korrekter Teilchenzahl unter Benutzung des konstruierten Energiefunktionals zu extrapolieren, solange die Teilchenzahlerhaltung nicht so stark verletzt ist, dass die Qualität des Energiefunktionals kompromittiert wird. An example of this is the conservation of particle numbers in electron systems. In many cases, for example in quantum chemistry, the ground state of a system should be determined with a fixed and known number of electrons. One initializes the initial state

In the Jordan-Wigner representation, the depolarization of the electron orbitals on the quantum computer causes the number of excitations to change and the number of particles in the test states is not maintained. In this case, a usual variational approach cannot find the ground state energy we are looking for because no state with the correct particle number can be found. In comparison, with the presented invention it is possible to extrapolate to the ground state energy at the correct particle number using the constructed energy functional, as long as particle number conservation is not violated to such an extent that the quality of the energy functional is compromised.

auf, da beide im Allgemeinen eine große Zahl von Quantengattern enthalten. Die vorgestellte Erfindung wird von diesen Fehlem wenig beeinflusst. Jeder Testzustand

der auf einem Quantencomputer mit zeitabhängigem Fehler präpariert wird, kann für die Konstruktion des Energiefunktionals herangezogen werden, auch wenn er nicht dem Zustand

) entspricht, solange die N-Darstellbarkeitsbedingungen nicht zu stark verletzt sind. Damit produziert auch eine Messung von valide Daten für die Konstruktion des Funktionals. Für den Fall, dass die N- Darstellbarkeitsbedingungen für einen bestimmten Zustand verletzt sind, so kann der Zustand auch einfach verworfen werden. Dies gilt insbesondere für die Bestimmung des Grundzustandes von Spin-Problemen, wie sie zum Beispiel für den Quantum-Approximate-Optimization-Algorithm (QOAO) notwendig ist. In Spin-Systemen ist jeder Zustand des Quantencomputers ein valider Zustand des simulierten Spin-Systems und kann zur Bestimmung des Grundzustandes benutzt werden. Another problem for conventional variational approaches is non-static errors. The errors on the quantum computer change between two preparations of test states, so that even with the same classical parameters, 6 different test states are prepared. Due to the fluctuations in the test states, the measured energy also fluctuates and the classic optimizer has problems finding the minimum. The majority of errors occur during the initialization of the start state and the unitary transformations

because both generally contain a large number of quantum gates. The invention presented is little influenced by these errors. Any test condition

which is prepared on a quantum computer with time-dependent error, can be used for the construction of the energy functional, even if it does not correspond to the state

) corresponds as long as the N-representability conditions are not violated too severely. This also produces a measurement of valid data for the construction of the functional. In the event that the N-representability conditions for a specific state are violated, the state can also simply be rejected. This applies in particular to determining the ground state of spin problems, such as is necessary for the Quantum Approximate Optimization Algorithm (QOAO). In spin systems, each state of the quantum computer is a valid state of the simulated spin system and can be used to determine the ground state.

The main advantage of the invention over usual variational approaches is the faster convergence of the method, i.e. with a small number of measurements on the quantum computer. Typical variational approaches require a large portion of the measurements for the final optimization steps, which only minimally improve the approximated ground state energy. Black box optimizers in particular require many steps at the end of the optimization to rule out the existence of a deeper local minimum. Additionally, both gradient-based and black-box optimizers require multiple runs with different starting parameters to have a high probability of obtaining a good result.

The high difficulty of classical optimization is understandable if you take a closer look at the idea of the VHA, for example. In the VHA, the unitary transformations in the preparation of the test state are chosen so that they correspond to the time evolution of an electron system under a gradual switching on of the interaction between electrons. The classic parameters 9 correspond to pseudo time steps in the time development. The classic optimizer should then find an optimized quasi-adiabatic time evolution that transfers the system from the non-interacting to the interacting ground state. This shows why it is not easy for the optimizer to find a clear optimum for parameters 9. By definition, an adiabatic time evolution can have any number of revolutions If you take paths through the Fock or Hilbert space, as long as it is adiabatic enough and the initial Hamiltonian corresponds to the non-interacting system and the final Hamiltonian corresponds to the interacting system, you will get the interacting ground state. Analogously, there can in principle be many parameter sets 6 that transform the initial state into the correct basic state or a very similar state.

In contrast, the invention uses physical observables to minimize the energy functional. On the one hand, the constructed energy functional can provide a better approximation for the ground state energy through extrapolation, and on the other hand, the gradients of the energy functional with respect to the observables in a feedback mechanism can accelerate the optimization of the classical parameters. Furthermore, with the help of the energy functional and methods of functional theory, stagnation in the minimization can be detected at an early stage and unnecessary measurements of the quantum register can be avoided.

wobei ( _i,X_i, Y_i,Z_i') die Identitäts- beziehungsweise Paulioperatoren sind, die auf das (/)-te Qubit wirken. (N) ist die Zahl der Qubits im Quantenregister, die benutzt werden, um die Basiszustände des simulierten Systems zu speichern. Die Erwartungswerte der einzelnen Produkte von Paulioperatoren können einfach durch Anwendung maximal einer Ein-Qubit-Rotation für jedes Qubit und mehrfacher projektiver Messung der Qubits in der Z-Basis gemessen werden. Anschließend wird der Erwartungswert der Energie des Systems im Testzustand durch Messungen des Quantencomputers bestimmt. Dazu werden “kompatible” (kommutierende) Teilbeiträge der Energie simultan ermittelt, um die Anzahl der benötigten Messungen zu optimieren. Aus der Gesamtheit aller - im Allgemeinen nicht kommutierenden - Teilbeiträgen, wird der Erwartungswert der Energie im Testzustand bestimmt. In an advantageous embodiment of the method, the Hamilton operator H of the physical system under investigation is rewritten into a sum of products of Pauli operators for all qubits of the quantum register of the test state,

where ( _i ,X _i , Y _i ,Z _i ') are the identity and Pauli operators, respectively, acting on the (/)th qubit. (N) is the number of qubits in the quantum register used to store the basis states of the simulated system. The expected values of the individual products of Pauli operators can be easily measured by applying a maximum of one-qubit rotation for each qubit and multiple projective measurements of the qubits in the Z-basis. The expected value of the energy of the system in the test state is then determined by measurements from the quantum computer. For this purpose, “compatible” (commutating) partial energy contributions are determined simultaneously in order to optimize the number of measurements required. The expected value of the energy in the test state is determined from the totality of all - generally non-commuting - partial contributions.

The central idea of the invention is not to use the classic parameters 9 of the test states for minimizing the energy - and thus the approximation of the ground state energy - but rather the measured expected values of the observables in relation to the test state, for example the local densities of the system . At the same time, an optimization of the measured observables can be carried out.

Based on the general ideas of a functional theory described above, a functional of the energy can be generated from the measured observables and the energy of a large number of test states generated by the quantum computer depending on the expected values of the observables for the system under consideration.

To construct the functional, not only the expected value of the energy of each test state must be measured, but also the expected values of the observables. If these two measurements are completely independent, the measurement of the observables results in considerable additional effort, which runs counter to the aim of the invention, namely to get by with fewer measurements than in the known variational approaches.

In order to construct a functional that provides a good basis for minimizing the energy, physically motivated observables are chosen. By measuring the expected energy value via the above-mentioned decomposition of the Hamilton operator into Pauli products, these ob- servables, which can also be broken down into Pauli products, are in most cases automatically measured and there is therefore no additional measurement effort.

wobei t die Hoppingstärke, U die Dichte-Dichte Wechselwirkungsstärke, p das chemische Potential und c_{x n} der elektronische Vernichter eines Elektrons im Orbital x mit Spin Up oder Down ist, so werden während der Messung von (H) automatisch die lokalen kinetischen Erwartungswerte (ct_acj _a + h. c. ) gemessen, um die gesamte kinetische Energie zu bestimmen, und die lokalen Dichten werden gemessen, um den Beitrag des chemischen Potentials zur Gesamtenergie zu erhalten. For example, if you look at the Hubbard model,

where t is the hopping strength, U is the density-density interaction strength, p is the chemical potential and c _xn is the electronic annihilator of an electron in orbital x with spin up or down, the local kinetic expectation values (ct _a cj _a + hc ) are measured to determine the total kinetic energy, and the local densities are measured to obtain the contribution of the chemical potential to the total energy.

Examples of observables that can be used to construct a functional are:

- Local densities: For example, the local densities of an electron system. This is where the greatest similarities lie with standard DFT, which uses density functionals.

- Local magnetization densities: The local magnetization density and/or magnetization current can be used to describe spin systems.

- Kinetic energies: The expected values of the local kinetic energies between different locations in a grid system. There are similarities here to so-called kinetic functional theories (see for example in Löpez-Sandoval, R. & Pastor, GM Densitymatrix functional theory of strongly correlated lattice fermions. Phys. Rev. B 66, 1551 18 (15 Oct. 2002), Tokatly, IV Time-dependent current density functional theory on a lattice. Phys. Rev. B 83, 035127 (3 Jan. 2011), Theophilou, I., Buchholz, F., Eich, FG, Ruggenthaler, M. & Rubio, A. Kinetic-Energy Density-Functional Theory on a Lattice. Journal of Chemical Theory and Computation 14. PMID: 29969552, 4072-4087 (2018) and Schmitteckert, P. Inverse mean field theories. Phys. Chem. Chem. Phys. 20, 27600-27610 (43 2018))

- One-particle reduced density matrix: Local densities, kinetic energy and general expectation values of the one-particle reduced density matrix, see for example Gilbert, T. L. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B 12, 21 11 (1975).

- Density-density correlators: The double population or local density-density correlation plays an important role in the description of highly correlated states, for example in the Hubbard model.

spielen eine wichtige Rolle in der Beschreibung von magnetischen Zuständen. - Spin-spin correlators: The correlators of the spin operators, in particular (S?S2) and

play an important role in the description of magnetic states.

spielt eine wichtige Rolle in der Beschreibung von Hochtemperatursupraleiter. - Pair hopping: Pair hopping

plays an important role in the description of high-temperature superconductors.

- abnormal 1RDM: For the description of superconductivity within the framework of a BCS theory, abnormal 1 RDM expected values (1 RDM = one-particle reduced density matrix) are of central importance.

- Entanglement entropies: Entanglement entropies are based on the partitioning of the system into local environments. They can be used to characterize the spread of information in the system.

- Embedding methods: Functional theories that are based on a self-consistent description of local clusters embedded in effective environments, for example DMET or (cluster) DMFT.

gleichbedeutend mit der Aussage, dass die Grundzustandsenergie durch den kleinsten Erwartungswert des Hamiltonoperators des Systems bezüglich aller Quantenzustände \ip) gegeben ist. In einem zweiten Schritt werden alle Quantenzustände durch ihre Erwartungswerte bestimmter, generischer Dichteoperatoren oder Observablen p_t, klassifiziert und die Optimierung zunächst innerhalb dieser Klassen durchgeführt,

In contrast to the usual DFT, there may be no Hohenberg-Kohn theorem for general functional theories, i.e. no clear mapping from ground state densities to corresponding coupling potentials. However, it is possible to define an energy functional using a so-called “constrained search” method. For this purpose, only the variational character of the ground state energy, Eo, is exploited, ie,

equivalent to the statement that the ground state energy is given by the smallest expectation value of the Hamiltonian of the system with respect to all quantum states \ip). In a second step, all quantum states are classified by their expected values of certain generic density operators or observables p _t and the optimization is initially carried out within these classes,

The minimization breaks down into terms that are explicitly given by the expectation values of the observables, represented in the aforementioned equation by Ei Pi * h _t , and a remainder, defined in the aforementioned equation as the infimum of the expectation value of H _w over all quantum states , which provide the required expected values {. Here H _w is the Hamilton operator without terms that explicitly link to the observable,

This operator is introduced to go from the infimum over test states to the infimum over expected values. This contribution is generally not explicitly known and therefore needs to be approximated. The choice of the observable p therefore significantly influences which components of the energy are approximately considered. In principle, it is possible to consider all observables relevant to the calculation of energy as fundamental variables and thus explicitly calculate all energy contributions. However, it should be noted that the N-representability conditions are generally not well known or require a complex numerical implementation. This means that the choice of observables allows complexity of the energy contributions to be traded for complexity in the N-representability conditions. By selecting the observables, the functional theory can be adapted to the specific problem. The actual energy minimization based on the measured data can be done in different ways.

Three possible strategies are suggested here:

- Envelope: If you plot the measured energy over the observables, you get a (potentially high-dimensional) point cloud. An example of a Hubbard model can be seen in Figure 1 below. This shows sampling data from a VHA optimization of the ground state of a 6 x 1 Hubbard model with periodic boundary conditions at half filling and U/t = 1. In the upper plot, the energy difference to the exact ground state energy E _gs is plotted against the deviation of the local energy from the exact value 0.5. The bottom plot shows the energy difference over the difference between the local kinetic energy and the exact local kinetic energy in the ground state. In both cases you can see point clouds of the measurement data, which condense as they converge to the exact basic state. From this high-dimensional point cloud, a convex envelope can be constructed, which represents an approximation of the inner optimization cycle in the constrained search method. The minimum of the so constructed th energy functional produces a new estimate for the ground state energy.

- Multidimensional fit: A multidimensional fit function can be created from the energies and the complete set of measured observables, for example with the help of machine learning techniques, the global minimum of which can be determined. The fit function for determining the minimum itself can be represented, for example, by a neural network that was trained on the measured data. The resulting energy functional can be minimized using classical optimization methods.

- Physically motivated functional: Furthermore, a physically motivated functional can be created for a specific system to be examined, also using already known approximations for specific models in the context of DFT. These functionals have parameters that can be approximately determined using the measured data. Finally, the minimum energy and thus the ground state energy can be calculated from the functionals with the fitted parameters.

One way to determine the large number of test states through whose measurements the functional is constructed, for example along the strategies suggested above, is to combine the construction of the functional with the well-known variational approaches. With a classic optimizer, the energy of the test states is optimized using the classic parameters 9. With every energy measurement, the observables are also measured and stored in order to gradually build up the functional.

This route is chosen to use as many low-energy test states as possible to construct the functional. When chosen purely at random In test states the spread would be too large to build a meaningful functional. However, it is not absolutely necessary to let the optimization of the classic parameters 9 run until convergence, as long as enough test states are evaluated to obtain the desired accuracy in the functional design. Furthermore, this can be successively improved

The classic parameters 9 can be functionally fed back into the optimization in order to accelerate it.

Claims

PATENT CLAIMS 1 . Method for determining a ground state energy of a physical system by means of a quantum computer, wherein at least one test state is prepared for the physical system on the quantum computer, characterized in that at least one expected value of an energy in relation to the test state and at least one expected value of an observable is determined and by means of of the at least one specific expected value of the observables, the ground state energy of the physical system is determined in that from the at least one specific expected value of the observables and the at least one energy in relation to the test state, a functional of the energy depends on the at least one expected value of the observables for the physical system is generated and the ground state energy is determined by applying energy minimization to the functional. 2. The method according to claim 1, characterized in that the at least one test state of a classic parameter 6 according to I '(ö)

is dependent, whereby the test state is determined by initializing a start state

and subsequent unitary transformation Uj ßi) is formed on a qubit register of the quantum computer. 3. The method according to claim 1 or 2, characterized in that the at least one test state is generated by a variational method, the parameters being optimized in the variational method in such a way that the expected value of the energy becomes smaller and this approaches a minimum. Method according to at least one of claims 1 to 3, characterized in that the at least one expected value of the energy with respect to the test state is measured as a sum of partial contributions according to (H) = i ht {Ai), where A _t is the measured operators and ht is a prefactor, and that at the same time by means of the measured operators A _t according to 0 = £ 0^), where 0 is the at least one observable and Oi are prefactors which can also be zero, the at least one expected value of the observable is determined. Method according to at least one of the preceding claims, characterized in that the Hamilton operator H is used to determine the at least one expected value of the energy of the physical system and the at least one observable in the form of a sum of products of Pauli operators for all qubits of the qubit register H =

for the Hamilton operator and according to 0=

Ilt i °a for which at least one observable is described, where ([ _L , X _L , Y _L , Z _L ) are the identity or Pauli operators acting on the (i)th qubit. Method according to at least one of the preceding claims, characterized in that a physical observable is used to generate the functional. Method according to claim 6, characterized in that the physical observable is a local density, a local magnetization density, a kinetic energy, a one-particle reduced density matrix, a density-density correlation, a spin-spin correlation, a pair hopping, a anomalous one-particle reduced density matrix, an entanglement entropy, an embedding method, or a combination thereof. Method according to at least one of the preceding claims, characterized in that the functional is defined via a constrained search method with which the ground state energy of the physical system is determined by minimizing the expected value of the energy via physically permitted quantum states. Method according to claim 8, characterized in that the quantum states are classified into specific, generic density operators or observables p by their expected values and an optimization is carried out within the respective class. Method according to at least one of the preceding claims, characterized in that the energy minimization is carried out by constructing an enveloping, convex structure on the functional. Method according to at least one of the preceding claims 1 to 9, characterized in that the energy minimization is carried out by a multidimensional adaptation to the functional. Method according to at least one of the preceding claims 1 to 9, characterized in that the energy minimization is carried out by creating a concrete, physically motivated functional.