Method for determining a ground state of a quantum system, method for preparing a ground state of a quantum system by means of a quantum computer, hybrid computer platform, adiabatic variational quantum algorithm

Abstract

The invention describes a method (100) for determining a ground state (1000) of a quantum system, wherein the quantum system comprises a simplified quantum system and a correction quantum system, wherein the method comprises the following steps: • providing o a variational quantum thermaliser (1010), comprising at least one quantum circuit (1011), and o a temperature value (1001); • determining (102) a thermal state (1020) of the simplified quantum system at the temperature value (1001) by means of the variational quantum thermaliser (1010); • determining a thermal state (10202) of the quantum system by performing a time evolution (1025) of the thermal state (1020) of the simplified quantum system, wherein the time evolution (1025) starts in the simplified quantum system and ends in the quantum system by time-dependent successive activation of the correction quantum system; • determining (103) a spectrum (1030) of the quantum system using the time-evolved thermal state (10202) of the quantum system; • determining (104) the ground state (1000) of the quantum system by selecting the state from the spectrum (1030) of the quantum system with which the lowest energy is associated as the ground state (1000) of the quantum system.

Description

[0001] Description

[0002] title

[0003] Method for determining a ground state of a quantum system, method for preparing a ground state of a quantum system using a quantum computer, hybrid computer platform, adiabatic variational quantum algorithm

[0004] State of the art

[0005] In “Proof of the Adiabatic Theorem” (M.Bornand and V.Fock, Z. Phys.51, 165(1928)), “On non-adiabatic processes in inhomogeneous fields” (J. Schwinger,

[0006] Phys. Rev.51, 648(1937)) and “On the adiabatic theorem of quantum mechanics” (T.Kato, JPSJ5, 435(1950)) describe a method for determining a ground state of a many-body system using adiabatic time evolution on a quantum computer or a classical computer.

[0007] Core and advantages of the invention

[0008] Finding the ground state of a quantum mechanical system is an important task in the context of atomistic material simulations and in the field of quantum chemistry.

[0009] An important task of currently available quantum computers is the calculation of the ground state of a quantum mechanical system. Currently available noisy intermediate-scale quantum (NISQ) computers are limited in their capabilities. Due to their limited size (small number of qubits) and inherent gate errors, such noisy NISQ computers only allow the execution of short quantum circuits, i.e., quantum circuits of shallow depth, and the results typically exhibit rather large error bars. Due to the noise and errors, most known algorithms (such as phase estimation) are not applicable. Due to the limitations of NISQ computers, many approaches are based on hybrid quantum-classical algorithms, i.e., algorithms where parts are executed on the quantum computer and parts on classical computers, such as the variational quantum eigensolver (VQE).Such algorithms can support the development of new materials. Currently, NISQ computers without error correction and with a moderate number of qubits (less than 100) are available. Nevertheless, these NISQ computers can already achieve an advantage in terms of speed and accuracy over conventional high-performance computers in the fields of materials simulation and quantum chemistry.

[0010] The VQE calculates the expectation value of a parameterized circuit and optimizes the parameters to minimize the energy. The global energy minimum is then assumed to be a good approximation to the ground-state energy. Optimization in the VQE becomes challenging for larger systems, as the optimizer can get stuck in local minima or in so-called Baren plateaus with vanishing gradients.

[0011] Currently, many quantum problems are solved with variational quantum algorithms using a hybrid quantum-classical approach. Such hybrid methods utilize a quantum computer, particularly an NISQ computer, to execute a classically infeasible subroutine, and a classical computer to solve the overarching problem. A NISQ computer typically comprises a small number of qubits, for example, less than or equal to 1000 qubits, in particular 50 to several hundred qubits. The number of real physical qubits of the NISQ generation of quantum computers would be insufficient for quantum-based error correction. The term "intermediate-scale" is intended to reflect the low computing power of these low-qubit, noise-sensitive quantum computers.

[0012] Many applications aim to find a ground state of a Hamiltonian (i.e., the lowest-energy state) using variations. In this approach, a quantum state (e.g., a wavefunction) is encoded using a variational approach in a quantum circuit with variational parameters 0; and the corresponding expectation value Et is measured. This measurement requires many (on the order of thousands) individual measurements to obtain a statistically well-converged expectation value.

[0013] After obtaining the expected value, a classical method, specifically a classical optimizer, is used to update the variational parameters 0;. For VQE, for example, the parameters are updated along a downward direction to minimize the expected value (energy) from the quantum circuit. The updated parameters 6 i+1 are then fed back to the quantum computer to calculate E i+1This iterative process is repeated until the converged ground state energy E n at the n-th iteration with its corresponding state, which is defined in the gate parameters 9 n is encoded.

[0014] The invention relates to a method for determining a ground state of a quantum system, a method for preparing a ground state of a quantum system by means of a quantum computer, a use of the method for determining a ground state of a quantum system for a material simulation, a hybrid computer platform and an adiabatic variational quantum algorithm.

[0015] The present invention relates to a variational algorithm based on a variational quantum thermalizer (VQT), which is used to determine a thermal state at a given temperature by optimizing the free energy (=objective function of the VQT optimizer) and to enable a reproducible initialization of this thermal state on the quantum computer. The present invention uses this VQT algorithm together with another class of algorithms based on adiabatic evolution to transform a quantum mechanical system from a simplified, i.e.In particular, to develop a classically solvable problem (described by a simplified Hamiltonian and hereinafter also referred to as a simplified quantum system) into a physically interesting problem (hereinafter also referred to as a quantum system, which can be described by a sum of a simplified Hamiltonian and a correction Hamiltonian). This approach usually breaks down when phase transitions occur on the way from the simple to the interesting system. Here, a method is proposed that uses an adiabatic variational quantum algorithm (an adiabatic evolution algorithm) based on the variational quantum thermalizer (VQT) and uses finite temperatures to overcome phase transitions. This approach is also referred to as "thermal adiabatic evolution."

[0016] A variational quantum thermalizer is a variational quantum algorithm, particularly executable or executed on a hybrid computer platform comprising a classical computer and a quantum computer, and used to determine a thermal state of the quantum system at a given temperature and / or the spectrum of the quantum system, in other words, to determine the possible energy values ​​that can be measured on the quantum system under consideration at the given temperature. In particular, the VQT additionally measures the entropy St in each minimization step. This can be achieved either by introducing a second measurement between two consecutively executed quantum circuits, by classical sampling from a Boltzmann machine, or by another classical method that generates samples from a tunable probability distribution.The state determined in this way describes the density matrix of the quantum system at the given temperature. By minimizing the free energy F. f = Et ~ TS t In each iteration step i, an approximation to the thermal state can be obtained. This iterative process is repeated until the converged free energy F n at the n-th iteration with its corresponding thermal state, which is defined in the gate parameters 9 n is encoded.

[0017] The invention is disclosed below with the features of the independent patent claims, which advantageously enables the ground state of a quantum system to be determined efficiently. One advantage of providing an improved method for determining the ground state of a quantum system is that it allows for more reliable and efficient material simulations, thus enabling, supporting, and, in particular, accelerating the development of new materials. This is achieved with a method according to claim 1 for determining a ground state of a quantum system.

[0018] The term "quantum system" refers to physical systems in which manifestations of quantum mechanics are visible. Examples of such manifestations are quantization of energy or other observables, interference of particle waves, non-locality, or quantum mechanical tunneling. Quantum systems encompass the entire microscopic world, such as elementary particles and atoms, but also electrical conductors with dimensions in the nanometer range, semiconductors, large and small molecules, and certain materials whose macroscopic properties are determined by quantum mechanical interactions on microscopic scales. In particular, the quantum system can be described by a Hamiltonian. In quantum mechanics, the Hamiltonian of a system is an operator that describes the total energy of that system, including kinetic energy and potential energy.Its spectrum, the energy spectrum of the system, comprises the eigenvalues ​​of the Hamiltonian, i.e., the energy eigenvalues. This is the set of possible results that can be obtained by measuring the total energy of the system. Due to its close relationship to the energy spectrum and the time evolution of a system, it is of fundamental importance for most formulations of quantum theory and quantum chemistry. Certain optimization problems, such as the combinatorial job shop problem, can also be reformulated so that the solution is given as a state of a corresponding Hamiltonian. Such problems can also be solved efficiently using a quantum algorithm or a hybrid quantum-classical algorithm.

[0019] Determining the ground state here specifically means determining an approximation of the actual ground state of the quantum system. In other words, a state is determined that at least approximates the actual ground state.

[0020] Executing a quantum circuit on a quantum computer specifically means that gates provided in the quantum circuit are executed on the qubits assigned to the initial quantum state, and measurements are performed on each of them, so that the execution of the quantum circuit serves to generate measurement results. Measuring the expected values ​​of observables is an essential component of variational quantum algorithms. This requires a large number of individual measurements for statistical convergence in order to meet precision requirements, such as chemical accuracy in applications to quantum chemistry calculations. In other words, executing the quantum circuit on the quantum computer involves executing the quantum circuit multiple times on the quantum computer. To give an order of magnitude for the number of measurements: It is usually true that a measurement with precision E requires a number of measurements of order 1 / E A2 required.

[0021] The method according to claim 1 enables the determination of a ground state of a quantum system,

[0022] • where the quantum system comprises a simplified quantum system and a correction quantum system (in other words: where the quantum system is divisible into a simplified quantum system and a correction quantum system); For example, in a Hubbard model, the simplified system can be constructed by setting the interaction parameter U of electrons in the same orbital to 0 and the correction term includes all components of the entire quantum system that depend on U. Another option is that all local terms of the entire quantum system form the simplified quantum system and the correction term accordingly includes the non-local terms. In a further embodiment, the simplified quantum system is chosen as a system that is only described by a Slater determinant and accordingly contains no correlations of the electrons / qubits. In this case, the correction term describes the correlation between the electrons orQubits. A simplified quantum system is understood, in particular, as a subsystem of the quantum system, where an additional correction term transforms the simplified quantum system into a quantum system. In particular, the simplified quantum system is a simpler system than the quantum system; in particular, it describes a problem that is simpler than the overall quantum system, and in particular, one that is even classically solvable.

[0023] The procedure includes the following steps:

[0024] • Providing a variational quantum thermalizer comprising at least one quantum circuit; In particular, this step can comprise providing at least one quantum circuit of the VQT. The provision can be effected, in particular, by input, by data transmission or wireless or wired data transmission, or by retrieval, for example, from a database. By providing this information, in particular, the information required for the method regarding the initial quantum state and the initial gate parameters, the temperature value, etc., is made available. The initial gate parameters, which are chosen at the beginning of the method, can be selected randomly or after solving a simple problem. This can, for example, be either a system described by a simplified Hamiltonian or the high-temperature limit of the Hamiltonian.Algorithms and applications that utilize quantum mechanical resources can be easily and efficiently written in the language of quantum logic circuits. A quantum circuit is a computational routine constructed from coherent quantum operations. Each horizontal line or wire in a quantum circuit represents a qubit, with the left end of the wire representing the original quantum data and the right end representing the final quantum data generated by the quantum circuit's computation. Operations on qubits are represented by boxes placed on these wires. Quantum gates are the elementary operations that a quantum computer can perform on its qubits. They are comparable to electronic gates, which perform the elementary operations of a classical computer. However, a quantum gate operates on quantum mechanical systems such as spin.Quantum operations are mathematically realized by matrix multiplication with unitary matrices. Unitary matrices are always invertible, and thus the input values ​​of a circuit can be reconstructed from the output values. For quantum gates that operate on two qubits (2-qubit gates), an interaction between the physical qubits in question is required. For spin qubits, this can occur, among other things, through exchange interactions. Atoms in an ion trap, for example, can exchange photons. For qubits based on superconducting circuits, these qubits can be manipulated, for example, via the applied voltage, the magnetic field, or via coupling to microwave resonators.The VQT is a variational quantum algorithm that is executable or executed, in particular, on a hybrid computer platform comprising a classical computer and a quantum computer, and can be used to determine a thermal state at a given temperature and / or the spectrum of the associated quantum system, in other words, to determine the possible energy values ​​that can be measured on the quantum system under consideration at the given temperature.

[0025] • Providing a temperature value; The temperature value describes in particular a temperature in Kelvin, whereby a value greater than zero, for example 1000K or room temperature, is selected as the initial temperature value.

[0026] • Determining a thermal state of the simplified quantum system at the temperature value using the variational quantum thermalizer; This determination can in particular comprise preparing the thermal state of the simplified quantum system using the variational quantum thermalizer. The thermal state is preferably determined by optimizing the free energy of the thermal state. This involves iteratively adjusting the gate parameters of the quantum gates of the VQT until a first termination criterion is met. In particular, this method step comprises storing the gate parameters that result from a final iteration of the VQT, which leads to the first termination criterion being met, or the gate parameters that result from the optimization after the last iteration of the VQT.In particular, storage takes place in an internal or external storage unit so that the gate parameters, or at least some of these gate parameters, can be made available by the storage unit for later preparation of the thermal state or at least a subset of the eigenstates. Determining the thermal state here means, in particular, determining an approximation of the actual thermal state of the simplified quantum system. In other words, a state is determined or prepared that at least approximates the actual thermal state. The thermal state p. Gibbs (also called Gibbs state) at the given temperature T has in particular the following form: e~isse E i where the probabilities are: , with Z = ie~ lßEi and ß = — fc ß T , where k B the Boltzmann constants. The Et denote the

[0027] Eigenenergies (spectrum) and \ <pt) die zugehörigen Eigenzustände. Vorzugsweise überführt der auf einem Quantencomputer, insbesondere NISQ-Computer, auszuführende Teil des VQT durch Anwenden eines VQT- Quantenschaltkreise, dessen Gatter durch die Gatterparameter einstellbar sind, die Basiszustände in eine Superposition der Basiszustände, wobei diese so erzeugten Zustände (e)) depend on the gate parameters 9, where 6 summarizes the gate parameters as a vector. This is done in particular starting from an initial density matrix, which is given by a probability distribution, and a set of basis states {| bt )}, where p t is the probability of the state \bt), and where p tdepends on the choice of quantum gates and can be adjusted by adjusting the gate parameters. Using the states (e) ), for example, the spectrum of the simplified quantum system can be determined by determining the expectation value of the simplified Hamiltonian using these states. In particular, the spectrum for each iteration step of the VQT and / or the spectrum for the final iteration step, which leads to the first termination criterion being reached, can be stored in a memory unit.

[0028] In addition to the at least one quantum circuit comprising a sequence of parameterized, unitary quantum gates that can be adjusted by selecting the gate parameters, the VQT includes a cost function, which in this case is the free energy. This part of the VQT algorithm is preferably implemented on a classical computer of the hybrid computer platform. Preferably, the VQT uses free energy as the cost function. The free energy can be calculated from the expectation value of the Hamiltonian of the simplified quantum system and the entropy, which can be determined from the probability distribution of the p;, where the energy and entropy depend on the gate parameters of the quantum circuit of the VQT.By minimizing the free energy on the classical computer, an optimized set of gate parameters is obtained, which is then passed to the quantum computer, allowing the quantum circuits to be re-executed with the optimized gate parameters. These iterations are repeated until the first termination criterion is met. Optimization to adjust the gate parameters can be performed, for example, using a gradient-free optimizer or a gradient-based optimizer by evaluating the free energy gradient on the quantum hardware, for example, using the parameter shift rule or finite differences.

[0029] For example, the VQT described in "Extending the Variational Quantum Eigensolver to Finite Temperatures" (Selisko et al., August 2022; arXiv:2208.07621) can be used in this step of the method. In this method, the probability distribution is provided by an upstream quantum circuit for a second quantum circuit, and a measurement between the first and second circuits provides initial measurement results from which the entropy can be determined. Another exemplary variant of a VQT is described in "Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm" (Verdon et al., October 2019; arXiv:1910.02071), which can be used alternatively here. Other known VQTs can also be used here.

[0030] In particular, a hybrid computer platform comprising at least one classical computer and at least one quantum computer is used to execute the VQT, wherein the optimization is preferably carried out on the classical computer and the execution of the quantum circuit and the execution of the measurements are preferably carried out on the quantum computer.• Determining a thermal state of the quantum system by performing a time evolution of the thermal state of the simplified quantum system, whereby the time evolution starts in the simplified quantum system and ends by time-dependent, successive activation of the correction quantum system in the quantum system. The underlying principle of the present method is based on the adiabatic theorem of quantum mechanics, which states that a quantum mechanical system remains, to a good approximation, in an eigenstate if the Hamiltonian explicitly depends on time but changes only slowly. In the present case, the quantum system is preferably described by a sum of a simplified Hamiltonian H. o , which describes the simplified quantum system, and a correction term Hamiltonian which describes the correction quantum system, where the correction term Hamiltonian is provided with a time-dependent prefactor a(t), which models the time-dependent successive switching on of the correction term Hamiltonian: At the beginning of the time evolution (t = 0) a takes the value zero and at the end of the time evolution (t = t end ) a takes the value one:

[0031] In other words, the time evolution models the slow activation of the correction term, assuming that the thermal state of the simplified quantum system evolves towards the thermal state of the quantum system through the adiabatic activation of the correction term during the time evolution, and that the thermal state of the quantum system is ultimately present. In particular, a time evolution quantum circuit is provided to perform the time evolution and executed on the quantum computer. The time evolution quantum circuit can be obtained, for example, by Trotterization, where the time t end into n intervals of time dt. The time evolution is now obtained by performing a time evolution at each time point with the Hamiltonian operator H G(tt). Assuming that dt is small, this stepwise time evolution can be realized by sequentially executing the time evolution with respect to each term of the Hamiltonian. Preparing the thermal state of the simplified quantum system can preferably also be understood in such a way that, in particular, only a subset of the eigenstates of the simplified quantum system, and not the entire thermal state, is prepared for the time evolution. For example, eigenstates of higher energies can be neglected. This embodiment reduces the circuit depth of the quantum circuit and reduces the necessary number of measurements, thus reducing the required quantum resources. Advantageously, fewer qubits are required, which leads to fewer errors. Furthermore, fewer shots are necessary, resulting in a shorter runtime.

[0032] • Determining a spectrum of the quantum system using the time-evolved thermal state of the quantum system; In particular, the expectation value of the Hamiltonian H G of the quantum system is determined using the thermal state of the quantum system previously determined through time evolution. For this purpose, the thermal state of the quantum system determined through time evolution is provided and / or prepared, and measurement results are generated. In particular, this process step is carried out on the quantum computer.

[0033] In particular, providing the spectrum can involve measuring the spectrum on the quantum computer. This allows the spectrum to be determined, i.e., in particular, a mapping between the measured qubit state sequence of the first measurement of the VQT and the resulting energy measured in the second measurement (expectation value of the overall Hamiltonian) can be determined. The eigenstate with the lowest energy provides a good approximation to the ground state. This ground state can be prepared using a quantum computer. When using the eigenstate described in "Extending the Variational Quantum Eigensolver to Finite Temperatures" (Selisko et al., August 2022. arXiv:2208).For the VQT described in (ref. 07621), the preparation is carried out as follows: Replace the first circuit of the VQT and the first measurement with a circuit that prepares the qubit state sequence associated with the lowest energy, and then apply the second quantum circuit with the gate parameters determined from the final gate parameters using the described method. This means, in particular, that not all gate parameters are necessarily used to measure the spectrum (here, only the gate parameters of the second quantum circuit). In particular, at least a selection or all gate parameters are used.

[0034] To determine the ground state, the spectrum can alternatively or additionally be retrieved or transmitted to the classical computer, and the ground state can be determined based on this spectrum.

[0035] • Determining the ground state of the quantum system by selecting the state from the previously determined spectrum of the quantum system to which the lowest energy is assigned as the ground state of the quantum system. In particular, the ground state can be stored and / or output. Preferably, the quantum circuit with the gate parameters that led to reaching the ground state (i.e., in particular, the gate parameters of the last iteration that led to the second termination criterion being met), as well as the associated initial state that results in the ground state after application of the quantum circuit, are output and / or stored. Alternatively or additionally, the associated ground state energy can also be output and / or stored. The output can, for example, be transmitted to a hybrid computer platform, a classical computer, a quantum computer, a cloud, and / or a display device.Alternatively or additionally, the output can be used as input of another algorithm, in particular a material simulation algorithm in which the quantum system is investigated for certain material properties.

[0036] In particular, the method may comprise a step: decomposing the quantum system into a simplified quantum system and a correction term. Alternatively, the quantum system already divided into a simplified quantum system and a correction term is provided to the method.

[0037] In particular, the method can be executed on a hybrid computer platform comprising a classical computer and a quantum computer for executing the quantum circuit. The classical computer can also be understood as a network of several classical computers, whereby the tasks of the classical computer can be distributed among one or more classical computers within the framework of the method, for example, depending on resource requirements. Quantum computers programmed using quantum circuits can, in principle, be constructed from any quantum technology capable of implementing single- and multi-qubit gate operations. Architectures based on, for example, superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are currently being actively developed.A quantum computer comprises a qubit arrangement, wherein the qubit arrangement comprises a plurality of physical qubits, which can preferably be provided with devices or units adapted to the technology with which the qubits are realized for initializing (e.g., initializing the qubit in a base state), manipulating (e.g., applying 1-qubit and / or 2-qubit gates) and / or reading the physical qubits.

[0038] The aforementioned method advantageously allows for crossing phase boundaries. This allows the application of this algorithm to a wider class of use cases. Furthermore, the use of quantum computing resources can be advantageously reduced, for example, by using fewer time points between t = 0 and t = t endmust be used. It should be noted that a "normal," i.e., the familiar, adiabatic expansion only works if the system is gapped, i.e., if the crucial eigenstates do not cross while the time parameter a transitions from 0 to 1. The method described here also allows the ground state to be reached in the end for ungapped systems, i.e., systems where a phase transition from a = 0 to a = 1 occurs. VQT plays a crucial role here, since a VQT can freely find and reassign the energy states for any value of a.

[0039] According to one embodiment, the Hamiltonian used in performing the time evolution is a sum of the simplified Hamiltonian and the correction term weighted with a time-dependent parameter. The time-dependent parameter is chosen such that

[0040] • the Hamiltonian at the beginning of the time evolution corresponds to the simplified Hamiltonian, • the Hamiltonian at the end of the time evolution corresponds to the sum of the simplified Hamiltonian and the correction term and

[0041] • the Hamiltonian of the time evolution includes successively increasing portions of the correction term between the beginning and the end of the time evolution.

[0042] As described above, various prior art VQTs are suitable for determining the thermal state at a constant temperature. According to one embodiment, the quantum circuit of the variational quantum thermalizer comprises a first quantum circuit and a second quantum circuit. Determining the thermal state of the simplified quantum system at the temperature value using the variational quantum thermalizer in this embodiment comprises the following steps:

[0043] 1) Providing the VQT;

[0044] 2) Initializing a quantum computer, which comprises preparing an initial quantum state and providing gate-parameter-based control signals for controlling quantum gates of the quantum circuit depending on the gate parameters. The initial quantum state specifies, in particular, which states the qubits of the quantum computer are assigned at the beginning of the process.

[0045] 3) Executing the first quantum circuit on the quantum computer to generate a first superposition state, comprising a superposition of several quantum states, each of which is assigned an amplitude. In particular, the following density matrix results after execution of the first circuit:

[0046] Where \bt) denotes the basis states that are brought into a superposition by executing the first quantum circuit.

[0047] 4) Determining first measurement results of the first quantum circuit, in particular by means of a conventional computer, which are suitable for determining the entropy and in particular storing the first measurement results; in particular, a plurality of first measurements are carried out, wherein in each of these measurements the superposition state collapses into a basis state of the first measurement, wherein the amplitude is a measure of the probability of collapsing into this basis state; the first measurement results can then be determined from the results of these first measurements. A plurality of first measurements are required in order to determine the first measurement results, in particular the probability for each basis state of finding oneself in this basis state after the first measurement.The goal of this first part of the procedure is therefore to generate a classical probability distribution from which the entropy can be determined. After the first measurement, the following density matrix results:

[0048] In particular, the results of each first measurement can be transmitted to a classical computer and / or a storage unit and, in particular, the results can be saved. In particular, the result of the subsequent second measurement from step 6) will be saved for each first measurement. The transmission can be understood in particular to mean that the results of each of the first measurements are transmitted to the classical computer and / or to a storage unit and the first measurement results are determined on the classical computer. The transmission of the results of the first measurements from the quantum computer to the classical computer can take place by means of a communication unit, wirelessly or with a cable, both via the Internet and locally to the classical computer. Alternatively or additionally, the measurement results can be saved from the quantum computer to a buffer, in particular to aThe quantum computer and the classical computer can be transferred to an external storage device, such as a cloud, from which the data can be retrieved by the classical computer. For example, the quantum computer can be part of a cloud computing platform to which the classical computer has access, allowing it to retrieve the quantum computer's measurement results from there. These are just a few examples of how the measurement results can be transmitted from the quantum computer to the classical computer.) Executing the second quantum circuit, the input of which forms a collapsed basis state after a first measurement of the first quantum circuit, on the quantum computer to generate a second superposition state; 6) Determining second measurement results of the second quantum circuit by means of a classical computer, which are suitable for determining an energy of the quantum system, in particular an energy spectrum of the quantum system, and in particular storing the second measurement results; This results in:.

[0049] 7) Determine the free energy from the first and second measurement results on the classical computer;

[0050] 8) Optimizing the free energy with respect to the gate parameters of the first quantum circuit and the gate parameters of the second quantum circuit on the classical computer;

[0051] 9) Check whether a first termination criterion is met:

[0052] • If no: transmit the optimized gate parameters to the quantum computer and repeat steps 1 to 9. The transmission of the gate parameters, which are the result of the optimization, from the classical computer to the quantum computer can be done wirelessly or via cable, either over the internet or locally. Alternatively or additionally, the gate parameters can be transmitted to a cloud computing platform that manages access to the quantum computer, which then forwards them to the quantum computer for reinitialization.

[0053] • If yes: Providing the gate parameters of the last iteration before the first termination criterion is met and providing the thermal state at the temperature value for the time evolution. Providing the gate parameters of the last iteration before the first termination criterion is met can be understood as providing the gate parameters used in the last iteration in the quantum circuit and / or providing the optimized gate parameters determined after executing the quantum circuit of the last iteration before the first termination criterion is met.

[0054] According to one embodiment, fulfillment of the first termination criterion can be affirmed if the magnitude of a difference between the optimized free energies of the last two iterations is less than or equal to a first limit and / or if the norm of a free energy gradient is less than a second limit. The first termination criterion describes a convergence criterion of the VQT and consequently terminates the iterations of the VQT, whereby the thermal state determined in the last iteration is considered to be the thermal state at the temperature used in the VQT (which is kept constant across the iterations of the VQT to determine the thermal state). In this embodiment, the convergence criterion is based on the free energy gradient or improvement.In this embodiment, the first termination criterion is reached when the free energy does not change by more than a certain percentage, for example, 10%, over the last two iterations of the VQT at the given temperature. The VQT is then terminated, and the time evolution continues. An alternative or complementary convergence criterion (first termination criterion) is based on the convergence of the gradients. If | |VF|| < g. toi , where g toiis a small, positive limit, the VQT is terminated and the time evolution continues. The time evolution can continue if one or all of the aforementioned criteria are met. Checking whether the first termination criterion is met is preferably performed on a classical computer. One advantage of using the aforementioned energy-based criterion is that a gradient-free optimizer can be used. Such optimizers require fewer quantum resources because the gradient does not need to be computed. When using a gradient-based optimizer, it is always advantageous to also consider the gradient convergence criterion, since no additional computational effort is required (the gradient is computed anyway).

[0055] According to one embodiment, the quantum circuit of the variational quantum thermalizer comprises a sequence of parameterized unitary quantum gates, wherein at least one of the unitary quantum gates has one or more gate parameters that are adaptable during the optimization.

[0056] In particular, the method for determining the ground state described above and its embodiments enable a simple and reproducible preparation of the ground state as described here: • Determining the ground state of the quantum system using one of the methods described above;

[0057] • Preparation of the ground state o by a preparation quantum circuit that generates a qubit state sequence associated with the lowest energy of the spectrum of the quantum system, o by applying the second quantum circuit comprising the gate parameters of the last iteration before fulfilling the first termination criterion to the qubit state sequence, and o by applying the time evolution.

[0058] According to one embodiment, the ground state energy can be determined and provided on the quantum computer.

[0059] According to one embodiment, the quantum system whose ground state is to be determined by the method is a many-body system, which can be described, for example, by a Hubbard Hamiltonian. The Hubbard model is an approximate model of a solid. It describes the behavior of electrons in a lattice assumed to be rigid. The repulsive Coulomb forces are only considered for those electrons that are located at the same lattice site. The kinetic energy component of the electrons is modeled by an overlap integral derived from the tight-binding model. Some examples of quantum systems that can be described by a Hubbard Hamiltonian are strongly correlated fermion systems, transition metals, mobile electron systems (e.g., ferromagnetism, antiferromagnetism, ferrimagnetism), and TT electron systems in quantum chemistry.One advantage is that the process accelerates and, in some cases, even enables the development and investigation of new materials. Furthermore, the properties of these new materials can be better adapted to the respective application.

[0060] In addition to applications in quantum chemistry and materials simulations, variational quantum algorithms can also be used to

[0061] To solve optimization problems, such as the combinatorial job shop problem, by reformulating the solution of the problem as a solution of a corresponding Hamiltonian.

[0062] These advantages also apply to the use of the method to determine the ground state of the quantum system.

[0063] A hybrid computer platform comprising a classical computer and a quantum computer for executing a quantum circuit, which are adapted such that the steps of the method described above can be carried out and / or that the use of the method for determining the ground state of the quantum system can be carried out, has the advantage that it can be used particularly efficiently for material simulation, preferably in addition to the advantages that result directly from the advantages of the method.Adaptation can be understood in particular as meaning that, for example, the hardware of the quantum computer can be tuned to the quantum circuit so that the mapping of the logical qubits of the quantum circuit to the physical qubits of the quantum computer is possible and preferably requires the addition of as few as possible to a negligible number of additional SWAP operations, which ensure the interaction of the physical qubits occupied by the logical qubits when executing, for example, 2-gate operations. In particular, the hardware of the quantum computer can be selected based on the provided quantum circuit, thus enabling an even more efficient and less noise-susceptible execution of the method. The hybrid computer platform can be understood in particular as comprising at least one classical computer and at least one quantum computer.

[0064] Preferably, it includes a cloud computing platform to which the classical computer has access, so that it can retrieve, in particular, the measurement results of the quantum computer from there.

[0065] According to one embodiment, a NISQ computer is used as a quantum computer.

[0066] A computer-readable storage medium on which the adiabatic variational quantum algorithm, which is an implementation of the method described above, is stored can in particular be controlled by the quantum computer and / or the classical computer to provide the adiabatic variational quantum algorithm.

[0067] Short description of the drawings

[0068] Embodiments of the invention are illustrated in the drawings and explained in more detail in the following description. Identical reference numerals in the figures denote identical or equivalent elements.

[0069] It shows

[0070] Fig. 1 is a flowchart of a procedure for determining a ground state of a quantum system;

[0071] Fig. 2 is a flowchart of a process for determining a thermal state using a VQT on a hybrid computer platform in the event that a first termination criterion is not met;

[0072] Fig. 3 is a flowchart of a method for preparing the ground state; Fig. 4 is a schematic representation of a hybrid computing platform.

[0073] Embodiments of the invention

[0074] Fig. 1 shows a flowchart of a process 100 for determining a ground state 1000 of a quantum system. An adiabatic variational quantum algorithm 1012 is stored on a computer-readable storage medium 203, for example, on a memory or in a cloud, and can be retrieved therefrom. In other words, the computer-readable storage medium 203 in this embodiment provides the adiabatic variational quantum algorithm 1012 for executing the method 100 on a hybrid computer platform 200, in particular comprising a classical computer and a quantum computer. In this embodiment, the storage medium 203 provides, in particular, the following:

[0075] • a variational quantum thermalizer 1010, hereinafter referred to as VQT, comprising at least one quantum circuit 1011,

[0076] • initial gate parameters 1022' of the quantum circuit 1011 of the VQT • a temperature value 1001 which specifies the temperature of the thermal state determined by executing the VQT, and

[0077] • a Hamiltonian of the quantum system, which comprises a sum of a simplified Hamiltonian and a correction term Hamiltonian (also called correction term for short).

[0078] Alternatively or additionally, the aforementioned data may also be provided by user input or by another computer platform and transmitted, for example, by data transmission or wireless or wired data transmission or by retrieval, for example from a database.

[0079] In the method 100 for determining the ground state 1000 of the quantum system, the adiabatic variational quantum algorithm 1012 in Fig.

[0080] 1 is carried out by a hybrid computer platform 200 comprising at least one quantum computer and at least one classical computer. This comprises the following steps: A thermal state 1020 of the simplified quantum system at the temperature value 1001 is prepared 102 by executing 1021 the VQT 1010. Executing 1021 the VQT 1010 comprises initializing the quantum computer, wherein the initialization comprises preparing an initial quantum state and providing control signals based on the initial gate parameters 1022' for controlling the quantum gates of the quantum circuit 1011 of the VQT 1010 depending on the initial gate parameters 1022'. The VQT 1010 uses the free energy as the objective function and minimizes it by optimization, preferably on the classical computer, where the objective function depends on the gate parameters.The gate parameters 1022 are iteratively adjusted to minimize the free energy. In other words, after each execution of the quantum circuit 1011, new gate parameters 1022 are determined by minimizing the objective function, which are then used for the quantum gates in the subsequent iteration when executing the quantum circuit 1011. Furthermore, further data 1023, which depend, for example, on the choice of the first termination criterion, can be provided by the VQT 1010 for a check 1024 as to whether a first termination criterion is met. After each execution 1021 of the quantum circuit 1011, a check 1024 is made as to whether the first termination criterion is met. If it is not met, the next iteration proceeds using the gate parameters 1022 determined in the optimization step. The first termination criterion can comprise one or more criteria.Furthermore, it can be agreed that if one, several, or all criteria are met, the first termination criterion is considered met. The criteria can be based, for example, on a change in free energy. Possible criteria include:

[0081] • If the free energy, which is determined from measurement results of the quantum circuit 1011, does not change by more than a certain percentage, e.g. 10%, over the last two iterations, the VQT 1010 is terminated or one or more further criteria, which may be included in the first termination criterion, are checked 1024.

[0082] • Another convergence criterion is based on the convergence of the gradients. If | |VF|| < g toi , where g toi is a small, positive limit, the VQT 1010 is terminated or one or more further criteria, which may be included in the first termination criterion, are checked 1024.

[0083] If the first termination criterion is met, the VQT 1010 ends and the thermal state 1020 determined by the VQT 1010 is provided for the next method step 1025. In particular, this provision includes that the thermal state 1020 is prepared on the quantum computer or information for preparing the thermal state 1020 of the simplified quantum system is provided, which information enables a preparation of the thermal state 1020 of the simplified quantum system on the quantum computer. This information can, for example, be the initial

[0084] Quantum state, the gate parameters 1022, which were determined or used in the last iteration of the VQT before the first termination criterion was met, as well as the associated quantum circuit 1011.

[0085] In the next step, the thermal state 1020 of the simplified quantum system is time-evolved 1025. In particular, the time evolution 1025 is carried out on the quantum computer, for example, by Trotterization. A modified Hamiltonian is used as the Hamiltonian for the time evolution, which in this embodiment is a sum of the simplified Hamiltonian and the correction term Hamiltonian. The correction term Hamiltonian is provided with a time-dependent prefactor that models a successive, slow activation of the correction term Hamiltonian. In particular, the slow activation is modeled by the time-dependent prefactor assuming the value zero at the beginning of the time evolution 1025 and the value one at the end of the time evolution 1025.In other words, the Hamiltonian at the beginning of the time evolution 1025 (time t = 0) is equal to the simplified Hamiltonian, and at the end of the time evolution 1025 (time t = 0) it is equal to the Hamiltonian of the quantum system, i.e., the sum of the simplified Hamiltonian and the correction term Hamiltonian (i.e., the prefactor assumes the value 1 at the end of the time evolution 1025). By slowly switching on the correction term, the thermal state 1020 of the simplified quantum system is transformed over time into the thermal state 10202 of the quantum system, so that at the end of the time evolution, the thermal state 10202 of the quantum system is present. Using this state 10202, a spectrum 1030 of the quantum system is determined 103. In other words, the expectation value of the Hamiltonian of the quantum system is determined using the thermal state 10202 determined by the method described above.From the spectrum 1030, the state with the lowest energy value is selected, which is considered the ground state. This determines the ground state 1000 of the quantum system 104. The selection is preferably performed on a classical computer. In addition, the corresponding ground state energy can also be specified, which then corresponds to the lowest energy in the spectrum 1030.

[0086] Fig. 2 shows a flowchart of the process of the method 100 for determining the thermal state 1020 using the VQT 1010 on a hybrid computer platform 200, in the event that the first termination criterion is not met, according to one embodiment. In this embodiment, an exemplary quantum circuit 1011 of the VQT 1010 is shown. Alternatively, other VQTs may also be used.

[0087] The quantum computer 202 is initialized 301. Initialization 301 involves preparing an initial quantum state. For this purpose, the qubits qi, ... , q n of the quantum computer into the initial quantum state. For example, all qubits are brought into the state |0) or all qubits into the state |1). Alternatively, the qubits qi, ... , q ninto an initial qubit state sequence, i.e., a first number of qubits is brought into the state |0) and a second number of qubits is brought into the state |1). An example of a qubit state sequence of five qubits, in which all qubits are in the state |0) is: 00000. The initialization further comprises providing control signals based on the gate parameters 1022', 1022 for controlling the quantum gates depending on the gate parameters 1022', 1022. A first quantum circuit 10111 is applied to this initial qubit state sequence, which comprises a number of unitary quantum gates, which can be parameterized, i.e., the i-th quantum gate depends on at least one parameter (p tIf the quantum gates are designed as rotation gates, for example, which cause a single-qubit rotation by an angle around one of the X, Y, or Z axes of the Bloch sphere, the angle is equal to the gate parameter 1022', 1022. The control signals depend on the technology of the quantum computer used. In the case of qubits based on superconducting circuits, the manipulation of the qubits can be achieved, for example, via the applied voltage, the magnetic field, or via coupling to microwave resonators, so that the control signals are configured, for example, to adjust the magnetic field and / or to adjust the frequency of the microwave resonators. In particular, the control signals can comprise electrical signals.

[0088] By executing the first quantum circuit 10111 on the quantum computer 202, a first superposition state is generated, which comprises a superposition of several quantum states, each of which is assigned an amplitude. Subsequently, a first measurement 3021 is performed. In particular, each of the n qubits is measured 3021 (indicated in Fig. 2 by the two lines and the indication of 1, ..., n measurements in the measurement symbols of the first measurement 3021). As a result of the measurement 3021, the superposition state generated by the first quantum circuit 10111 collapses into one of the several possible measurement states (= basis states).In particular, the result of the first measurement 3021 comprises a qubit state sequence which is a sequence of 0 and 1, where 0 means that the corresponding qubit of the state sequence is in state |0) after the first measurement 3021 and where 1 means that the corresponding qubit of the state sequence is in state |1) after the first measurement 3021.

[0089] After the first measurement 3021, the qubit state sequence is used as input for the second quantum circuit 10112. The second quantum circuit 10112 comprises a number of unitary quantum gates, which can be parameterized, i.e., the i-th quantum gate depends on a parameter 0;. If the quantum gates are designed, for example, as rotation gates, which, for example, cause a one-qubit rotation by an angle about one of the axes X, Y, or Z of the Bloch sphere, the angle is equal to the gate parameter 1022', 1022. By applying the second quantum circuit 10112, a second superposition state is generated. After executing the second quantum circuit 10112, a second measurement 3022 is performed. In particular, each of the n qubits is measured 3022 (indicated in Fig. 2 by the two lines and the indication of 1 ,... ,n measurements in the measurement symbols of the second measurement 3022).Here the expectation value of the Hamiltonian of the quantum system is determined.

[0090] By repeatedly executing the first measurement 3021 and the second measurement, as described above, 1000 times in succession, first measurement results 3021' and second measurement results 3022' can be determined. For example, the results of the repeatedly executed first and second measurements 3021, 3022 are each stored in a storage unit, and the stored results are retrieved from the storage unit by the conventional computer 201 to determine the first measurement results 3021' and second measurement results 3022'. The first measurement results 302T comprise, in particular, a probability distribution which allows a statement to be made about the probability of being in the respective measurement state after the first measurement 3021. The first measurement results 3021' are determined from the repeatedly executed first measurements 3021 preferably on a conventional computer 201, particularly preferably in a cloud.For example, the results of the multiple first measurements 3021 are each stored in a storage unit, and the stored results are retrieved from the storage unit by the classical computer 201 to determine the first measurement results 302T. The second measurement results 3022' enable the determination of an energy of the simplified quantum system, in particular an energy spectrum 1030 of the simplified quantum system. From the first measurement results 302T and the second measurement results 3022', the entropy S 3021" and the energy E 3022" of the simplified quantum system can first be calculated, or the free energy 3020 can be calculated directly from the first measurement results 302T and the second measurement results 3022' according to the following equation:.

[0091] F = E - TS where T is the temperature 1001 provided to the VQT in the method 100 disclosed by way of example in the embodiment shown in Fig. 1.

[0092] This free energy is used as the objective function of an optimizer 302. The optimization 302 for adjusting the gate parameters 1022', 1022 can be performed, for example, using a gradient-free optimizer or a gradient-based optimizer by evaluating the gradient of the free energy on the quantum hardware, for example, using the parameter shift rule. In this embodiment, the optimized gate parameters 1022 include both the gate parameters of the first quantum circuit 10111 (p t (simplified, the (p t be written as a vector < >, whose components each have the (p tare) and the gate parameters of the second quantum circuit 10112 are 0; (for simplification, the 0i can be written as a vector 0, whose components are each the 0;). Subsequently, a check 1024 is performed to determine whether the first termination criterion is met. In this exemplary embodiment, the result of the check is no; therefore, the optimized gate parameters 1022 are transmitted to the quantum computer 202, and the previously described steps are repeated, with the quantum circuits 10111, 10112 using the optimized gate parameters 1022.

[0093] If the first termination criterion is met, the VQT 1010 ends, and the thermal state 1020 determined by the VQT 1010 is provided for the next method step (see Fig. 1, time evolution 1025). In particular, this provision includes preparing the thermal state 1020 on the quantum computer or providing information for preparing the thermal state 1020 of the simplified quantum system, which information enables preparation of the thermal state 1020 of the simplified quantum system on the quantum computer. In the embodiment shown in Fig. 2, this information can, for example, include the following:

[0094] • the initial quantum state, • the gate parameters 1022 of the second quantum circuit 10112, which were determined or used in the last iteration of the VQT before the first termination criterion was met,

[0095] • the qubit state sequence which is present after the first measurement and which is associated with the lowest energy of the spectrum, and preferably a preparation quantum circuit which generates this qubit state sequence.

[0096] After the time evolution 1025, the spectrum 1030 of the Hamiltonian of the quantum system is determined based on the time-evolved thermal state 10202. This results in a mapping between the measured qubit state sequence in the first measurement of the VQT and the resulting energy measured in the second measurement. The eigenstate with the lowest energy provides a good approximation to the ground state. This ground state 1000 can preferably be prepared by replacing the first circuit 10111 of the VQT 1010 from Fig. 2 and the first measurement 3021 with a circuit (preparation quantum circuit) that prepares the qubit state sequence associated with the lowest energy, and then applying the second VQT circuit 10112 and the time evolutions.

[0097] Fig. 3 shows a flowchart of a method 400 for preparing the ground state 1000 on a quantum computer 202. In particular, the ground state 1000 of the quantum system can be prepared as the initial state of the quantum computer 202. The method 400 comprises the following steps:

[0098] • Determining the ground state 1000 of the quantum system, for example by means of the method 100 shown in Fig. 1 and the VQT shown in Fig. 2;

[0099] • Preparation 401 of the ground state 1000, wherein the preparation 401 comprises in particular the following steps: o Applying a preparation quantum circuit 402 that generates a qubit state sequence 4002 associated with the lowest energy of the spectrum; o Applying the second quantum circuit 10112 to this state, wherein the second quantum circuit 10112 comprises the optimized gate parameters 1022 of the final iteration to the qubit state sequence 4002, and o Applying the time evolution 1025.

[0100] Fig. 4 is a schematic representation of a hybrid computer platform 200 comprising the classical computer 201 and the quantum computer 202, which are adapted to execute the method 100, as described by way of example in Fig. 1 and Fig. 2, to determine the ground state 1000 of the quantum system. The classical computer 201 determines the gate parameters 1022 during optimization 302 as part of executing the VQT 1010. In particular, the classical computer 201 provides the gate parameters 1022 for the quantum computer 202. The temperature 1001 is used by the classical computer to determine the free energy during optimization 302.The quantum computer uses the provided gate parameters 1022 and performs the first measurements 3021 and the second measurements 3022, thus providing the classical computer 201 with the data for determining the first measurement results 302T and the second measurement results 3022', which the classical computer uses together with the temperature 1001 to perform the optimization 302. The time evolution 1025 for determining the thermal state 10202 of the quantum system and the determination 103 of the spectrum 1030 are preferably also performed on the quantum computer 202.

Claims

Claims 1. A method (100) for determining a ground state (1000) of a quantum system, wherein the quantum system comprises a simplified quantum system and a correction quantum system, and wherein the method comprises the following steps: • Providing o a variational quantum thermalizer (1010) comprising at least one quantum circuit (1011), and o a temperature value (1001); • Determining (102) a thermal state (1020) of the simplified quantum system at the temperature value (1001) by means of the variational quantum thermalizer (1010); • Determining a thermal state (10202) of the quantum system by carrying out a time evolution (1025) of the thermal state (1020) of the simplified quantum system, wherein the time evolution (1025) starts in the simplified quantum system and ends by time-dependent successive switching on of the correction quantum system in the quantum system; • Determining (103) a spectrum (1030) of the quantum system using the time-evolved thermal state (10202) of the quantum system, • Determining (104) the ground state (1000) of the quantum system by selecting the state from the spectrum (1030) of the quantum system to which the lowest energy is assigned as the ground state (1000) of the quantum system.

2. Method (100) according to claim 1, wherein • the simplified quantum system is described by a simplified Hamiltonian, • the correction quantum system is described by a correction term, and • the quantum system is described by a sum of the simplified Hamiltonian and the correction term.

3. The method (100) of claim 2, wherein a Hamiltonian used in performing the time evolution (1025) comprises a sum of the simplified Hamiltonian and the correction term weighted with a time-dependent parameter, the time-dependent parameter being such that • the Hamiltonian at the beginning of the time evolution (1025) corresponds to the simplified Hamiltonian, • the Hamiltonian at the end of the time evolution (1025) corresponds to the sum of the simplified Hamiltonian and the correction term and • the Hamiltonian of the time evolution (1025) includes successively increasing portions of the correction term between the beginning and the end of the time evolution (1025).

4. The method (100) according to any one of the preceding claims, wherein the quantum circuit (1011) of the variational quantum thermalizer (1010) comprises a first quantum circuit (10111) and a second quantum circuit (10112), and wherein determining (102) the thermal state (1020) of the simplified quantum system at the temperature value (1001) by means of the variational quantum thermalizer (1010) comprises the following steps: 1) Initializing (301) a quantum computer (202), which comprises preparing an initial quantum state and providing control signals based on gate parameters (1022', 1022) for controlling quantum gates of the quantum circuit (1011) as a function of the gate parameters (1022', 1022); 2) executing the first quantum circuit (10111) on the quantum computer (202) to generate a first superposition state comprising a superposition of a plurality of quantum states, each of which is associated with an amplitude; 3) determining first measurement results (302T) of the first quantum circuit (10111) which are suitable for determining the entropy (3021") and in particular storing the first measurement results (302T); 4) executing the second quantum circuit (10112), the input of which forms a collapsed basis state after a first measurement of the first quantum circuit (10111), on the quantum computer (202) to generate a second superposition state; 5) determining (3022) second measurement results (3022') of the second quantum circuit (10112) which are suitable for determining an energy (3021") of the quantum system, and in particular storing the second measurement results; 6) Determining the free energy (3020) from the first and second measurement results (3021", 3022") on the classical computer (201); 7) optimizing the free energy (3020) with respect to the gate parameters of the first quantum circuit (10111) and the gate parameters of the second quantum circuit (10112) on the classical computer (201); 8) Check (1024) whether a first termination criterion is met: • If no: transmit the optimized gate parameters (1022) to the quantum computer (202) and repeat steps 1) to 8) • If yes: Provide the gate parameters of the last iteration before fulfilling the first termination criterion and provide the thermal state at the temperature value for the time evolution (1025).

5. The method (100) according to any one of the preceding claims, wherein the quantum circuit (1011) of the variational quantum thermalizer (1010) comprises a sequence of parameterized unitary quantum gates, wherein at least one of the unitary quantum gates has one or more gate parameters (1022', 1022).

6. Method (400) for preparing a ground state (1000) of a quantum system by means of a quantum computer (202), comprising the following steps: • Determining the ground state (1000) of the quantum system using a method (100) according to one of the preceding claims, in particular claim 4; • Preparation (401) of the ground state (1000) o by a preparation quantum circuit (402) which generates a qubit state sequence (4002) which is assigned to the lowest energy of the spectrum (1030) of the quantum system, o by applying the second quantum circuit (10112) comprising the gate parameters (1022) of the last iteration before fulfilling the first termination criterion to the qubit state sequence (4002) and o by applying the time evolution (1025).

7. The method (100, 400) according to any one of the preceding claims, wherein the quantum system is described by a many-body Hamiltonian of a material.

8. The method (100, 400) according to any one of the preceding claims, wherein the Hamiltonian operator of the quantum system is a Hamiltonian operator for the Hubbard model.

9. Use of the method (100, 400) according to one of the preceding claims for a material simulation, wherein a Hamiltonian of the quantum system is a many-body Hamiltonian which describes the material to be simulated.

10. A hybrid computer platform (200) comprising a classical computer (201) and a quantum computer (202) for executing a quantum circuit (1011), which are adapted such that the steps of the method (100, 400) according to any one of claims 1 to 8 can be carried out and / or the use of the method (100, 400) according to claim 9 can be carried out.

11. The hybrid computer platform (200) of claim 10, wherein the quantum computer (202) is a NISQ computer.

12. Adiabatic variational quantum algorithm (1012) which causes a hybrid computer platform (200) comprising a classical computer (201) and a quantum computer (202) for executing a quantum circuit, in particular a hybrid computer platform (200) according to one of claims 10 or 11, to carry out the method (100, 400) according to one of claims 1 to 8.

13. A computer-readable storage medium (203) on which the adiabatic variational quantum algorithm (1012) according to claim 12 is stored.

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