Determining the ground state of a quantum system

Abstract

The invention relates to a method (100) for determining a ground state (1001) of a quantum system, comprising the steps: providing (101) a hybrid variational quantum algorithm and executing (102) the hybrid variational quantum algorithm (1010) on a hybrid computer platform, comprising a non-quantum computer and a quantum computer for executing the quantum circuit (1011), the execution of the hybrid variational quantum algorithm (1010) comprising the following: determining (1025) the gate parameters (1020) by solving the differential equation, comprising the equation matrix and the equation vector, on the non-quantum computer, and determining (1026) the imaginary development over time of the gate parameters (1020), determined by solving the differential equation, for a time step by applying the Runge-Kutta method. The invention also relates to a hybrid computer platform, to a hybrid variational quantum algorithm, and to a use of the method for material simulation.

Description

[0001]R.407999 - 1 - Description Title Method for determining a ground state of a quantum system, hybrid computer platform, hybrid variational quantum algorithm State of the art In "Variational approach-based quantum simulation of imaginary time evolution" (McArdle et al., npj Quantum Information (2019) 5:75; https: / / doi.org / 10.1038 / s41534-019-0187-2) a method for determining a ground state of a many-body system is described which uses a hybrid system of quantum computer and classical computer. Core and advantages of the invention Finding the ground state of a quantum mechanical system is an important task in the context of atomistic materials simulations and in the field of quantum chemistry. In the past, many algorithms have been developed that use classical computers to tackle this problem.However, the properties of materials cannot be calculated with sufficient accuracy on conventional high-performance computers. On a quantum computer, however, fundamentally new methods can be used that allow for highly accurate simulation. Such simulations can support the development processes of new materials. Currently, noisy intermediate-scale quantum computers (NISQs) without error correction and with a moderate number of qubits (~100) are available. Nevertheless, even these NISQ computers can achieve a significant advantage in terms of speed and accuracy in materials simulation and quantum chemistry compared to conventional high-performance computers. Currently available NISQ quantum 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. Therefore, hybrid quantum-classical algorithms such as the Variational Quantum Eigensolver (VQE) or Variational Quantum Imaginary Time Evolution (VarQITE) are viable approaches to employing NISQ computers. However, these methods also suffer from divergence or slow convergence due to the inherent noise of the NISQ results. The invention relates to a method for determining a ground state of a quantum system, a use of the method for determining a ground state of a quantum system for a materials simulation, a hybrid computing platform, a hybrid variational quantum algorithm, and a computer-readable storage medium.Hybrid methods use 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 capacity of these low-qubit, noise-sensitive quantum computers.Given the resource limitations, low reliability, and high variability of physical properties such as coherence time or error rates of NISQ computers, a method for determining a ground state of a quantum system is proposed here. This method allows the imaginary time evolution to be mapped onto the available qubits in such a way that the probability of successful calculations R.407999 - 3 - is increased or maximized compared to known methods. In particular, the disclosed method allows the imaginary time evolution to be simulated on a quantum computer using a hybrid quantum-classical variational algorithm. In particular, the method is feasible using currently available NISQ computers because it requires only a quantum circuit with a small depth, i.e., a small number of quantum operations.The invention is disclosed below with the features of the independent patent claims, which advantageously makes it possible to make a hybrid variational quantum algorithm, in particular a VarQITE algorithm, (a) more efficient in terms of the number of time steps and (b) improve the numerical stability of the algorithm on current NISQ computers. This is achieved in particular by adapting the algorithm to the hybrid computer platform comprising a classical computer and a quantum computer in such a way that it exploits the strengths of the quantum computer and the classical computer while simultaneously taking into account the aforementioned shortcomings of the quantum computer.An 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, which comprises the following steps: A hybrid variational quantum algorithm is provided, comprising: a Hamiltonian operator, in particular, where: ^ =. where and ℎ ^represent the real coefficients and the observables, respectively; 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, includes the eigenvalues ​​of the Hamiltonian, i.e., the energy eigenvalues. This is the set R.407999 - 4 - 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 with a quantum algorithm or a hybrid quantum-classical algorithm. oan approach comprising a sequence of parameterized unitary quantum gates, ^ where each unitary quantum gate has a gate parameter, ^ where a test state approximating a time-evolved quantum state at a time ^, where ^ describes an imaginary time, is obtained by applying the approach to an initial quantum state; The mathematical concept of imaginary time evolution can be used to find the ground state of a many-body problem. The imaginary time evolution operator of a many-body system described by a Hamiltonian ^ is given by the non-unitary operator ^. ^^, where ^ denotes imaginary time. The imaginary time evolution is therefore described by a non-unitary operator, so that a direct mapping of the imaginary time evolution to a quantum circuit is not possible. Therefore, the actual imaginary-time-evolved quantum state | ^ (^) ^ described by a test state ^^ (^(^))^, where ^ = (^ ^ (^), ^ ^ (^), … , ^ ^ (^)), the gate parameters ^ ^ (^) of the unitary quantum gates as a vector. The test state is constructed in such a way that it can be calculated by applying the parameterized unitary quantum gates to the initial quantum state | 0 ^ results in: |^ (^(^))^ = |^ (^)^ = ^ ^ (^ ^ )^ ^^^ (^ ^^^ ) … ^ ^ (^ ^)|0^. a differential equation for the gate parameters, which results from the application of the McLachlan variational method to the test state, and which comprises an equation matrix comprising at least one matrix element and an equation vector comprising at least one vector element; R.407999 - 5 - This differential equation can be determined in particular from the equation of the McLachlan variational principle for the imaginary time evolution. By projecting the equation onto a subspace of the Hilbert space, which can be achieved with the aforementioned approach - in other words, by replacing the actual imaginary time-evolved quantum state | ^ ( ^ )^ through the test state | ^ (^) ^ - the following equation results: ^ ^ = ^^ ( ^ ) |^ | ^ ( ^ )^ Now the derivatives can be carried out and the equation can be solved for ^̇ ^be sorted, whereby by exploiting the fact that the test state is normalized, i.e. ^ ( ^ )^ = 1, some dates become zero, resulting in the following differential equation: with: and The aim is to determine the parameters and ^ ^ efficiently determined on the quantum computer. ^ ^^ describe the matrix elements of the equation matrix ^ with ^ rows and ^ columns, ^ ^ describes the vector elements of the matrix vector ^ with ^ components and ^̇ ^Describes the vector element of the vector of gate parameters derived with respect to ^, which comprises ^ components (= number of gate parameters). oa quantum circuit for determining the equation matrix and the equation vector; algorithms and applications that use quantum mechanical resources can be easily and efficiently written in the language of logical quantum circuits. A quantum circuit is a computational routine constructed from coherent quantum operations. Each R.407999 - 6 - 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 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 with 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 ​​can be reconstructed from the output values ​​of a circuit. For quantum gates that operate on two qubits (2-qubit gates), an interaction between the physical qubits in question is required. With spin qubits, this can occur, among other things, through exchange interactions. Atoms in an ion trap, for example, can exchange photons. With 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.Furthermore, the method according to claim 1 comprises executing the hybrid variational quantum algorithm 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, wherein the tasks of the classical computer can be distributed among one or more classical computers within the scope 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, for example, on 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 R.407999 - 7 - 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), for manipulating (e.g., applying 1-qubit and / or 2-qubit gates), and / or for reading the physical qubits. Executing the hybrid variational quantum algorithm comprises the following steps: 1) Providing the quantum circuit for the quantum computer, comprising the initial quantum state and initial gate parameters; Providing the quantum circuit can be done 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, which together define the test state at the start time, is made available. 2) Initializing the quantum computer, comprising preparing the initial quantum state, which preferably has a non-vanishing overlap with the ground state of the system, and providing control signals based on the gate parameters for controlling the quantum gates depending on the gate parameters when executing the quantum circuit on the quantum computer. If the quantum gates are designed, for example, as rotation gates, which, for example, cause a single-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. The control signals depend on the technology of the quantum computer used.For qubits based on superconducting circuits, the qubits can be manipulated, 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 the frequency of the microwave resonators. In particular, the control signals can comprise electrical signals. 3) Execution of the quantum circuit on the quantum computer; In other words, the gates provided in the quantum circuit are executed on the qubits assigned to the initial quantum state, and measurements are performed at the end of each run, so that the execution of the quantum circuit serves to generate measurement results, which are used to determine the equation matrix and the equation vector. Measuring the expected values ​​of observables is an essential component of variational quantum algorithms.This requires a large number of measurements for statistical convergence to meet precision requirements, such as chemical accuracy in applications to quantum chemistry calculations. In other words, running the quantum circuit on the quantum computer involves running the quantum circuit multiple times. To give an order of magnitude for the number of measurements: it is usually true that a measurement with precision ^ requires a number of measurements of order 1 / ^. ^required. 4) Transmitting measurement results from the quantum computer to the classical computer to determine the at least one matrix element of the equation matrix and the at least one vector element of the equation vector of the differential equation; in particular, the measurement results represent the individual matrix elements of the equation matrix and the vector elements of the equation vector. In particular, the classical computer determines an expected value of the measurement results of the quantum computer for each matrix element and each vector element, where the expected value can be, for example, a mean value, an error-reduced mean value, or an error-suppressed mean value. These expected values ​​are then used to solve the differential equation as an equation matrix and a vector matrix. 5) Determining the gate parameters ^ by solving the following differential equation: comprising the equation matrix and the equation vector, on the classical computer; In particular, the differential equation is solved by numerically solving the differential equation by discretizing the time variable ^ in time steps of length ^^. In particular, the solution describes the gate parameters at the time ^ of the iteration, which results from the addition of the already iterated time steps ^^. R.407999 - 9 - 6) Determining the imaginary time evolution of the gate parameters ^ determined by solving the differential equation for a time step by applying the Runge-Kutta method; In particular, the time evolution of the gate parameters ^ for a time step ^^ is approximated using the Runge-Kutta method. In particular, the second-order Runge-Kutta method can be used for differential equations of the form ^ ^ = ^ ( ^, ^(^) ) , as in the following case, where ^ dem ^ and ^ ^^^(^) of the right side, so ^ ( ^, ^(^) ) corresponds to, can be used, where the time evolution for a time step ^^ can be determined as follows: 1 ^(^ + ^^) = ^(^) + ^^ (^ ^ + ^ ^ ) 2 with ^ ^ = ^ ( ^, ^(^) ) and ^ ^ = ^ ( ^ + ^^, ^ ( ^ ) + ^^^ ^ ) Alternatively, the fourth-order Runge-Kutta method can be used for a differential equation of the form ^ ^ = ^(^, ^(^)), as in the following case, where ^ is the ^ and ^ ^^ ^(^) corresponds to the right-hand side, i.e. ^(^, ^(^)), where the time evolution for one time step ^^ can be determined as follows: with ^ ^ = ^ ( ^ + ^^, ^ ( ^ ) + ^^^ ^ )Compared to the Euler method, which requires only one function evaluation, the second-order Runge-Kutta method requires twice as many function evaluations, and the fourth-order Runge-Kutta method requires four times as many. However, it turns out that using one of the two Runge-Kutta methods mentioned above can reduce the noise susceptibility of the results compared to using the Euler method, and thus the accuracy and reliability can be increased by the method proposed here. Furthermore, the convergence of the method is improved compared to using the Euler method, as will be shown in the following examples. In other words, the number of iteration steps is lower, while maintaining a similar accuracy to the Euler method, i.e.The time steps can be chosen larger than with the Euler method, and still achieve similarly accurate or even more accurate results. A further advantage is that this allows the number of control cycles of the quantum computer to be reduced. This is advantageous because, as described above, the quantum computer must perform a large number of measurements in each iteration step in order to determine the equation matrix and the equation vector with high accuracy. 7) Transmit the imaginary time-evolved gate parameters ^(^ + ^^) to the quantum computer and repeat steps 2) to 6) up to a final time step. After step 5) of the final time step, the ground state is determined on the classical computer or the quantum computer using the gate parameters determined in step 5) of the final time step by solving the differential equation.In particular, the present method is based on the idea that every initial quantum state that exhibits a non-vanishing overlap with the ground state corresponds, through imaginary time evolution in the limit ^ → ∞, to the ground state of the quantum system, which is described by the Hamiltonian underlying the method. In particular, the gate parameters ^ can then be prepared on the quantum computer to determine the ground state. Alternatively, however, the quantum computer can also be simulated on a classical computer for this purpose. I believe that's correct. The ground-state wavefunction is encoded in the converged (classical parameter) thetas. An observable of the ground state (such as the ground-state energy) would then preferably be calculated on the quantum computer. But it could also be simulated on a classical computer.The measurement results can be transmitted from the quantum computer to the classical computer using a communication unit, either wirelessly or via cable, both over the internet and locally to the classical computer. Alternatively or additionally, the measurement results can be transferred from the quantum computer to a buffer, in particular to a storage device external to the quantum computer and the classical computer, 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 measurement results from the quantum computer. These are just a few examples of how the measurement results can be transmitted from the quantum computer to the classical computer.The transmission of the imaginary-time-evolved gate parameters from the classical computer to the quantum computer using the Runge-Kutta method can be done wirelessly or via cable, either over the internet or locally. Alternatively or additionally, the imaginary-time-evolved 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. According to one embodiment, the ground-state energy can be determined and provided on the quantum computer. According to one embodiment, the quantum system whose ground state is to be determined using 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 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 π-electron systems in quantum chemistry. One advantage is that the method accelerates the development and investigation of new materials R.407999 - 12 - and in some cases even makes them possible. Furthermore, the properties of these new materials can be better adapted to the respective application.In addition to their application in quantum chemistry and materials simulations, variational quantum algorithms can also be used to solve optimization problems, such as the combinatorial job-shop problem, by reformulating the solution to the problem as the solution of a corresponding Hamiltonian. These advantages also apply to the use of the method to determine the ground state of the quantum system. A hybrid computer platform comprising a classical computer and a quantum computer for executing a quantum circuit, adapted to perform the steps of the method described above, has the advantage that, in addition to the advantages directly resulting from the advantages of the method, it can be used particularly efficiently for materials simulation.Adaptation can be understood in particular as meaning that, for example, the hardware of the quantum computer can be tuned to the quantum circuit, allowing the mapping of the logical qubits of the quantum circuit to the physical qubits of the quantum computer. This can preferably be achieved by adding as few as possible, or even negligible, additional SWAP operations to ensure the interaction of the physical qubits occupied by the logical qubits when executing, for example, two-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. Alternatively or additionally, the quantum circuit or the choice of approach can be adapted to the hardware of the quantum computer.According to one embodiment, an NISQ computer is used as a quantum computer. This is possible despite the susceptibility to noise of NISQ computers, since the method is more robust against noise through the use of the Runge-Kutta method. A computer-readable storage medium on which the hybrid variational quantum algorithm is stored can be controlled, in particular, by the quantum computer and / or the classical computer to provide the hybrid variational quantum algorithm. Brief Description of the Drawings Exemplary embodiments of the invention are illustrated in the drawings and are explained in more detail in the following description. Identical reference numerals in the figures denote identical or equivalent elements. They show: Fig. 1 shows a flowchart of a method for determining a ground state of a quantum system Fig.2 shows a diagram illustrating a comparison of the lattice parameters using a simplified example system for the differential equation as a function of the time steps according to the second and fourth order Runge-Kutta method, as well as the Euler method. FIG. 3 shows a diagram illustrating a comparison of the lattice parameters using a simplified example system for the differential equation as a function of the time steps according to the second and fourth order Runge-Kutta method, as well as the Euler method, with noise. FIG. 4 shows a schematic representation of a hybrid computer platform. Embodiments of the invention. FIG. 1 shows a flowchart of a procedure for a method 100 for determining a ground state 1001 of a quantum system. A hybrid variational quantum algorithm 1010 is stored on a computer-readable storage medium 203, for example on a memory, in a cloud, etc. R.407999 - 14 - and can be retrieved from it.In other words, in this embodiment, the computer-readable storage medium 203 provides the hybrid variational quantum algorithm 1010 for executing the method 100. The hybrid variational quantum algorithm 1010 comprises: o a Hamiltonian operator; o an approach comprising a sequence of parameterized unitary quantum gates, ^where each unitary quantum gate has a gate parameter, ^where a test state for approximating a time-evolved quantum state at a time ^, where ^ describes an imaginary time, results from applying the approach to an initial quantum state.oa differential equation for the gate parameters, which results from the application of the McLachlan variational method to the test state and which comprises an equation matrix comprising at least one matrix element and an equation vector comprising at least one vector element; oa quantum circuit 1011 for determining the equation matrix and the equation vector. An exemplary quantum circuit for measuring the matrix element ^. ^,^for determining the ground state of hydrogen is shown in Fig. S3 of "Supplementary Information: Variational approach-based quantum simulation of imaginary time evolution" (McArdle et al. npj Quantum Information (2019) 5:75; https: / / doi.org / 10.1038 / s41534-019-0187-2). In particular, providing 101 the hybrid variational quantum algorithm also includes providing the quantum circuit 1011. The method 100 further includes executing 102 the hybrid variational quantum algorithm 1010 on a hybrid computer platform comprising a classical computer 201 and a quantum computer 202 for executing the quantum circuit 1011.The execution of the hybrid variational quantum algorithm 1010 is distributed between the quantum computer 202 and the classical computer, with particular consideration given to the strengths and limitations of the respective computer when distributing the steps in order to enable an efficient, resource-saving, yet reliable execution of the method 100. The following steps are executed on the quantum computer 202: 1) Providing 1021 the quantum circuit for the quantum computer 202, comprising the initial quantum state and initial gate parameters; R.407999 - 15 - 2) Initializing 1022 the quantum computer 202, comprising preparing the initial quantum state, and providing control signals based on the gate parameters for controlling the quantum gates depending on the gate parameters when executing the quantum circuit 1011 on the quantum computer 202; 3) Executing 1023 the quantum circuit 1010; In particular, these three steps are executed multiple times to increase the accuracy of the measurement results. All measurement results are transmitted 1024 to the classical computer 201. Preferably, the classical computer 201 forms expected values ​​of each matrix element and each vector element, which are then subsequently used when solving the differential equation.The following steps are performed on the classical computer 201: 4) Determining 1025 the gate parameters 1020 by solving the differential equation, comprising the equation matrix comprising the previously determined matrix elements and the equation vector comprising the previously determined vector elements; 5) Determining 1026 the imaginary time evolution of the gate parameters 1020 determined by solving the differential equation for a time step by applying the Runge-Kutta method. In particular, the second- or fourth-order Runge-Kutta method can be used for this purpose.Subsequently, the imaginary time-evolved gate parameters 1021 are transmitted 1027 to the quantum computer 202, and steps 2) to 5) are carried out again up to a final time step, wherein after step 4) of the final time step, the ground state on the classical computer or the quantum computer is determined 103 using the gate parameters 1020 determined in step 5) of the final time step. In particular, the gate parameters 1020 determined in step 4) of the final time step can be transmitted to the quantum computer 202, and a measurement of an observable can be carried out there using the test state, which is determined as follows: applying the quantum gates of the approach, comprising the gate parameters 1020 of the final time step, which were determined in step 4) of the final time step, to the initial state, and measuring an observable using this test state, which corresponds to R.407999 - 16 - corresponds to the ground state 1001 of the quantum system defined by the provided Hamiltonian. For example, the ground-state energy (expectation value of the Hamiltonian) can be determined. Fig. 2 shows a diagram with the x-axis 300' representing imaginary time ^ and the y-axis 300'' representing the solution to the differential equation ^ determined using different methods. ^ = sin(^) ^∙ ^ with the initial condition ^(^ = 0) = 2. The differential equation describes a greatly simplified system which has comparable properties to the differential equation of the method 100 explained in Fig.1 and which is used below to illustrate the quality of the solution of the previously proposed method 100. The diagram shows, on the one hand, the exact solution 301 ^(^) as a solid curve. Furthermore, the solution 304, which uses the Euler method for the imaginary time evolution in step 5) in the method 100 shown in Fig.1 for a time step ^^ (hereinafter referred to as Euler solution 304), is shown, whereby for each time step the corresponding determined function value is shown as a cross and neighboring crosses are connected to one another in pairs by straight lines. The size of the time steps is chosen here to be ^^ = 0.2.Furthermore, the solution 303, which uses the second-order Runge-Kutta method for the imaginary time evolution in step 5) in the method 100 shown in Fig. 1 for a time step ^^ (hereinafter referred to as the RK2 solution 303), is entered, wherein for each time step the corresponding determined function value is entered as a square and neighboring squares are connected to each other in pairs by straight lines. The size of the time steps is chosen here to be ^^ = 0.4, i.e. twice as large as in the Euler solution 304. In addition, the solution 302, which uses the fourth-order Runge-Kutta method for the imaginary time evolution in step 5) in the method 100 shown in Fig. 1 for a time step ^^ (hereinafter referred to as the RK4 solution 302), is entered, wherein for each time step the corresponding determined function value is entered as a star and neighboring stars are connected to each other in pairs by straight lines.The size of the time steps is chosen here to be ^^ = 0.8, which is twice as large as the RK2 solution 303 and four times as large as the Euler solution 304. In other words, the total time R.407999 - 17 - ^. ^^^ in the case of the RK2 solution 303, it is divided into half as many time steps and in the case of the RK4 solution 302, even into a quarter of the time steps compared to the Euler solution 304. Nevertheless, the RK2 solution and the RK4 solution represent the exact solution 301 better, especially when approaching ^ ^^^ . This means that by using the Runge-Kutta method, iterations can be saved compared to the Euler method and a solution can still be found which, especially when approaching ^ ^^^ = 7.2 the exact solution approximates 301 better than the solution according to the Euler method. Since in particular the quality of the solution at ^ ^^^relevant for the quality and reliability of the determination of the ground state 1001, it can be seen here that the Runge-Kutte method requires significantly fewer time steps and thus iterations and still allows a more accurate determination of the ground state 1001. In the following table, the errors compared to the exact solution 301 for the various approximations in the time evolution (Euler solution 304, RK2 solution 303 and RK4 solution 302) for different time step values ​​^^ are compared: Time evolution Time step Abs. t ^^ Error Euler 0.001 8.82 × 10 ^^ RK2 0.001 2.43 × 10 ^^ RK4 0.001 4,10 × 10 ^^^ Euler 0.01 8.71 × 10 ^^ RK2 0.01 2.43 × 10 ^^ RK4 0.01 8.36 × 10 ^^ Euler 0.1 7.7934 R.407999 - 18 - RK2 0.1 2.39 × 10 ^^ RK4 0,1 9,48 × 10 ^^ Euler 0.2 13.9145 RK2 0.4 3.4800 RK4 0.8 5.39 × 10 ^^As can be seen from the table, the choice of the time step size ^^ affects the convergence behavior of the three approximation methods. It should be noted that the RK4 solution 302, even with a time step size of ^^ = 0.8, is still somewhat closer to the exact solution than the Euler solution 304 with a time step size of ^^ = 0.01. In other words, the method 100 shown in Fig. 1, using the fourth-order Runge-Kutta method with 1 / 80 of the iterations, comes close to the number of iterations when using the Euler method, with a similar accuracy of the resulting solution. Fig. 3 shows a diagram on whose x-axis 300' the imaginary time ^ and on whose y-axis 300'' the solution of the differential equation ^ determined using different methods ^ = sin(^) ^ ∙ ^ with the initial condition ^ ( ^ = 0 )= 2. The differential equation describes a highly simplified system which has comparable properties to the differential equation of the method 100 explained in Fig.1 and which is used below to illustrate the quality of the solution of the previously proposed method 100. In contrast to the diagram shown in Fig.2, in Fig.3, noise in the form of an additive uniform noise on ^ ^ , scaled by a factor of 0.5. In this example, the noise to which NISQ computers are subject is represented in the form of additive uniform noise on ^ ^simulated. In the diagram, the exact solution 301 ^(^) is entered as a solid curve. Furthermore, the solution 304, which uses the Euler method for the imaginary time development in step 5) in the method 100 shown in Fig.1 for a time step ^^, is entered (hereinafter referred to as Euler solution 304), R.407999 - 19 - where for each time step the corresponding determined function value is entered as a cross and neighboring crosses are connected to each other in pairs by straight lines. The size of the time steps is chosen here to be ^^ = 0.2. Furthermore, the solution 303, which uses the imaginary time development in step 5) in the method shown in Fig.1, the second-order Runge-Kutta method is used for a time step ^^ (hereinafter referred to as the RK2 solution 303), with the corresponding determined function value being entered as a square for each time step and neighboring squares being connected to one another in pairs by straight lines. The size of the time steps is chosen here to be ^^ = 0.4, i.e. twice as large as in the Euler solution 304. In addition, the solution 302, which uses the fourth-order Runge-Kutta method for the imaginary time evolution in step 5) in the method 100 shown in Fig. 1, for a time step ^^ (hereinafter referred to as the RK4 solution 302), is entered, with the corresponding determined function value being entered as a star for each time step and neighboring stars being connected to one another in pairs by straight lines.The size of the time steps is chosen here to be ^^ = 0.8, which is twice as large as the RK2 solution 303 and four times as large as the Euler solution 304. In other words, the total time ^. ^^^ in the case of the RK2 solution 303, it is divided into half as many time steps and in the case of the RK4 solution 302, even into a quarter of the time steps compared to the Euler solution 304. Nevertheless, the RK2 solution and the RK4 solution represent the exact solution 301 better, especially when approaching ^ ^^^= 7.2. The result is a similar picture to that in Fig. 2: here too, the RK2 solution 303 and the RK4 solution 302 are closer to the exact solution 301 than the Euler solution 304. In the following table, different strengths of the uniform noise are chosen and the errors compared to the exact solution 301 for the various approximations in the time evolution (Euler solution 304, RK2 solution 303 and RK4 solution 302) are compared (the time step size is constant for all calculations as ^^ = 0.0001 and statistical averages were taken over 100 runs): Time Evolution Noise Abs. n Error R.407999 - 20 - Euler 0.001 8.83 × 10 ^^ RK2 0.001 1,53 × 10 ^^ RK4 0.001 1.29 × 10 ^^ Euler 0.01 8.83 × 10 ^^ RK2 0.01 1.03 × 10 ^^ RK4 0.01 9.55 × 10 ^^ Euler 0.1 8.91 × 10 ^^ RK2 0.1 7.30 × 10 ^^ RK4 0.1 3.40 × 10 ^^ Euler 0.5 1.06 × 10 ^^ RK2 0,55,25 × 10 ^^ RK4 0.5 7.30 × 10 ^^ The table shows that RK2 and RK4 are significantly more robust against noise than the Euler method, and thus method 100, when using an NISQ computer, despite the fact that this is very noisy, delivers significantly more accurate results when using RK2 or RK4. It is clear that for all noise levels, the error in the Euler solution is two orders of magnitude larger than with RK2 and RK4. The following table shows the influences of different step sizes and different noise levels on the error (a statistical average was taken over 5000 runs): R.407999 - 21 - Time Evolution Time Step Noise Abs. t ^^ n Error Euler 0.01 0.1 8.71 × 10 ^^ RK2 0.02 0.1 9,58 × 10 ^^ RK4 0.04 0.1 8.66 × 10 ^^ Euler 0.01 0.5 8.61 × 10 ^^ RK2 0.02 0.5 1,59 × 10 ^^ RK4 0.04 0.5 2.16 × 10^^ Euler 0.01 1.0 8,84 × 10 ^^ RK2 0.02 1.0 2.18 × 10 ^^ RK4 0.04 1.0 8.36 × 10 ^^ Here, too, it becomes clear that in all cases, RK2 and RK4 are superior to the Euler method in terms of error susceptibility. Figure 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 Figure 1, to determine the ground state 1001 of the quantum system.

Claims

R.407999 - 22 - Claims 1. Method (100) for determining a ground state (1001) of a quantum system, comprising the steps of: ^ providing (101) a hybrid variational quantum algorithm (1010), comprising: o a Hamiltonian operator; o an approach comprising a sequence of parameterized unitary quantum gates, ^ each unitary quantum gate having a gate parameter, ^ a test state for approximating a time-evolved quantum state at a time ^ , where ^ describes an imaginary time, resulting from applying the approach to an initial quantum state; o a differential equation for the gate parameters, which results from applying the McLachlan variational method to the test state, and which comprises an equation matrix comprising at least one matrix element and an equation vector comprising at least one vector element;oa quantum circuit (1011) for determining the equation matrix and the equation vector; oexecuting (102) the hybrid variational quantum algorithm (1010) on a hybrid computer platform (200) comprising a classical computer (201) and a quantum computer (202) for executing the quantum circuit (1011), wherein executing the hybrid variational quantum algorithm (1010) comprises the following steps: 1) providing (1021) the quantum circuit for the quantum computer (202), comprising the initial quantum state and initial gate parameters; 2) initializing (1022) the quantum computer (202), comprising preparing the initial quantum state, and providing control signals based on the gate parameters for controlling the quantum gates as a function of the gate parameters when executing the quantum circuit (1011) on the quantum computer (202); R.407999 - 23 - 3) executing (1023) the quantum circuit (1010) on the quantum computer (202); 4) transmitting (1024) measurement results of the quantum computer (202) to the classical computer (201) for determining the at least one matrix element of the equation matrix and the at least one vector element of the equation vector of the differential equation; 5) determining (1025) the gate parameters (1020) by solving the differential equation, comprising the equation matrix and the equation vector, on the classical computer; 6) determining (1026) the imaginary time evolution of the gate parameters (1020) determined by solving the differential equation for a time step by applying the Runge-Kutta method; 7) transmitting (1027) the imaginary time-evolved gate parameters (1021) to the quantum computer (202) and repeating steps 2) to 6) up to a final time step,wherein after step 5) of the last time step, the ground state is determined (103) on the classical computer (201) or the quantum computer (202) using the gate parameters (1020) determined in step 5) of the last time step.

2. The method (100) according to claim 1, wherein a two-stage or a four-stage Runge-Kutta method is used in determining (1026) the imaginary time evolution.

3. The method (100) according to any one of the preceding claims, wherein the Hamiltonian is a many-body Hamiltonian of a material.

4. The method (100) according to any one of the preceding claims, wherein the Hamiltonian is a Hamiltonian for the Hubbard model.

5. The method (100) according to any one of the preceding claims, wherein the ground state energy is determined and provided on the quantum computer (202).

6. Method (100) according to one of the preceding claims, wherein the solution of the differential equation is carried out by a numerical method.R.407999 - 24 - 7. The method (100) according to any one of the preceding claims, wherein, after transmitting (1024) the measurement results of the quantum circuit (1011), the at least one matrix element of the equation matrix and the at least one vector element of the equation vector of the differential equation are calculated on the classical computer (201), and the results of this calculation are used in solving the differential equation.

8. The method (100) according to any one of the preceding claims, wherein the ground state (1001) is determined by applying the approach comprising the gate parameters (1020) determined in step 5) of the last time step to the initial quantum state.

9. The method according to any one of the preceding claims, wherein the initial quantum state has a non-vanishing overlap with the ground state (1001) of the system. 10.Use of the method for determining a ground state (1001) of a quantum system according to one of the preceding claims for a material simulation, wherein the Hamiltonian is a many-body Hamiltonian that describes the material.

11. 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) according to one of claims 1 to 9 can be carried out.

12. The hybrid computer platform (200) according to claim 11, wherein the quantum computer (202) is a NISQ quantum computer.

13. Hybrid variational quantum algorithm (1010) comprising a hybrid computer platform (200) comprising a classical computer and a quantum computer (200) for executing a quantum circuit (1010), in particular a hybrid computer platform (200) according to one of the claims. R.407999 - 25 - 11 and 12, to execute the method (100) according to one of claims 1 to 9.

14. A computer-readable storage medium (203) on which the hybrid variational quantum algorithm (1010) according to claim 13 is stored.

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