Method for determining expected value of observable measurement and method for determining ground state energy

Abstract

The invention relates to a method for determining an error-corrected desired value of an observable measurement of a quantum system by means of a classical computer, comprising the following steps: providing a first desired value of the observable measurement dependent on a first distance metric, wherein the first distance metric is indicative of a spatial separation between the first set of qubits and the second set of qubits; providing at least one second desired value, which is observably measured and depends on the second distance metric; determining an adapted fitting function, comprising the steps of: 0 providing a fitting function, the fitting function comprising at least one fitting parameter, and the fitting function being dependent on a distance metric; 0, fitting a fitting parameter of the fitting function to the first expected value and the second expected value; determining an error-corrected desired value of the observable measurement by evaluating the adapted fitting function for the distance metric using a value greater than the first distance metric and the second distance metric; an error-corrected desired value is provided.

Description

Background Technology

[0001] In “Variational ansatz-based quantum simulation of imaginary timeevolution” (McArdle et al., npj Quantum Information (2019) 5:75; https: / / doi.org / 10.1038 / s41534-019-0187-2), a method for determining the ground state of a many-body system is described, which utilizes a hybrid system consisting of quantum and classical computers.

[0002] Currently available quantum computers, especially those based on superconducting circuits, are showing an increase in the number of physical qubits available. IBM's currently available Osprey quantum computer chip contains 433 qubits. This chip is even surpassed by IBM's Condor chip, which has 1121 qubits.

[0003] Currently available quantum computers are not fully error-corrected, resulting in inherent errors and noise in any quantum circuits executed on these computers. Therefore, such quantum computers are often referred to as NISQ (noisy intermediate-scale quantum) technology. Currently available NISQ quantum computers are limited in their capabilities. Due to their finite size (few physical qubits) and inherent gate errors, these noisy NISQ computers only allow the execution of short quantum circuits, i.e., shallow-depth quantum circuits, and the results typically have a considerable error bar. An example of shallow-depth quantum circuits is a hybrid quantum-classical algorithm, such as the variational quantum eigenvalue solver (VQE). However, due to the challenges posed by inherent noise, variational algorithms such as VQE cannot efficiently utilize large numbers of qubits.

[0004] Core and advantages of this invention Finding the ground state of a quantum mechanical system is an important task related to atomic-level materials simulation and in the field of quantum chemistry. In the past, many algorithms have been developed to solve this problem using classical computers. However, the properties of materials cannot be calculated with sufficiently high accuracy on traditional high-performance computers.

[0005] An example of algorithms that utilize quantum computers to solve quantum problems is the variational quantum algorithm, particularly the variational quantum eigensolver (VQE) based on a quantum-classical hybrid approach. These algorithms are used, for example, to determine the ground state (i.e., the lowest energy state) of a quantum system. In this method, the quantum state (e.g., the wave function) is encoded using variational methods with variational parameters. In quantum circuits, the expected value of the Hamiltonian operator describing or approximating the quantum system (i.e., measuring the observable) is measured. To obtain a statistically well-converged expected value, this measurement requires a large number (on the order of thousands) of individual measurements (shots).

[0006] After determining the expected value, the classical method is used to update the variational parameters. For example, for VQE, the parameters are updated along the descent direction to minimize the expected value (energy) obtained from the quantum circuit. The updated parameters... This feedback was then sent to the quantum computer to obtain... This iterative process is repeated until the ground state energy is obtained in the nth iteration. and its corresponding quantum state. The quantum state of the nth iteration is encoded in the variational parameters. In this context, it corresponds to the ground state of the quantum system, or describes an approximation of the ground state.

[0007] Variational quantum eigenfunction solvers (VQEs) are particularly useful in quantum chemistry and materials science. They are hybrid algorithms that utilize both classical and quantum computing. VQEs are used to determine the ground-state energy and wavefunction of complex quantum systems and provide insights into the behavior of complex molecules and materials. VQEs compute the expectation of a parameterized circuit and optimize the parameters to minimize the energy. The global energy minimum is then considered a good approximation of the ground-state energy. For larger systems, optimization in VQEs becomes challenging because the optimizer can get stuck in local minima or the so-called plateau of vanishing gradients. A more efficient approach is to use a quantum computer to determine the expectation of the system with respect to observables (e.g., the system's Hamiltonian operator) and then use a classical optimizer to refine the parameters of the method.

[0008] VQE combines quantum computing with classical optimization techniques. To this end, Hamiltonian operators are provided to describe the properties of the quantum system under study, such as the electronic configuration of its molecular structure. For example, a Hamiltonian operator can be represented as a linear combination of Pauli operators. A carefully chosen method or parameterized quantum circuit can approximate the ground state of the system. At the heart of VQE lies its hybrid methodology. Classical optimizers are used to tune or optimize the parameters of the method, aiming to minimize the energy expectation of the Hamiltonian operator, which is crucial for reaching the system's ground state.

[0009] In this interaction between quantum and classical computers, VQE iterates between measuring the expectation value of the Hamiltonian operator on the quantum computer and using classical algorithms to optimize the parameters of the quantum circuit (i.e., variational parameters).

[0010] Another problem arises when using quantum algorithms with a large number of qubits, due to the increasing challenges that come with system size. For VQE, this is, for example, the variational parameters. (usually through a set of angles) The number of variational parameters (represented by the method) increases unfavorably with the complexity of the method and the number of qubits. Furthermore, the inherent noise of NISQ devices makes gradient-based optimization inefficient.

[0011] In particular, due to the aforementioned problems, as well as the inherent errors of NISQ computers and the limitations of classical optimization algorithms, this type of variational quantum classical method can only be used for shallow quantum circuits and only for a few dozen variational parameters. A quantum computer is implemented using only a small number of physical qubits. Therefore, the performance of a quantum computer containing a large number of physical qubits (such as "Osprey" (433 physical qubits) or "Condor" (1121 physical qubits)) cannot be fully / efficiently utilized. Thus, in practice, only a small fraction of the physical qubits of a quantum computer are used to encode quantum circuits, while the remaining qubits remain inactive during the execution of the quantum circuits on the quantum computer.

[0012] Furthermore, in order to obtain statistically well-converged measurement results when executing quantum circuits, a large number of individual measurements (shots) are required, which introduces another problem.

[0013] The aforementioned problems primarily limit the potential for using an increasing number of physical qubits on quantum computers, especially NISQ computers. One possibility for utilizing more qubits lies in encoding multiple small quantum circuits into multiple clusters / qubit groups on a single quantum computer. These smaller quantum circuits can be identical, allowing for parallelization of the number of individual measurements. In this way, multiple individual measurements can be performed in parallel by executing these identical quantum circuits in a single run of the quantum computer.

[0014] Alternatively or supplementarily, these smaller quantum circuits can also be different from each other, for example, to measure different observables in parallel, or to obtain gradients along different directions, as is required, for example, in the VQE algorithm.

[0015] While this may not necessarily handle larger systems, it reduces the total computation time for handling smaller systems.

[0016] The main problem with the parallelization schemes described above is "crosstalk" between qubit groups, which leads to correlated and nonlocal errors, even between different, independent clusters of quantum circuits encoded on a quantum computer. To address this source of error, it is proposed to spatially separate these clusters on the quantum computer to reduce crosstalk errors. This enables the reduction of noise in individual measurements on quantum computers with a large number of physical qubits (especially more than 100 or 400) with efficient computation time. Summary of the Invention

[0017] This invention proposes a method that, by determining multiple clusters of physical qubits (= subsets of the physical qubits of a quantum computer; hereinafter also referred to as qubit groups), where each cluster is suitable for executing quantum circuits, enables the following: • Utilizing the larger number of physical qubits in quantum computers, and • Parallelize the execution of quantum circuits (especially quantum circuits for VQE).

[0018] Preferably, the present invention enables more efficient use of quantum computers with a large number of physical qubits (especially more than 100 or 400) by allowing multiple quantum circuits to execute in parallel on the quantum computer. This increases the number of measurements per run as the number of clusters / qubit groups used increases, or reduces the number of quantum computer runs required to produce the same number of measurements. In particular, the present invention reduces crosstalk between qubit groups, thereby improving the reliability of measurement results.

[0019] In other words, the present invention advantageously utilizes a large number of qubits by enabling the parallel execution of many smaller quantum circuits on a large quantum computer, for example, to reduce noise in individual measurements. Crosstalk errors are mitigated, in particular, by spatially separating clusters of small, parallel-coded quantum circuits on the same quantum computer and by other methods to reduce crosstalk between qubits.

[0020] A quantum circuit is a computational routine constructed from coherent quantum operations. Each horizontal line or wire in a quantum circuit represents a qubit, where the left end of the wire represents the initial quantum data, and the right end represents the final quantum data produced by the computation through the quantum circuit. Operations on qubits are represented by boxes placed on these wires. A quantum gate is a fundamental operation that a quantum computer can perform on its qubits. Quantum gates are analogous to electron gates that perform fundamental operations on classical computers. For a quantum gate operating on two qubits (2-qubit gate), an interaction between the physical qubits in question is required. In the case of spin qubits, this is particularly achieved through exchange interactions. For example, atoms in an ion trap can exchange photons. For qubits based on superconducting circuits, these qubits can be manipulated, for example, by applying a voltage, a magnetic field, or by coupling with a microwave resonant cavity. Hereinafter, quantum circuit refers to the quantum circuit used for physical qubits. Compared to quantum circuits used for logical qubits, these quantum circuits can have other quantum gates, especially SWAP gates, which may be supplemented in quantum circuits used for physical qubits so that logical qubits can be transferred to interacting physical qubits when they participate in common quantum operations.

[0021] The present invention relates to a method for determining the error-corrected expected value of an observable quantity of a quantum system, a method for determining the ground state energy, a hybrid computer platform, and an error reduction algorithm.

[0022] This method is based on a parallelization scheme in which the physical qubits of a quantum computer are divided into independent qubit clusters, and these qubit clusters are spaced apart from each other and encode independent quantum circuits. The spatial spacing (which is achieved, for example, by arranging physical qubits that are not needed to execute the encoded quantum circuits between the qubit clusters) reduces the interaction between the qubit clusters.

[0023] Crosstalk errors can often occur even between different, independent sets of qubits (or between quantum circuits encoded therein). The advantage of the method presented below is that it avoids or at least reduces crosstalk between qubit sets, which leads to correlated and nonlocal errors. Therefore, the method according to claim 1 is particularly capable of producing more reliable measurement results.

[0024] In particular, this invention enables efficient utilization of the available physical qubits of a quantum computer. Specifically, it allows the physical qubits in a qubit array to be divided into qubit groups, each of which can be used to execute a quantum circuit. Therefore, potentially (but not necessarily) identical quantum circuits can be executed in parallel. This reduces the number of quantum computer runs while maintaining the same number of measurement results. In this way, parallelization of individual measurements (shots) with respect to observables or gradient descent directions of VQE offers significant advantages when using a large number of physical qubits on a NISQ computer.

[0025] This is achieved by the method described in claim 1 for determining the error-corrected expected value of observables of a quantum system using a classical computer.

[0026] The term "quantum system" refers to a physical system in which quantum mechanical phenomena are observable. Examples of such phenomena include the quantization of energy or other observable quantities, interference of particle waves, nonlocality, or quantum tunneling. Quantum systems encompass the entire microscopic world, such as elementary particles and atoms, as well as electrical conductors, semiconductors, macromolecules, and specific materials whose macroscopic properties are determined by quantum mechanical interactions at the microscopic scale, with dimensions in the nanometer range. A quantum system can be described, in particular, by the Hamiltonian operator. In quantum mechanics, the Hamiltonian operator of a system is the operator that describes the total energy (including kinetic and potential energy) of the system. Its spectrum, the energy spectrum of the system, includes the eigenvalues ​​of the Hamiltonian operator, which are the energy eigenvalues. This is a set of possible results that can be obtained by measuring the total energy of the system.

[0027] As an observable measure, the Hamiltonian operator of a quantum system can be used, in particular. Specifically, this can be a many-body Hamiltonian operator describing or approximating a material. This method can be used, in particular, to determine the ground-state energy of a material system.

[0028] Observables, especially quantum mechanical observables, can be understood as the quantity to be measured and the associated operator that operates in the state space (Hilbert space). Examples of observables are energy (belonging to the Hamiltonian operator of the quantum system), position coordinates, momentum coordinates, and the spin component of a particle. Observables assign values ​​corresponding to the eigenvalues ​​of the operator to the result of a particular measurement. A key difference between classical and quantum mechanical observables is that several pairs of quantum mechanical observables cannot be measured simultaneously. If the operators of two quantum mechanical observables do not commute, then measuring the first operator alters the quantum state in a way incompatible with subsequent measurements of the second observable, and vice versa. Quantities that can be simultaneously and precisely determined are called commutative observables; they have the property that the order of their operators in the product can be interchanged without changing the result. Observables that cannot be simultaneously and arbitrarily precisely measured are also called complementary observables.

[0029] To determine the expected value of an observable, a large number of individual measurements are performed on a quantum computer, and then, preferably on a classical computer, the expected value of the observable is determined from these individual measurements. This requires a large number (e.g., on the order of thousands) of individual measurements for statistical convergence to meet accuracy requirements, such as chemical accuracy when applied to quantum chemical calculations. To give the order of magnitude of the number of measurements, it is generally appropriate to use: in terms of accuracy... Measuring observables requires an order of magnitude of The number of measurements. In particular, a single individual measurement includes the following steps: • At least a portion of the qubits in a quantum computer are initialized in an initial quantum state. • Applying quantum circuits to this initial quantum state to prepare a quantum state with respect to the desired value of its measurable observable, and • Measure the observables of this quantum state.

[0030] Quantum computers, programmable via quantum circuits, can in principle be built using any quantum technology capable of implementing single-qubit and multi-qubit gate operations. Architectures based on superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are currently under active development.

[0031] A quantum computer includes a qubit array, wherein the qubit array comprises a plurality of physical qubits, which are preferably configured by means of devices or units adapted to the technology used to implement the qubits for initialization (e.g., initializing the qubits to a base state), manipulation (e.g., applying 1-bit gates and / or 2-bit gates), and / or readout of the physical qubits.

[0032] As previously mentioned, crosstalk errors can be particularly prevalent when executing at least two quantum circuits in parallel on a quantum computer's qubit array. This method is specifically capable of correcting for such crosstalk error distortion in the expected value to determine the error-corrected expected value.

[0033] Parallel execution of quantum circuits specifically refers to the ability, by manipulating / calling a quantum computer, to perform a number of individual measurements in parallel (this number, for example, corresponds to the number of qubit groups in the case of identical quantum circuits), thereby determining the desired value with fewer quantum computer calls compared to performing individual measurements serially. In other words, during parallel execution, at least one quantum gate of the first quantum circuit is executed concurrently with the quantum gate of the second quantum circuit.

[0034] The method according to claim 1 particularly includes the following steps: • Provides a first expected value that can be measured, which depends on a first distance metric, wherein the first distance metric describes the spatial separation between a first set of qubits and a second set of qubits; wherein, in order to provide a measurement result for determining the first expected value, quantum circuits have been executed in parallel with each other on the first set of qubits and the second set of qubits on the quantum computer. This distance metric can be, in particular, a value of spatial distance, such as the distance between the geometric centroids or centers of mass of the physical qubits in a qubit cluster. Alternatively or supplementarily, the distance metric can include the sum of minimum distances between clusters, taking into account qubit connectivity. Alternatively or supplementarily, the fidelities and connectivity of the qubits used in each qubit cluster can be considered as additional parameters in the distance metric, especially to avoid qubits with very high noise levels. This distance metric can be, in particular, in micrometers ( Please specify in units of nanometers (nm).

[0035] A qubit group is understood in particular as a set or cluster of physical qubits in a quantum computer's qubit array, which are functionally, and especially spatially, separate from the other physical qubits in the qubit array. Specifically, the physical qubits of this qubit group are suited to execute quantum circuits, and the connectivity of the physical qubits in the qubit group is designed to enable the execution of such quantum circuits. Furthermore, the interactions of these qubits with the other qubits in the qubit array are reduced compared to their interactions with other members of the qubit group itself.

[0036] Functional partitioning can be understood, in particular, as combining a subset of the physical qubits of a quantum computer's qubit array into a qubit group suitable for executing quantum circuits, and this qubit group, especially due to its spatial separation (but especially through its functional separation) from the physical qubits of the qubit array that do not belong to this qubit group, can be considered as a quantum computer independent of these qubits. Functional partitioning into qubit groups specifically means that the physical qubits of different qubit groups preferably do not interact. This is achieved, in particular, through the spatial separation of the qubits. In particular, qubit groups can be chosen to be spaced apart from each other on the qubit array. Spatial separation can be achieved, for example, by arranging physical qubits that are not associated with any qubit group (i.e., physical qubits not required for executing quantum circuits) between qubit groups, thereby forming a boundary for the qubit groups. Functionality here specifically means that the physical qubits of a qubit group are particularly necessary for executing quantum circuits encoded on that qubit group, and are therefore grouped to perform a common function. The claims propose partitioning into at least two qubit groups, i.e., particularly including more than two or more than three qubit groups. The measurement results used to determine the first expected value are generated, in particular, by a quantum computer. These measurement results are also provided, especially by a quantum computer. The first expected value is determined, particularly on a classical computer, based on the measurement results of the individual measurements.

[0037] • Provide at least one second expected value for the observable, the second expected value depending on a second distance metric, wherein the second distance metric describes the spatial separation between the first and second qubit groups, wherein quantum circuits have been executed in parallel on the first and second qubit groups on a quantum computer to provide the measurement results for determining the second expected value; the measurement results for determining the second expected value are generated, in particular, by the quantum computer. These measurement results are also provided, in particular, by the quantum computer. The second expected value is determined, in particular, on a classical computer, based on the measurement results of the individual measurements.

[0038] • Determine the adapted fitting function, following these steps: ○ Provide a fitting function, which includes at least one fitting parameter and depends on a distance metric; curve fitting (or fitting) is a technique used to attempt to fit a given mathematical model function—the fitting function—to the data points (here, the expected value related to the distance metric) as well as possible. The simplest case of a fitting function is probably determining a fitting line, where the first-order polynomial... The coefficients k and l, especially the independent variable This can represent the reciprocal of the distance metric d. The fitting parameters k and l can be determined such that the sum of the squared distances between the data points takes the minimum possible value. ○ Fit the fitting parameters of the fitting function to the first and second expected values; • The expected value of the observable after error correction is determined by evaluating the fitted function for distance metrics that are greater than the first and second distance metrics. This step is based on the consideration that crosstalk error is distance-dependent and that crosstalk error between qubit groups is expected to decrease as the distance between qubit groups increases, such that extrapolating to a distance metric larger than the expected value used for fitting can reduce crosstalk error. In particular, extrapolation can be made to a virtual, infinite distance metric. When the fitted function is expressed as being correlated with the inverse of the distance metric, the inverse of the distance metric is compared. To plot the expected values ​​under different distance metrics d. Then, this extrapolation corresponds to... The fitting function is evaluated when the reciprocal of the distance metric approaches zero or the distance metric approaches infinity. • Provides an error-corrected expected value. This provision can be made, in particular, through data transmission (wireless or wired data transmission), display, output, or storage (e.g., in a database).

[0039] One advantage of this method is that it enables the execution of multiple quantum circuits (identical, at least partially identical, and / or distinct quantum circuits) on a physical qubit array of a quantum computer, where hardware-specific characteristics, such as the connectivity of the physical qubits of the qubit array, their corresponding fidelity, and the requirements imposed by the quantum circuits, can be considered to reduce crosstalk errors.

[0040] This method specifically includes providing more than two expected values ​​for distance metrics that are different from each other, and taking them into account when determining the fitting parameters of the fitting function.

[0041] This method is based on a mitigation scheme, also known as "infinite distance extrapolation." In this case, clusters (= qubit groups) are arranged at different, especially gradually increasing, distances (characterized by a distance metric) to determine the desired value, thereby assessing the effect of crosstalk and correcting the result based on the extrapolated error term within the limit of infinite cluster spacing.

[0042] In one implementation, the distance metric is given by the particularly minimal shortest path distance between groups of qubits. In square or cubic lattices, the shortest path distance is reduced to the Manhattan distance, or Manhattan metric. The latter is a metric where the distance between two points is defined as the sum of the absolute differences between their individual coordinates. Two points and Manhattan norm between Defined as .

[0043] If the qubit connectivity of some qubits deviates from a square / cubic lattice, it may be possible to adapt the Manhattan norm accordingly by considering the missing connectivity (through breadth-first search or depth-first search algorithms). From this, the minimum distance with reduced connectivity can be derived again. As the sum of individual path lengths. In the most general case, physical qubits and their connectivity can be represented as an undirected graph, thus allowing various shortest path algorithms (such as Dijkstra's algorithm) to determine the path. .

[0044] In one implementation, the distance metric is given as the Euclidean distance between the geometric centroids of the qubit groups.

[0045] According to one implementation, when determining the error-corrected expected value, the fitted function is evaluated for the case where the distance metric takes infinite values. One advantage is that, in the limiting case where the distance between qubit groups is infinite, crosstalk error no longer exists.

[0046] According to one implementation, providing the first expected value and the second expected value includes the following steps: • Provide a first distance metric and at least a second distance metric; in particular, this step includes providing more than two distance metrics. • Perform the following steps on the first and second distance metrics: ○ The quantum computer's qubit array is functionally divided into a first qubit group and at least one second qubit group, and their spatial separation is determined by a preset distance metric; ○ Provides a mapping from quantum circuits to the first and second qubit sets; ○ Perform the following steps in parallel on a quantum computer: • Initialize the first qubit set and execute a quantum circuit using the first qubit set to produce a first measurement result; and • Initialize the second qubit set and execute a quantum circuit using the second qubit set to produce a second measurement result; By using the first and second measurement results, the expected value of the observable quantity, dependent on a preset distance metric, is determined. Specifically, for each distance metric, multiple individual measurements (especially thousands) are performed using a quantum computer, and the expected value of each distance metric is determined separately from these numerous individual measurements. This latter step is preferably performed on a classical computer.

[0047] Providing a mapping from quantum circuits to qubit groups specifically includes assigning the qubits listed in the quantum circuit to the physical qubits of the quantum computer. Furthermore, providing may include providing an initial quantum state for the corresponding qubit group and gate parameters for manipulating quantum gates to execute the quantum circuit. This provision can be made, in particular, through input, data transmission (wireless or wired data transmission), or retrieval (e.g., from a database).

[0048] Initializing a qubit set specifically means preparing an initial state for each qubit set and providing control signals according to the gate parameters to manipulate the quantum gates when executing the quantum circuit on a quantum computer; if the quantum gate is configured, for example, as a rotation gate, which causes a single qubit to rotate by an angle about the X, Y, or Z axis of a Bloch sphere, then that angle is the gate parameter. The control signals depend on the technology of the quantum computer used. In the case of qubits based on superconducting circuits, the qubits can be manipulated, for example, by applying a voltage, a magnetic field, or coupling with a microwave resonant cavity, such that the control signals in this case are configured, for example, to set the magnetic field and / or the frequency of the microwave resonant cavity. The control signals can in particular include electrical signals. The desired value obtained benefits from the fact that crosstalk occurs at different locations in the quantum circuit and can therefore cancel each other out. Thus, improved results, especially with reduced crosstalk errors, can be achieved compared to the desired value determined from measurements without an insertion delay value.

[0049] Quantum states can be prepared, for example, by applying quantum circuits to an initial quantum state on a quantum computer. When initializing a quantum computer, i.e., setting the initial quantum state, the quantum bits of the quantum computer are placed in the initial quantum state. For example, all the quantum bits are placed in the initial state. The state or all qubits are placed Alternatively, the qubits are placed in the initial qubit state sequence, that is, the first number of qubits are placed in... The second number of qubits is placed in the state, while the second number of qubits is placed in the state. State. A sequence of five qubit states (where all qubits are in state). An example of a quantum state is 00000. Initialization also includes providing a control signal, which is preset, in particular, by a quantum circuit for preparing a quantum state whose observable desired value is to be determined. During the preparation of the quantum state, quantum gates manipulate the state of the quantum computer's qubits according to the quantum gates set in the quantum circuit, such that the prepared quantum state exists after the execution of the quantum circuit.

[0050] In one implementation, the fitting function includes linear functions, polynomial functions, and / or exponential functions. An example of a polynomial fitting function is as follows: in, It is a function of the distance metric d. is the fitting parameter, and k is the number of expected values ​​for different distance metrics (especially, k>1).

[0051] In one implementation, the fitting function is a function of the reciprocal of the distance metric, i.e. In other words, the fitting function is represented as a function of the inverse of the distance: .

[0052] In one implementation, the fitting function is represented as a function of the inverse of the distance: Furthermore, when determining the error-corrected expected value, a zero value is used for the reciprocal of the distance metric. The fitted function is evaluated based on the following conditions. Specifically, through... To approximate the expected value under an infinite distance metric.

[0053] Furthermore, a method for determining the ground state energy of a quantum system is proposed, wherein the method is applied to reduce errors when determining the Hamiltonian operator (especially the expectation value of the Hamiltonian operator of the quantum system) in the ground state of the quantum system, so as to obtain a ground state energy with reduced errors.

[0054] In particular, the associated ground-state energy can be output and / or stored. The output can be made, for example, by transmitting it to a hybrid computing platform, classical computer, quantum computer, cloud, and / or display device. Alternatively or additionally, the output can be used as input to another algorithm, particularly for materials simulation algorithms, where quantum systems are used to study certain material properties.

[0055] Finding the ground state of a quantum mechanical system is an important task related to atomic-level materials simulation and the field of quantum chemistry.

[0056] In one implementation, the quantum system whose ground-state energy is to be determined is a many-body system, which can be described, for example, by the Hubbard-Hamilton operator. The Hubbard model is an approximate model of solids. It describes the behavior of electrons in a lattice that is considered rigid. Here, only the repulsive Coulomb force between electrons residing at the same lattice position is considered. The kinetic energy part of the electron is modeled by the overlap integral from the tight-binding model. Some examples of quantum systems that can be described by the Hubbard-Hamilton operator are strongly correlated fermionic systems, transition metals, mobile electron systems (e.g., ferromagnetic, antiferromagnetic, and subferromagnetic), and quantum chemistry. Electronic systems. One advantage is that this method thus accelerates the development and research of new materials, and in part makes it possible. Furthermore, the properties of these new materials can therefore be better adapted to their respective applications.

[0057] The advantages mentioned above also apply to using the aforementioned method for materials simulation, where the Hamiltonian operator of the quantum system is a many-body Hamiltonian operator describing the material to be simulated.

[0058] A hybrid computing platform is an example of a noisy quantum computing platform, comprising a classical computer and a quantum computer for executing quantum circuits, adapted to enable the execution of the steps of the above-described method and / or to enable the application of the method in materials simulation. The advantage is that this quantum computing platform can be used particularly efficiently for materials simulation. Adaptation can be understood, in particular, as the quantum computer hardware being coordinated with the quantum circuits, enabling the mapping of logical qubits of the quantum circuits to physical qubits of the quantum computer, and preferably, using as few as possible, zero additional SWAP operations to ensure the interaction between the physical qubits occupied by the logical qubits when performing, for example, two-gate operations. In particular, the quantum computer hardware can be selected based on the provided quantum circuits, thereby executing the method in a more efficient and noise-insensitive manner. A hybrid computing platform can be understood, in particular, as comprising at least one classical computer and at least one quantum computer. Preferably, it includes a cloud computing platform accessed by the classical computer, so that the classical computer can, in particular, retrieve the measurement results from the quantum computer. In particular, the quantum computer has more than 400 physical qubits. In particular, quantum computers are configured to execute at least two or more quantum circuits in parallel during a single call.

[0059] In one implementation, the NISQ computer is used as a quantum computer.

[0060] Furthermore, an error reduction algorithm is proposed that enables a hybrid computer platform, including classical computers and quantum computers for executing quantum circuits, to perform at least one of the aforementioned methods.

[0061] A computer-readable storage medium storing an error reduction algorithm as an implementation of the aforementioned method, which can be manipulated, in particular, by a quantum computer and / or a classical computer, to provide the error reduction algorithm. Attached Figure Description

[0062] Embodiments of the present invention are shown in the accompanying drawings and described in detail below. The same reference numerals in the drawings denote the same or functionally identical elements.

[0063] In the attached diagram: Figure 1 A schematic diagram of the connectivity of the physical qubits of a quantum computer with 127 qubits according to a first embodiment is shown, in which groups of qubits are drawn; Figure 2A schematic diagram of the physical qubit connectivity of a quantum computer with 127 qubits according to a second embodiment is shown, in which the qubit group is drawn; Figure 3 A flowchart is shown for a method for determining the error-corrected expected value of an observable quantity of a quantum system; Figure 4 A schematic diagram of a hybrid computing platform is shown. Detailed Implementation

[0064] Figure 1 A schematic diagram of a hardware-specific connectivity graph 200 for the physical qubits 0, ..., 126 of a quantum computer is shown. In this embodiment, the physical qubits 0, ..., 126 are arranged in a heavy-hex-gitter lattice, for example, used in IBM's superconducting quantum computer (127-qubit Eagle processor). The physical qubits are numbered row by row. The connectivity graph to which the quantum computer's qubit array belongs comprises a total of 127 physical qubits 0, ..., 126, which are represented as circles or nodes in the schematic diagram and are numbered to identify the physical qubits. In the hardware-specific connectivity graph, the nodes of the physical qubits 0, ..., 126 configured to interact are interconnected, in particular, by edges. In other words, connectivity graph 200 provides information about the number and connectivity of the physical qubits 0, ..., 126.

[0065] In this embodiment, a quantum circuit with twelve physical qubits should be implemented. In principle, such a quantum circuit can be implemented ten times in parallel on a 127-qubit Eagle processor, resulting in n=10 clusters (qubit groups) on the connectivity graph. However, such a dense clustering of qubit groups leads to significant crosstalk errors. Therefore, a smaller number of qubit groups to be implemented, namely five, is chosen. This reduces crosstalk between qubit groups 201, 202, 203, 204, and 205. Various possibilities exist for mapping these clusters onto the connectivity graph.

[0066] In this embodiment, the five qubit groups 201, 202, 203, 204, and 205 are positioned on the physical qubits 0, ..., 126 of the quantum computer as follows: • The first qubit group 201 includes physical qubits numbered 0, 1, 2, 3, 4, 14, 15, 18, 19, 20, 21, and 22. The second qubit group 202 includes physical qubits numbered 8, 9, 10, 11, 12, 16, 17, 26, 27, 28, 29, and 30. • The third qubit group 203 includes physical qubits numbered 62, 63, 64, 65, 66, 72, 73, 81, 82, 83, 84, and 85. • The fourth qubit group 204 includes physical qubits numbered 96, 97, 98, 99, 100, 109, 110, 114, 115, 116, 117, and 118. • The fifth qubit group 205 includes physical qubits numbered 104, 105, 106, 107, 108, 111, 112, 122, 123, 124, 125, and 126.

[0067] In general, each qubit group 201, 202, 203, 204, and 205 comprises the same number of physical qubits 0, ..., 126, i.e., twelve each. Furthermore, these individual qubit groups have the same structure in terms of the connectivity of their contained physical qubits 0, ..., 126. At the center of each qubit group 201, 202, 203, 204, and 205, the geometric centroid 2002 is marked, where some distance metric 2001 is exemplarily labeled, here being the Euclidean distance (= L2 distance).

[0068] Each of these qubit groups can execute independent quantum circuits in parallel, enabling them to be used in a single call. Figure 1 When a quantum computer is used, it can perform five individual measurements of an observable, thus providing measurement results from the five individual measurements.

[0069] Figure 2 and Figure 1 The only difference is the distance metric. In this embodiment, the shortest path metric 2000 between qubit groups 201, 202, 203, 204, and 205 is plotted.

[0070] Figure 3 A flowchart of a method 300 for determining, with error correction, the expected value of an observable quantity of a quantum system using a classical computer is shown, comprising the following steps: • Provides a first expected value 3011 that is observable 301, the first expected value depending on first distance measures 2000, 2001, wherein the first distance measures 2000, 2001 describe the spatial separation between the first qubit group 201 and the second qubit group 202, wherein in order to provide a measurement result for determining the first expected value, quantum circuits have been executed in parallel on the first qubit group 201 and the second qubit group 202 on the quantum computer respectively; • Provide at least one second expected value 3012 for the observable 301, the second expected value depending on the second distance measures 2000, 2001, wherein the second distance measures 2000, 2001 describe the spatial separation between the first qubit group and the second qubit group, wherein in order to provide the measurement results for determining the second expected value, quantum circuits have been executed in parallel on the first qubit group and the second qubit group on the quantum computer respectively. • Determine the fitted function 3022 after adaptation, including the following steps: ○ Provide a 3020 fitting function, which includes at least one fitting parameter and depends on distance metrics 2000 and 2001; ○ Fit the fitting parameters 3021 of the fitting function to the first expected value 3011 and the second expected value 3012 of 3021; • By evaluating the fitted function 3022 for distance metrics 2000 and 2001 with values ​​greater than the first distance metric 2000 and 2001 and greater than the second distance metric 2000 and 2001, the expected value of the observable 303 after error correction is determined. • Provides an expected value of 3040 after error correction.

[0071] In particular, providing the first expected value 3011 and the second expected value 3012 may include the following steps: • Provide a first distance metric and at least a second distance metric; • Perform the following steps on the first distance metric: ○ The quantum computer's qubit array is functionally divided into a first qubit group and at least one second qubit group, and their spatial separation is determined by a first distance metric; ○ Provides a mapping from quantum circuits to the first and second qubit sets; ○ Perform the following steps in parallel on a quantum computer: • Initialize the first qubit set (201) and execute a quantum circuit using the first qubit set to produce (404) the first measurement result; and • Initialize the second qubit set (202) and execute a quantum circuit using the second qubit set to produce a second measurement result; ○ The expected value of the observable quantity, depending on the first distance metric, is determined by using the results of the first and second measurements; • Perform the following steps on the second distance metric: ○ The quantum computer's qubit array is functionally divided into a first qubit group and at least one second qubit group, and their spatial separation is determined by a second distance metric; ○ Provides a mapping from quantum circuits to the first and second qubit sets; ○ Perform the following steps in parallel on a quantum computer: • Initialize the first qubit set (201) and execute a quantum circuit using the first qubit set to produce (404) the first measurement result; and • Initialize the second qubit set (202) and execute a quantum circuit using the second qubit set to produce a second measurement result; ○ The expected value of the observable quantity, which depends on the second distance metric, is determined by using the results of the first and second measurements. • Perform the above steps for other distance metrics that are different from the first and second distance metrics.

[0072] Figure 4 This is a schematic diagram of a hybrid computing platform 400, which is an example of a noisy quantum computing platform on which operations such as... Figure 3 The exemplary method 400 is shown in the figure. The hybrid computing platform 400 includes a classical computer 401 and a quantum computer 402, which are adapted to perform operations such as... Figure 3 The method 400 is illustrated in the example and described above. In particular, the hybrid computer platform outputs an error-corrected expected value 3040, such as the ground state energy of a quantum system.

Claims

1. A method (300) for determining, with the aid of a classical computer (401), the expected value of an observable quantity of a quantum system after error correction, the method comprising the steps of: • Provides a first expected value (3011) of observable (301), the first expected value depending on a first distance metric (2000, 2001), wherein the first distance metric (2000, 2001) describes the spatial separation between a first qubit group (201) and a second qubit group (202), wherein in order to provide a measurement result for determining the first expected value, quantum circuits have been executed in parallel on the first qubit group (201) and the second qubit group (202) on a quantum computer, respectively. • Provide at least one second expected value (3012) of the observable quantity (301), the second expected value depending on a second distance metric (2000, 2001), wherein the second distance metric (2000, 2001) describes the spatial separation between the first qubit group and the second qubit group, wherein in order to provide a measurement result for determining the second expected value, quantum circuits have been executed in parallel on the first qubit group and the second qubit group on the quantum computer, respectively. • Determine the (302) adapted fitting function (3022) by the following steps: ○ Provide a (3020) fitting function, the fitting function including at least one fitting parameter, and the fitting function depending on the distance metric (2000, 2001). ○ The fitting parameters (3021) of the fitting function are adapted to the first expected value (3011) and the second expected value (3012) of (3021); • The adapted fitting function (3022) is evaluated for values ​​greater than the first distance metric (2000, 2001) and greater than the second distance metric (2000, 2001) to determine (303) the error-corrected expected value of the observable; • Provide the error-corrected expected value (3040) described in (304).

2. The method (300) according to claim 1, wherein when determining the error-corrected expected value (3040), the adapted fitting function (3022) is evaluated for the case of using infinite values ​​for the distance metric (2000, 2001).

3. The method (300) according to any one of the preceding claims, wherein providing the first expected value (3011) and the second expected value (3012) comprises the following steps: • Provide the first distance metric (2000, 2001) and at least the second distance metric (2000, 2001); • Perform the following steps on the first distance metric (2000, 2001) and the second distance metric (2000, 2001): ○ The quantum computer's qubit array is functionally divided into a first qubit group and at least one second qubit group, and their spatial separation is determined by a preset distance metric (2000, 2001); ○ Provide a mapping of the quantum circuit to the first qubit group and the second qubit group; The following steps are performed in parallel on the quantum computer: • Initialize the first qubit set (201) and execute the quantum circuit using the first qubit set to produce a first measurement result; and • Initialize the second qubit group (202) and execute the quantum circuit using the second qubit group to produce the second measurement result; ○ The expected value (3011, 3012) of the observable quantity is determined by using the first and second measurement results, depending on the preset distance metric (2000, 2001).

4. The method according to any one of the preceding claims, wherein the fitting function includes a linear function, a polynomial function, and / or an exponential function.

5. The method according to any one of the preceding claims, wherein the fitting function is a function of the reciprocal of the distance metric (2000, 2001).

6. The method of claim 5, wherein when determining the error-corrected expected value (3040), the adapted fitting function (3022) is evaluated with respect to the case where the reciprocal of the distance metric is zero.

7. A method for determining the ground state energy of a quantum system, wherein the method according to any one of the preceding claims is used to reduce errors when determining the expected value of the Hamiltonian operator of the quantum system in the ground state of the quantum system, so as to obtain an error-reduced ground state energy.

8. The method according to any one of the preceding claims, wherein the quantum system is described by a many-body Hamiltonian operator of the material.

9. The method according to claim 7 or 8, wherein the Hamiltonian operator of the quantum system is a Hamiltonian operator for the Hubbard model.

10. The application of the method according to any one of the preceding claims in materials simulation, wherein the Hamiltonian operator of the quantum system is a many-body Hamiltonian operator describing the material to be simulated.

11. A hybrid computer platform (400) comprising a classical computer (401) and a quantum computer (402) for executing quantum circuits, adapted to enable the steps of the method (300) according to any one of claims 1 to 9 and / or enable the application of the method according to claim 10.

12. The hybrid computer platform (400) according to claim 11, wherein the quantum computer (402) is an NISQ computer.

13. An error reduction algorithm that causes a hybrid computer platform (400), particularly the hybrid computer platform (400) according to claim 11 or 12, to perform the method (300) according to any one of claims 1 to 9, the hybrid computer platform comprising a classical computer (401) and a quantum computer (402) for performing quantum circuits.

14. A computer-readable storage medium having stored thereon the error reduction algorithm according to claim 13.

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