Methods for determining the error-corrected expected value of an observable quantity, methods for determining the ground state energy, hybrid computer platform, error reduction algorithms

Abstract

This invention provides a method for determining the error-corrected expectation value of an observable quantity in a quantum system using a classical computer, a hybrid computer platform, an error reduction algorithm, and a storage medium. [Solution] Method 300 provides a first expectation value 3011 of an observable that depends on a first distance size indicating spatial separation between first and second qubit groups, and at least one second expectation value 3012 of an observable that depends on a second distance size 301, determines a fitting function 3022 that includes at least one fitting parameter, depends on the distance size, and adapts the fitting parameter to the first and second expectation values ​​302, determines an error-corrected expectation value of the observable by evaluating the fitting function for the case where the distance size is greater than the first distance size and greater than the second distance size 303, and provides an error-corrected expectation value 3040.

Description

[Technical Field]

[0001] Background technology The paper “Variational Anzatz-based quantum simulation of imaginary time evolution” (by McArdle et al., npj Quantum Information (2019) 5:75; https: / / doi.org / 10.1038 / s41534-019-0187-2) describes a method for determining the ground state of a many-particle system using a hybrid system consisting of a quantum computer and a classical computer. [Background technology]

[0002] Currently available quantum computers, particularly those based on superconducting circuits, are showing an increasing number of available physical qubits. IBM's currently available "Osprey" quantum computer chip contains 433 qubits. Even this chip is surpassed by IBM's "Condor" chip, which has an additional 1121 qubits.

[0003] Because currently available quantum computers are not fully error-corrected, this is linked to the inherent errors and noise in each quantum circuit executed on these computers. Therefore, such quantum computers are often referred to as NISQ (noisy intermediate-scale quantum) technology. Currently available NISQ quantum computers have limitations in their capabilities. Based on their limited size (small number of physical qubits) and inherent gate errors, such noisy NISQ computers can only execute short quantum circuits, i.e., shallow-depth quantum circuits, and the results are usually accompanied by fairly large error bars. Examples of such shallow-depth quantum circuits include quantum-classical hybrid algorithms such as variational quantum eigenvalue solvers (VQEs).

[0004] However, variational algorithms like VQE cannot efficiently utilize a large number of qubits due to challenges arising from inherent noise.

[0005] Core and advantages of the present invention Finding the ground state of a quantum mechanical system is a crucial challenge in the fields of atomic-level materials simulation and quantum chemistry. Many algorithms using classical computers have been developed to address this problem. However, conventional high-performance computers cannot calculate material properties with sufficient accuracy.

[0006] One example of an algorithm for solving quantum problems using quantum computers is the variational quantum algorithm, particularly the variational quantum eigenvalue solver (VQE) based on a quantum-classical hybrid approach. These methods are used, for example, to determine the ground state (i.e., the state with the lowest energy) of a quantum system. In this approach, the quantum state (e.g., wave function) is transformed into a variational parameter within a quantum circuit using a variational approach.

number

[0007] After the expected value is determined, variational parameters

number

number

number

[0008] Variational quantum eigenvalue solvers (VQEs) are particularly applicable in quantum chemistry and materials science. They are hybrid algorithms that utilize both classical and quantum computers. This hybrid algorithm is used to determine the ground state energy and wave function of complex quantum systems, providing insights into the behavior of complex molecules and materials. VQEs calculate the expectation value of a parameterized circuit and optimize the parameters to minimize the energy. The general-purpose energy minimum is then considered a good approximation of the ground state energy. Optimization in VQEs presents challenges for large systems because the optimizer may encounter local minima or so-called bar plateaus where the gradient vanishes. Using a quantum computer, observable quantities, such as the expectation value of the system with respect to the system's Hamiltonian operator, are determined, and a classical optimizer is used to improve the parameters of the approach.

[0009] VQE combines classical optimization techniques with quantum computing. For this purpose, the properties of the quantum system being examined are provided, such as the Hamiltonian operator that describes the electronic configuration of the molecular structure. For example, the Hamiltonian operator is represented as a linear combination of Pauli operators. A carefully chosen approach or parameterized quantum circuit approximates the ground state of the system. The core of VQE lies in its hybrid technique. The classical optimizer is used to adapt or optimize the parameters of the approach. The goal is to minimize the energy expectation value of the Hamiltonian operator that is essential to reach the ground state of the system.

[0010] In this interaction between the quantum computer and the classical computer, VQE iterates between measuring the expectation value of the Hamiltonian operator on the quantum computer and using classical algorithms for optimizing the quantum circuit parameters (variational parameters).

[0011] A further problem when using quantum algorithms with a large number of qubits arises from the challenges associated with increasing system size. In the case of VQE, this problem is, for example, the variational parameters, generally represented by a set of angles Θ

Number

[0012] Particularly, based on the aforementioned problems, as well as the errors inherent in NISQ computers and the limitations of classical optimization algorithms, such variational quantum-classical approaches use shallow-depth quantum circuits, on the order of a few tens of variational parameters

Number

[0013] Moreover, further problems arise because a large number of shots (individual measurements) are required to obtain statistically well-converged measurement results during the execution of a quantum circuit.

[0014] The above problems substantially limit the usability of an increasing number of physical qubits on quantum computers, especially NISQ computers. One means of making more use of these qubits consists of encoding multiple small quantum circuits into multiple clusters / qubit groups on a single quantum computer. These small-scale quantum circuits may be considered identical, and thus, with this solution, parallelization regarding multiple individual measurements is possible. In this way, a large number of individual measurements can be carried out in parallel by executing these identical quantum circuits only once on the quantum computer.

[0015] Alternatively or additionally, these small-scale quantum circuits may be different from each other, for example, to measure various observables in parallel and obtain gradients along various directions as required by, for example, the VQE algorithm.

[0016] This does not necessarily enable the processing of large-scale systems, but in the case of small-scale systems, it is possible to shorten the overall calculation time.

Prior Art Documents

Non-Patent Documents

[0017] [Non-Patent Document 1] “Variational Anzatz-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) [Overview of the Initiative] [Problems that the invention aims to solve]

[0018] The main problem in the parallelization methods described above is the occurrence of crosstalk between qubit clusters, which leads to correlated nonlocal errors even between various independent quantum circuit clusters encoded on a quantum computer. To address the cause of this error, it has been proposed to spatially separate clusters to reduce crosstalk errors on the quantum computer, thereby enabling noise reduction in individual measurements on quantum computers with a large number of physical qubits, especially more than 100 or 400 physical qubits, in efficient computation time. [Means for solving the problem]

[0019] In this invention, the following applies: - To make effective use of a larger number of physical qubits in quantum computers, and, -By finding multiple physical qubit clusters (a subset of the physical qubits of a quantum computer, also referred to as a qubit group hereafter) that are suitable for executing quantum circuits, the execution of quantum circuits, especially VQE quantum circuits, can be parallelized. A method to make this possible is proposed.

[0020] Advantageously, the present invention enables the more efficient use of a large number of quantum computers, particularly those with more than 100 or more than 400 physical qubits, by allowing multiple quantum circuits to be executed in parallel on a quantum computer. This increases the number of measurement results per path for the number of clusters / qubit groups used, or reduces the number of quantum computer executions required to produce the same number of measurement results. In particular, the present invention makes it possible to reduce crosstalk between qubit groups and improve the reliability of measurement results.

[0021] In other words, the present invention advantageously enables the utilization of a large number of qubits, for example, by allowing a large number of smaller quantum circuits to be executed in parallel on a large quantum computer to reduce noise in individual measurements. Crosstalk errors are reduced, in particular, by spatially separating clusters of parallel encoded small quantum circuits on the same quantum computer, and furthermore, crosstalk between qubits is reduced.

[0022] A quantum circuit is a computational routine constructed from coherent quantum operations. Each horizontal line or wire within a quantum circuit represents a qubit, where the left end of the wire represents the original quantum data and the right end represents 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 fundamental operations that a quantum computer can perform on qubits. They are comparable to electronic gates that perform the fundamental operations of a classical computer. For quantum gates operating on two qubits (two-qubit gates), interaction between the physical qubits is required. In the case of spin qubits, this can occur particularly via exchange interactions. For example, atoms in an ion trap can exchange photons. In the case of qubits based on superconducting circuits, these qubits can be manipulated, for example, via applied voltage, magnetic field, or coupling to a microwave resonator. Hereinafter, "quantum circuit" refers to a quantum circuit for physical qubits. These quantum circuits may include additional quantum gates, particularly swaps, compared to quantum circuits for logic qubits, and these swaps may be intercepted within quantum circuits for physical qubits to transition logic qubits to interacting physical qubits if they are involved in common quantum operations.

[0023] The present invention relates to a method for determining the error-corrected expectation value of an observable quantity in a quantum system, a method for determining the ground state energy, a hybrid computer platform, and an error reduction algorithm.

[0024] This method is based on a parallelization scheme that divides the physical qubits of a quantum computer into independent groups of qubits (= clusters). In this case, the qubit groups are separated from each other and encode independent quantum circuits. For example, spatial separation achieved by placing physical qubits that are not necessary for the execution of the encoded quantum circuits between the qubit groups reduces the interaction between the qubit groups.

[0025] Typically, crosstalk errors can occur between various independent groups of qubits or between the quantum circuits encoded therein. The advantage of the method presented below is to avoid, or at least reduce, the occurrence of "crosstalk" between groups of qubits that leads to correlated nonlocal errors. This enables the method according to claim 1 to generate particularly reliable measurement results.

[0026] In particular, the present invention enables the efficient utilization of available physical qubits in an available quantum computer. Specifically, the present invention makes it possible to divide the physical qubits of a qubit array into qubit groups, each of which can be used for the execution of a quantum circuit. This allows for the parallel execution of the same quantum circuit in some cases, although it is not essential. This reduces the number of quantum computer executions while maintaining the number of measurement results. In this way, shots (individual measurements), observables, or gradient descent directions for VQE can be parallelized, which is a significant advantage when using a large number of physical qubits on a NISQ computer.

[0027] This is achieved by the method according to claim 1 for determining the error-corrected expectation value of an observable quantity in a quantum system using a classical computer.

[0028] The term "quantum system" refers to a physical system in which the manifestations of quantum mechanics are visible. Examples of such manifestations include the quantization of energy and other observable quantities, particle-wave interference, nonlocality, or quantum mechanical tunneling. Quantum systems encompass not only elementary particles and atoms, but also the entire microscopic world, including nanometer-sized conductors, semiconductors, macromolecules, and certain materials whose macroscopic properties are determined at the microscopic scale by quantum mechanical interactions. Quantum systems can be described in particular by the Hamiltonian operator. In quantum mechanics, the Hamiltonian operator of a system is an operator that describes the total energy of the system, including kinetic and potential energy. Its spectrum, i.e., the energy spectrum of the system, contains the eigenvalues ​​of the Hamiltonian operator, i.e., energy eigenvalues. This is the set of possible results that can be obtained by measuring the total energy of the system.

[0029] As observable quantities, in particular, the Hamiltonian operator of a quantum system can be used. Specifically, this could be a multi-particle Hamiltonian operator describing or approximating matter. In particular, this method can be used to determine the ground state energy of a material system.

[0030] Observable quantities, especially in quantum mechanics, can be understood as a measure and the operator acting in the state space assigned to that measure, i.e., Hilbert space. Examples of observable quantities include energy (the operator to which it belongs is the Hamiltonian operator of the quantum system), position coordinates, impulse coordinates, and the spin component of a particle. Observable quantities assign values ​​corresponding to the eigenvalues ​​of the operator to a particular measurement result. A crucial difference between classical quantities and quantum mechanical observable quantities is that some pairs of quantum mechanical observable quantities cannot be measured simultaneously. If the operators of two quantum mechanical observable quantities are not commutative, then a measurement of the first operator changes the quantum state in a manner incompatible with subsequent measurements of the second observable quantity, and vice versa. Measures that can be determined simultaneously and accurately are called commutative observable quantities, and they have the property that the result does not change even if the order of their operators in the product is changed. Observable quantities that cannot be measured simultaneously with arbitrary precision are also called complementary observable quantities.

[0031] To determine the expected value of an observable quantity, numerous discrete measurements are performed on a quantum computer, and then, from these discrete measurements, the expected value of the observable quantity is preferably determined on a classical computer. This requires a large number of discrete measurements, e.g., thousands, for statistical convergence to satisfy accuracy requirements, such as chemical accuracy when applied to quantum chemical calculations. To illustrate the scale of the number of measurements: typically, to measure an observable quantity with accuracy ε, 1 / ε 2 A certain number of measurements are considered necessary. In particular, individual measurements involve the following steps: - A step of initializing at least a portion of the qubits of a quantum computer into an initial quantum state, - The steps include applying a quantum circuit to an initial quantum state to prepare a quantum state in which the expectation value of an observable quantity is to be measured, - A step of measuring an observable quantity for the quantum state, Includes.

[0032] In principle, a quantum computer programmable using quantum circuits can be constructed from any quantum technology capable of realizing single-qubit and multi-qubit gate operations. Currently, architectures based on superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are being actively developed.

[0033] A quantum computer includes a qubit array, in which case the qubit array includes a plurality of physical qubits, which are preferably assumed to be initialized (e.g., initialized to a ground state), manipulated (e.g., applied to a 1-qubit gate and / or a 2-qubit gate), and / or read out using devices or units appropriately adapted to the techniques for realizing qubits.

[0034] As mentioned above, when at least two quantum circuits are executed in parallel on a quantum computer's qubit array, crosstalk errors can occur, in particular. This method makes it possible to correct expectation values ​​that have been altered by such crosstalk errors, in order to determine the error-corrected expectation values.

[0035] Parallel execution of quantum circuits specifically means that, through the driving control / calling of quantum computers, for example, the number of individual measurements equivalent to the number of qubits in the case of the same quantum circuit can be achieved in parallel. Therefore, the expected value can be obtained with fewer quantum computer calls than when individual measurements are performed sequentially. In other words, during parallel execution, at least one quantum gate of the first quantum circuit is executed in time overlap with the quantum gate of the second quantum circuit.

[0036] The method according to claim 1 is particularly the following: - A step of providing a first expectation value of an observable that depends on a first distance size, wherein the first distance size represents the spatial separation between a first qubit group and a second qubit group, and one quantum circuit is executed on the first qubit group and the second qubit group, respectively, in parallel with each other on a quantum computer to provide a measurement result for determining the first expectation value.

[0037] The distance size may be, in particular, a value of spatial distance, for example, the distance between the geometric or mass centers of the physical qubits of a qubit group. Alternatively or supplementally, the distance size may include the sum of the minimum distances between clusters, taking into account the connectivity of the qubits. Alternatively or supplementally, the fidelities of the qubits used in individual qubit groups and their connectivity can be considered together as additional parameters in the distance size, particularly to avoid noisy qubits. In particular, the distance size can be expressed in micrometers (μm) or nanometers (nm).

[0038] A qubit group is understood to mean, in particular, a group or cluster of physical qubits within a quantum computer's qubit array, which are functionally, and especially spatially, isolated from other physical qubits in the qubit array. In particular, the physical qubits of a qubit group are suitable for executing quantum circuits, and the connectivity of the physical qubits of a qubit group is configured to execute quantum circuits. Furthermore, qubit interactions with further qubits in the qubit array are reduced compared to interactions with members of the qubit group itself.

[0039] Functional partitioning is understood to mean, in particular, the integration of a subset of the physical qubits of a quantum computer's qubit array into a group of qubits suitable for executing quantum circuits. This group of qubits, particularly through its spatial separation, and especially its functional separation, can be considered an independent quantum computer consisting of the physical qubits of the qubit array that do not belong to this group. Functional partitioning into qubit groups specifically means that the physical qubits of different qubit groups preferably do not interact with each other. This is achieved, in particular, through the spatial separation of qubits. In particular, qubit groups can be selected to be spaced apart from each other on a qubit array. Spatial separation can be achieved, for example, by placing physical qubits that are not assigned to any qubit group (i.e., not required for the execution of quantum circuits) between qubit groups, thereby forming a kind of boundary between the qubit groups. Functional here specifically means that the physical qubits of a qubit group are particularly required for the execution of the quantum circuit encoded by that qubit group, and are thus grouped to perform a common function. The claims propose a partition into at least two qubit groups, i.e., a partition into three or more or four or more qubit groups. The measurement results of the individual measurements used to determine the first expectation value are generated in particular by a quantum computer. In particular, they are also provided by a quantum computer. The first expectation value is determined on a classical computer from the measurement results of the individual measurements.

[0040] The method according to claim 1 is particularly the following: - A step of providing at least one second expectation value of an observable that depends on a second distance size, wherein the second distance size represents the spatial separation between a first qubit group and a second qubit group, and one quantum circuit is executed on the first qubit group and the second qubit group, respectively, in parallel with each other on a quantum computer to provide a measurement result for determining the second expectation value.

[0041] The measurement results of the discrete measurements used to determine the second expectation value are generated, in particular, by a quantum computer. Specifically, they are also provided by a quantum computer. The second expectation value is determined on a classical computer, in particular, from the measurement results of the discrete measurements.

[0042] The method according to claim 1 is particularly the following: - Substeps below: - A substep that includes at least one fitting parameter and provides a fitting function that depends on the distance size; Here, curve fitting (also called curve fitting) is a technique for fitting a given mathematical model function (fitting function) as closely as possible to data points (in this case, expected values ​​that depend on the distance size). The simplest example of a fitting function is the determination of a regression line. In this case, the coefficients k and l of the linear polynomial are expressed as follows: f(d inv ,k,l)=kd inv +l In particular, the independent variable d inv This can be expressed as the reciprocal of the distance size d. The fitting parameters k and l can be determined to the values ​​that minimize the sum of the squares of the distances from the data points. - Substeps to fit the fitting parameters of the fitting function to the first and second expected values; This includes the step of determining a fitted function that includes [a specific function].

[0043] The method according to claim 1 is particularly the following: - The process includes the step of determining the error-corrected expected value of the observable by evaluating the fitted function for the case where the distance size is greater than the first distance size and greater than the second distance size.

[0044] This step is based on the consideration that the crosstalk error is distance-dependent, and since the crosstalk error between qubit groups is expected to decrease as the separation distance between qubit groups increases, an extrapolation to a distance size larger than the distance size of the expected value used in fitting is provided, meaning that the reduction of the crosstalk error can be achieved. In particular, it can be extrapolated to a virtual infinite distance size. If the fitting function is formulated as a dependence on the reciprocal of the distance metric, the expected values at various distance sizes d are plotted against the reciprocal of the distance size d inv = 1 / d. Therefore, this extrapolation corresponds to the evaluation of the fitting function when d inv = 1 / d approaches 0, that is, when the reciprocal of the distance metric approaches zero or the distance metric approaches infinity.

[0045] The method according to claim 1 particularly includes the following steps: - providing an error-corrected expected value.

[0046] This provision can be particularly carried out by data transfer, or by wireless or wired data transmission, display, output, or storage in a database, for example.

[0047] The advantage of this method is that it enables the execution of multiple quantum circuits (identical quantum circuits, at least partially identical quantum circuits, and / or different quantum circuits from each other) on an array of physical qubits of a quantum computer. In this case, hardware-specific features, such as the connectivity of the physical qubits of the qubit array caused by the quantum circuits, their respective qualities, requirements, etc., can be considered for the reduction of the crosstalk error.

[0048] In particular, this method includes providing three or more expected values for different distance sizes and considering them when determining the fitting parameters of the fitting function.

[0049] This method is based on a relaxation scheme that can also be called "infinite distance extrapolation." Here, in order to evaluate the effects of crosstalk and correct the results based on an extrapolated error term at the limit of infinite cluster distance, clusters (= qubit groups) are arranged to determine their expected values ​​at various distances, particularly increasing distances, characterized by their distance size from one another.

[0050] According to one embodiment, the distance size is given by the shortest path distance between groups of qubits, in particular the smallest shortest path distance. This shortest path distance is reduced to the Manhattan distance or Manhattan metric in a square or cubic lattice. The latter is a metric in which the distance between two points is defined as the sum of the absolute values ​​of the differences between their individual coordinates.

number

number

number

number

[0051] According to one embodiment, the distance size is given as the Euclidean distance between the geometric centroids of the qubit group.

[0052] According to one embodiment, when determining the error-corrected expectation value, if the distance size takes an infinite value, the adapted fitting function is evaluated. One advantage is that, in the limit case of infinite distance between qubit groups, crosstalk errors no longer exist.

[0053] According to one embodiment, the step of providing a first expected value and a second expected value includes the following steps: - A step of providing a first distance size and at least a second distance size; in particular, this step also includes providing three or more distance sizes. -For the first and second distance sizes, the following substeps apply: - A substep of functionally dividing the quantum computer's qubit array into a first group of qubits and at least one second group of qubits whose spatial separation is determined by a predetermined distance size; - A substep that provides a mapping of the quantum circuit to the first and second sets of qubits; - Further substeps on the quantum computer: - A further substep involves initializing a first set of qubits and executing a quantum circuit using the first set of qubits to generate a first measurement result, - A further substep involves initializing a second set of qubits and executing a quantum circuit using the second set of qubits to generate a second measurement result, Substeps that execute in parallel; - A substep to determine the expected value of an observable that depends on a predetermined distance size, using the first and second measurement results; The steps involve performing, in particular, for each distance size, multiple individual measurements, especially thousands of individual measurements, using a quantum computer, and from these numerous individual measurements, one expectation value is determined for each distance size. The latter is preferably performed on a classical computer.

[0054] The step of providing a mapping of a quantum circuit to a set of qubits includes, in particular, assigning the qubits listed in the quantum circuit to the physical qubits of a quantum computer. Furthermore, this providing step may include the initial quantum state for each set of qubits, and gate parameters for driving and controlling the quantum gates for executing the quantum circuit. This providing step can be carried out, in particular, by input, data transfer, wireless or wired data transmission, or, for example, by retrieval from a database.

[0055] Initialization of a group of qubits specifically means preparing an initial state for each group of qubits and providing control signals to drive and control the quantum gates based on the gate parameters, depending on those gate parameters, when executing the quantum circuit on a quantum computer. If a quantum gate is configured as a rotation gate that causes a rotation of one qubit by a certain angle around one of the X, Y, or Z axes of a Bloch sphere, this angle is equivalent to 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 operation of the qubits can be performed, for example, via applied voltage, magnetic field, or coupling in a microwave resonator, and therefore the control signals are configured, for example, to set the magnetic field and / or the frequency of the microwave resonator. In particular, the control signals may include electrical signals. The resulting expected values ​​benefit from the fact that crosstalk occurs at various positions in the quantum circuit and can therefore be canceled out. Thus, it is possible to achieve results that are improved, in particular, show a reduced crosstalk error compared to the expected values ​​determined from the measurement results without the insertion of delay values.

[0056] Preparing quantum states can be done, for example, by applying quantum circuits to the initial quantum state on a quantum computer. During the initialization of a quantum computer, in other words, during the setting of the initial quantum state, the quantum computer's qubits are brought to the initial quantum state. For example, all qubits are brought to state |0>, or all qubits are brought to state |1>. Alternatively, the qubits are brought to an initial qubit state sequence, i.e., the first number of qubits are brought to state |0>, the second number of qubits are brought to state |1>, and so on. In the example of a qubit state sequence consisting of five qubits, if all qubits are |0>, the result is 00000. This initialization step also includes providing control signals, particularly by quantum circuits, that are pre-set to prepare the quantum state in which the observable expectation value is to be determined. In preparing a quantum state, quantum gates manipulate the state of the quantum computer's qubits according to the quantum gates provided within the quantum circuit, thereby creating a prepared quantum state after the execution of the quantum circuit.

[0057] According to one embodiment, the fitting function includes a linear function, a polynomial function, and / or an exponential function. An example of a polynomial fitting function is given by the following equation:

number

[0058] According to one embodiment, the fitting function is a function of the reciprocal of the distance size, i.e., d inv = 1 / d. In other words, the fitting function can be expressed as a function of the reciprocal of the distance: f(1 / d).

[0059] In one embodiment, the fitting function is expressed as a function of the reciprocal of the distance: f(1 / d), and when determining the error-corrected expected value, if the reciprocal of the distance size takes the value of 0 (f(1 / ∞) → f(0)), the fitted function is evaluated. In particular, the expected value at infinite distance size is approximated by f(0).

[0060] Furthermore, a method for determining the ground state energy of a quantum system has been proposed. This method is applied to reduce errors when determining the Hamiltonian operator in the ground state of a quantum system, particularly the expectation value of the Hamiltonian operator in the quantum system, in order to obtain an error-reduced ground state energy.

[0061] In particular, the ground state energy to which it belongs can be output and / or stored. The output can be transmitted, for example, to a hybrid computer platform, classical computer, quantum computer, cloud, and / or display device. Alternatively or supplementally, the output can be used as input to further algorithms, in particular to algorithms for material simulations that examine specific material properties of quantum systems.

[0062] Finding the ground state of a quantum mechanical system is an important challenge in relation to atomic materials simulations and in the field of quantum chemistry.

[0063] According to one embodiment, the quantum system whose ground state energy is to be determined is, for example, a many-particle system that can be described by the Hubbard-Hamiltonian operator. The Hubbard model is an approximate model of solids. It describes the behavior of electrons in a lattice that is considered a rigid body. In this model, repulsive Coulomb forces are considered only for electrons remaining in the same lattice position. The proportion of the electron's kinetic energy is modeled by an overlap integral derived from a strongly coupled model. Some examples of quantum systems that can be described by the Hubbard-Hamiltonian operator include strongly correlated fermionic systems, transition metals, mobile electron systems (e.g., ferromagnetism, antiferromagnetism, ferrimagnetism), and π-electron systems in quantum chemistry. Therefore, the advantage of this method is that it can accelerate the development and research of new materials, and in some cases make it possible for the first time. Furthermore, it allows for better adaptation of the properties of these new materials to their respective applications.

[0064] The aforementioned advantages also arise when the above method is used for material simulations, in which case the quantum system's Hamiltonian operator is a multi-particle Hamiltonian operator that describes the material to be simulated.

[0065] A hybrid computing platform, an example of a quantum computing platform with noise, includes a classical computer and a quantum computer that runs quantum circuits, which are adapted so that the steps of the method described above are mandated and / or so that the use of the method is mandated for materials simulation, which has the advantage of being able to be used efficiently, in particular for materials simulation. Adaptation can be understood as meaning, for example, that the hardware of the quantum computer can be tuned to a quantum circuit, thereby making it possible to map the logic qubits of the quantum circuit to the physical qubits of the quantum computer, preferably this can be accomplished with as few and as close to zero additional swap operations as possible to ensure the interaction of the physical qubits assigned to the logic qubits, for example when performing a two-gate operation. In particular, the hardware of the quantum computer can be selected based on the provided quantum circuit, thereby enabling the execution of a more efficient and noise-resistant method. A hybrid computing platform can be understood in particular as including at least one classical computer and at least one quantum computer. Preferably, the hybrid computing platform includes a cloud computing platform accessible to the classical computer, from which measurement results from the quantum computer can be retrieved, in particular. In particular, quantum computers have more than 400 physical qubits. Specifically, quantum computers are configured to execute at least two or more quantum circuits in parallel when called.

[0066] In one embodiment, a NISQ computer is used as the quantum computer.

[0067] Furthermore, an error reduction algorithm is proposed that causes a hybrid computer platform, including a classical computer and a quantum computer for executing quantum circuits, to perform at least one of the methods described above.

[0068] A computer-readable storage medium containing an error reduction algorithm, which is an implementation of the aforementioned method, can be driven and controlled by a quantum computer and / or classical computer, in particular, for providing the error reduction algorithm.

[0069] Embodiments of the present invention are shown in the drawings and will be described in more detail in the following specification. Identical reference numerals in the figures represent elements that function identically or equivalently. [Brief explanation of the drawing]

[0070] [Figure 1] This is a schematic diagram showing the connectivity graph for the physical qubits of a 127-qubit quantum computer, where the qubit group according to the first embodiment is shown. [Figure 2] This is a schematic diagram showing the connectivity graph for the physical qubits of a 127-qubit quantum computer, which is represented by the qubit group according to the second embodiment. [Figure 3] This flowchart shows a method for determining the error-corrected expectation value of an observable quantity in a quantum system. [Figure 4] This is a schematic diagram illustrating a hybrid computer platform. [Modes for carrying out the invention]

[0071] Examples of the Invention Figure 1 shows a schematic diagram of a hardware-specific connectivity graph 200 of the physical qubits 0, ..., 126 of a quantum computer. In this embodiment, the physical qubits 0, ..., 126 are arranged in a heavy hex grid, for example, as used in IBM®'s superconducting quantum computer (a 127-qubit Eagle processor). The physical qubits are numbered sequentially in this diagram, row by row. The connectivity graph belonging to the quantum computer's qubit array contains a total of 127 physical qubits 0, ..., 126, which are shown as circles or nodes in the schematic diagram and are numbered sequentially for identification. Nodes of physical qubits 0, ..., 126 configured to interact with each other are connected to each other, particularly by edges, in the hardware-specific connectivity graph. In other words, the connectivity graph 200 provides information about the number and connectivity of the physical qubits 0, ..., 126.

[0072] In this embodiment, a quantum circuit with 12 physical qubits should be implemented. In principle, such a quantum circuit can be implemented 10 times in parallel on an Eagle processor with 127 qubits. This corresponds to n=10 clusters (= qubit groups) on a connectivity graph. However, such a dense collection of qubit groups can lead to significant crosstalk errors. Therefore, the number of qubit groups to be implemented is selected to be smaller, specifically 5. This reduces the crosstalk between qubit groups 201, 202, 203, 204, and 205. There are many different ways to map these clusters onto a connectivity graph.

[0073] In this embodiment, five qubit groups 201, 202, 203, 204, and 205 are arranged on the physical qubits 0, ..., 126 of the quantum computer as follows: - The first group of qubits, 201, includes physical qubits numbered 0, 1, 2, 3, 4, 14, 15, 18, 19, 20, 21, and 22. - The second group of qubits, 202, includes physical qubits numbered 8, 9, 10, 11, 12, 16, 17, 26, 27, 28, 29, and 30. - The third group of qubits, 203, includes physical qubits numbered 62, 63, 64, 65, 66, 72, 73, 81, 82, 83, 84, and 85. -The fourth group of qubits, 204, includes physical qubits numbered 96, 97, 98, 99, 100, 109, 110, 114, 115, 116, 117, and 118. -The fifth group of qubits, 205, includes physical qubits numbered 104, 105, 106, 107, 108, 111, 112, 122, 123, 124, 125, and 126.

[0074] Overall, each qubit group 201, 202, 203, 204, and 205 has the same number of physical qubits 0, ..., 126, specifically 12 physical qubits each. Furthermore, independent qubit groups have the same structure with respect to the connectivity of the physical qubits 0, ..., 126 they contain. At the center of each qubit group 201, 202, 203, 204, and 205, their respective geometric centroids 2002 are marked, and in this case, exemplarily, several distance sizes 2001, here Euclidean distances (=L2 distances), of the geometric centroids are plotted.

[0075] Since each of these qubit groups can execute independent quantum circuits in parallel, when the quantum computer in Figure 1 is invoked, five individual measurements of observable quantities can be performed, and therefore, measurement results from these five individual measurements can be provided.

[0076] Figure 2 differs from Figure 1 only in terms of distance size. In this embodiment, the shortest path metric 2000 between the qubit group 201, 202, 203, 204, and 205 is represented.

[0077] Figure 3 shows a flowchart of Method 300 for determining the error-corrected expectation value of an observable quantity in a quantum system using a classical computer, and the steps are as follows: -Step 301 provides a first expectation value 3011 of an observable that depends on a first distance size 2000,2001, where the first distance size 2000,2001 represents the spatial separation between a first qubit group 201 and a second qubit group 202, and step 301 is performed on a quantum computer, with one quantum circuit each running in parallel with the first qubit group 201 and the second qubit group 202, in order to provide a measurement result for determining the first expectation value. - Step 301 provides at least one second expectation value 3012 of an observable that depends on a second distance size 2000,2001, where the second distance size 2000,2001 represents the spatial separation between the first qubit group and the second qubit group, and step 301 is performed on a quantum computer, with one quantum circuit each running in parallel with the first qubit group and the second qubit group to provide a measurement result for determining the second expectation value. - Substeps below: - Substep 3020 provides a fitting function that includes at least one fitting parameter and depends on distance size 2000,2001, - Substep 3021, which adapts the fitting parameters of the fitting function to the first expected value 3011 and the second expected value 3012, Step 302 includes determining the adapted fitting function 3022, -Step 303 determines the error-corrected expected value of the observable by evaluating the fitted function 3022 in the case where the distance size 2000,2001 is greater than the first distance size 2000,2001 and greater than the second distance size 2000,2001. - Step 304 provides the error-corrected expected value 3040, Includes.

[0078] In particular, the steps to provide the first expected value 3011 and the second expected value 3012 are as follows: - A step of providing a first distance size and at least a second distance size, -For the first distance size, the following substeps: - A substep of functionally dividing the qubit array of a quantum computer into a first group of qubits and at least one second group of qubits whose spatial separation is determined by a first distance size, - A substep that provides a mapping of the quantum circuit to the first and second sets of qubits, - Further substeps on the quantum computer: - A further substep involves initializing the first set of qubits 201 and executing a quantum circuit using the first set of qubits 404 to generate the first measurement result, -A further substep involves initializing the second set of qubits 202 and executing a quantum circuit using the second set of qubits to generate a second measurement result, Substeps that execute in parallel, - A substep to determine the expected value of an observable that depends on the first distance size using the first and second measurement results, The steps to perform, -For the second distance size, the following substeps: - A substep of functionally dividing the qubit array of a quantum computer into a first group of qubits and at least one second group of qubits whose spatial separation is determined by a second distance size, - A substep that provides a mapping of the quantum circuit to the first and second sets of qubits, - Further substeps on the quantum computer: - A further substep involves initializing the first set of qubits 201 and executing a quantum circuit using the first set of qubits 404 to generate the first measurement result, -A further substep involves initializing the second set of qubits 202 and executing a quantum circuit using the second set of qubits to generate a second measurement result, Substeps that execute in parallel, - A substep to determine the expected value of an observable that depends on the second distance size using the first and second measurement results, The steps to perform, - The step of performing the above steps for a further distance size different from the first and second distance sizes, It may include.

[0079] Figure 4 is a schematic diagram of a hybrid computer platform 400, which is an example of a noisy quantum computer platform capable of performing Method 400 as illustrated in Figure 3. This hybrid computer platform 400 includes a classical computer 501 and a quantum computer 402 adapted to perform Method 400 as illustrated in Figure 3 and described above. In particular, the hybrid computer platform outputs an error-corrected expectation value 3040, for example, the ground state energy of the quantum system.

Claims

1. A method (300) for determining the error-corrected expectation value of an observable quantity in a quantum system using a classical computer (401), comprising the following steps: - Step (301) provides a first expectation value (3011) of an observable that depends on a first distance size (2000, 2001), wherein the first distance size (2000, 2001) represents the spatial separation between a first qubit group (201) and a second qubit group (202), and step (301) is performed on a quantum computer with one quantum circuit each running in parallel with the first qubit group (201) and the second qubit group (202) in order to provide a measurement result for determining the first expectation value. - Step (301) provides at least one second expectation value (3012) of an observable quantity that depends on a second distance size (2000, 2001), wherein the second distance size (2000, 2001) indicates a spatial separation between the first qubit group and the second qubit group, and one quantum circuit is executed on the quantum computer on the first qubit group and the second qubit group in parallel with each other to provide a measurement result for determining the second expectation value; - Substeps below: - A substep (3020) that provides a fitting function that includes at least one fitting parameter and depends on the distance size (2000, 2001), - A substep (3021) to adapt the fitting parameters of the fitting function to the first expected value (3011) and the second expected value (3012), The steps include: determining a fitted function (3022) including, - In the case where the distance size (2000, 2001) is greater than the first distance size (2000, 2001) and greater than the second distance size (2000, 2001), the error-corrected expected value of the observable is determined by evaluating the adapted fitting function (3022) (303), - Step (304) of providing the error-corrected expected value (3040), Method (300), including the method (300).

2. When determining the error-corrected expected value (3040), if the distance size (2000, 2001) takes a value that is infinite, the fitted function (3022) is evaluated. The method according to claim 1 (300).

3. The step of providing the first expected value (3011) and the second expected value (3012) is as follows: - The step of providing the first distance size (2000, 2001) and at least the second distance size (2000, 2001), - With respect to the first distance size (2000, 2001) and the second distance size (2000, 2001), the following substeps apply: - A substep of functionally dividing the qubit array of the quantum computer into a first group of qubits and at least one second group of qubits whose spatial separation is determined by a predetermined distance size (2000, 2001), - A substep that provides a mapping of the quantum circuit to the first group of qubits and the second group of qubits, - Further substeps on the aforementioned quantum computer: - A further substep of initializing the first group of qubits (201) and executing a quantum circuit using the first group of qubits to generate a first measurement result, - A further substep of initializing the second group of qubits (202) and executing a quantum circuit using the second group of qubits to generate a second measurement result, Substeps that execute in parallel, - A substep of determining the expected values ​​(3011, 3012) of observable quantities that depend on the preset distance sizes (2000, 2001) using the first and second measurement results, The steps to perform, including, The method according to claim 1 or 2 (300).

4. The fitting function includes linear functions, polynomial functions, and / or exponential functions. The method according to any one of claims 1 to 3.

5. The fitting function is a function of the reciprocal of the distance size (2000, 2001). The method according to any one of claims 1 to 4.

6. When determining the error-corrected expected value (3040), if the adapted fitting function (3022) has a reciprocal of the distance size of zero, the adapted fitting function (3022) is evaluated. The method according to claim 5.

7. A method for determining the ground state energy of a quantum system, wherein the method according to any one of claims 1 to 6 is applied to reduce errors in determining the expectation value of the Hamiltonian operator of the quantum system in the ground state of the quantum system in order to obtain an error-reduced ground state energy.

8. The aforementioned quantum system is described by the multi-particle Hamiltonian operator of matter. The method according to any one of claims 1 to 7.

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

10. The Hamiltonian operator of a quantum system is a multi-particle Hamiltonian operator that describes the matter to be simulated. Use of the method according to any one of claims 1 to 9 for material simulation.

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

12. The aforementioned quantum computer (402) is a NISQ computer. The hybrid computer platform (400) according to claim 11.

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

14. A computer-readable storage medium storing the error reduction algorithm described in claim 13.

Images