Method for the functional partitioning of a qubit arrangement of a quantum computer, method for parallel running of at least one first quantum circuit and one second quantum circuit on a quantum computer, and use of the method for determining a ground state of a quantum system

Abstract

The invention relates to a method (300) for the functional partitioning of a qubit arrangement of a quantum computer: providing (301) information relating to at least one first quantum circuit and a second quantum circuit, which are provided for parallel running on the quantum computer; providing (302) a number of qubit groups specifying the number of quantum circuits that can be run in parallel; providing (303) an optimization function, which depends on a distance measurement between the qubit groups in the qubit arrangement; optimizing (304) the optimization function with regard to the distance measurement between the qubit groups; providing the positions of the qubit groups in the qubit arrangement. The invention also relates to a method for the parallel running of at least one first quantum circuit and one second quantum circuit on a quantum computer.

Description

[0001] R. 409940

[0002] Description

[0003] title

[0004] Methods for the functional partitioning of a qubit arrangement of a quantum computer, methods for parallel execution on a quantum computer, use of the method for determining a ground state

[0005] State of the art

[0006] In “Variational approach-based guantum simulation of imaginary time evolution” (McArdle et al., npj Quantum Information (2019) 5:75; https: / / doi.org / 10.1038 / s41534-019-0187-2) a method for determining a ground state of a many-body system is described, which uses a hybrid system of quantum computer and classical computer.

[0007] Currently available quantum computers, especially those based on superconducting circuits, show an increase in the available number of physical qubits. IBM's currently available "Osprey" quantum computer chip comprises 433 qubits. This is even surpassed by IBM's "Condor" chip with 1121 qubits.

[0008] Currently available quantum computers are not fully error-corrected, resulting in errors and noise inherent in any quantum circuitry running on these computers. R. 409940

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[0010] 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 limited size (small number of physical qubits) and inherent gate errors, such noisy NISQ computers only allow the execution of short quantum circuits, i.e., shallow quantum circuits, and the results typically exhibit fairly large error bars. An example of such a shallow quantum circuit is hybrid quantum-classical algorithms, such as the variational quantum eigensolver (VQE).

[0011] However, variation algorithms such as VQE cannot efficiently exploit this large number of qubits due to challenges arising from the inherent noise.

[0012] Key and advantages of the invention

[0013] Finding the ground state of a quantum mechanical system is an important task in atomistic materials simulations and in the field of quantum chemistry. Many algorithms using classical computers have been developed in the past to address this problem. However, the properties of materials cannot be calculated with sufficient accuracy on conventional high-performance computers.

[0014] An example of algorithms for solving quantum problems that utilize quantum computers are variational quantum algorithms, in particular variational quantum eigensolvers (VQEs), which are based on a hybrid quantum-classical approach. These are used, for example, in determining a ground state of a quantum system (i.e., the state with the lowest energy). (See R. 409940)

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[0016] In this approach, a quantum state (e.g., a wave function) is encoded with a variational approach in a quantum circuit with variation parameter 0^ and the corresponding expectation value E k The Hamiltonian operator (measurable observable), which describes or approximates the quantum system, is measured. This measurement requires many (on the order of thousands) individual measurements (shots) to obtain a statistically well-converged expectation value.

[0017] After an expected value has been determined, a classical method is used to update the variation parameters. For the VQE, for example, the parameters are updated along a downward direction to minimize the expected value (energy) from the quantum circuit. The updated parameters k+1 are then fed back to the quantum computer to process E k+1to obtain. This iterative process is repeated until the converged ground state energy E is reached. n The quantum state of iteration n is obtained with its corresponding quantum state. The quantum state of iteration n is encoded in the variation parameters 0^ and corresponds to the ground state of the quantum system, or rather, describes an approximation of the ground state.

[0018] The Variational Quantum Eigensolver (VQE) is used particularly in quantum chemistry and materials science. It is a hybrid algorithm that utilizes both classical and quantum computers. It is used to determine ground-state energies and wave functions of complex quantum systems and provides insights into complex molecular and material behaviors. The VQE computes the expectation value of a parameterized circuit and optimizes the parameters to minimize the energy. The global energy minimum is then considered a good approximation of the ground-state energy. Optimization in the VQE becomes challenging for larger systems, as the R. 409940

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[0020] Optimizers can get stuck in local minima or in so-called bar plateaus with vanishing gradients. Using a quantum computer, the system's expectation value with respect to an observable, for example, the system's Hamiltonian operator, is determined, and a classical optimizer is used to improve the parameters of the approach.

[0021] VQE combines quantum computing with classical optimization techniques. For this purpose, a Hamiltonian operator is provided that describes properties of the investigated quantum system, such as the electron configuration of a 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 key to VQE lies in its hybrid methodology. A classical optimizer is used to adjust or optimize the parameters of the approach. The goal is to minimize the expectation value of the Hamiltonian operator's energy, which is crucial for reaching the system's ground state.

[0022] In this interplay between quantum computer and classical computer, VQE iterates between measuring the expectation value of the Hamiltonian operator on the quantum computer and using classical algorithms to optimize the quantum circuit parameters (=variation parameters).

[0023] Another problem with using quantum algorithms for a large number of qubits arises from the increasing challenges with system size. For VQE, for example, this includes the number of variation parameters.

[0024]

[0025] ^, generally denoted by a set of angles 0 which are unfavorably related to the R. 409940

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[0027] The complexity of the approach and the number of qubits scale. Furthermore, the inherent noise of NISQ devices leads to inefficient gradient-based optimizations.

[0028] Particularly due to the aforementioned problems, as well as errors inherent in NISQ computers and the limitations of classical optimization algorithms, such variational quantum-classical approaches can only be used for quantum circuits of low depth and only for a few dozen variational parameters.

[0029]

[0030] ^ and a small number of physical qubits. Therefore, the power of quantum computers that include a large number of physical qubits, such as "Osprey" (433 physical qubits) or "Condor" (1121 physical qubits), cannot be fully / efficiently utilized. In practice, only a small number of the quantum computer's physical qubits are used to encode the quantum circuit, and the remaining qubits remain inactive during the quantum circuit's execution on the quantum computer.

[0031] Furthermore, another problem arises from the large number of shots (individual measurements) required to obtain statistically well-converged measurement results when designing quantum circuits.

[0032] The problems mentioned above essentially limit the potential for using an ever-increasing number of physical qubits on quantum computers, especially NISQ computers. One way to utilize more of these qubits is to encode multiple small quantum circuits in several clusters / qubit groups on a single quantum computer. These smaller quantum circuits can be identical, so this solution allows for parallelization over the number of individual measurements. On R. 409940

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[0034] In this way, many individual measurements can be performed in parallel by executing these identical quantum circuits in a single pass.

[0035] Alternatively or additionally, these smaller quantum circuits can be different from each other, for example to measure different observables in parallel, or to obtain gradients along different directions, as are needed for VQE algorithms, for example.

[0036] Although this does not necessarily enable the processing of larger systems, it reduces the overall computing time for smaller systems.

[0037] The main problem with the parallelization solutions mentioned above is the occurrence of crosstalk between qubit groups, leading 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 below to spatially separate the clusters on the quantum computer to reduce crosstalk errors, thereby enabling the reduction of noise in individual measurements on quantum computers with many, especially more than 400 physical qubits, with efficient computation time.

[0038] The present invention therefore proposes methods which

[0039] • making use of a larger number of physical qubits of a quantum computer and

[0040] • enables the parallelization of the execution of quantum circuits, in particular quantum circuits of a VQE, by using multiple clusters of physical qubits (= subsets of the physical qubits of the quantum computer; hereinafter also referred to as qubit groups R. 409940

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[0042] (designated) are determined, which are each suitable for executing a quantum circuit.

[0043] make possible.

[0044] Advantageously, the invention enables more efficient use of quantum computers with a large number, in particular more than 400 physical qubits, by allowing multiple quantum circuits to be executed in parallel on the quantum computer, thus increasing the number of measurement results per pass with the number of clusters / qubit groups used, or reducing the number of passes of the quantum computer required to generate the same number of measurement results.

[0045] In other words, the present invention advantageously enables the utilization of a large number of qubits by allowing many smaller quantum circuits to be executed in parallel on a large quantum computer, for example, to reduce the noise of individual measurements. Crosstalk errors are particularly reduced by spatially separating clusters of parallel-encoded small quantum circuits on the same quantum computer.

[0046] A quantum circuit is a computational routine built from coherent quantum operations. Each horizontal line or wire in a quantum circuit represents a qubit, with the left end of the wire representing the original quantum data and the right end the final quantum data generated by the quantum circuit's computation. Operations on qubits are represented by boxes placed on these wires.

[0047] Quantum gates are the elementary operations that a quantum computer can perform on its qubits. They are R. 409940

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[0049] Comparable to electronic gates, which perform the elementary operations of a classical computer. For quantum gates operating on two qubits (2-qubit gates), an interaction between the physical qubits in question is required. In the case of spin qubits, this can occur, among other things, via exchange interactions. Atoms in an ion trap, for example, can exchange photons. In the case of qubits based on superconducting circuits, the manipulation of these qubits can be achieved, for example, via the applied voltage, the magnetic field, or coupling to microwave resonators. In the following, "quantum circuit" refers to a quantum circuit for physical qubits. These quantum circuits may contain additional quantum gates, especially swaps, compared to quantum circuits for logic qubits, which may be further differentiated within the quantum circuit for physical qubits.to be added to transport logical qubits to interacting physical qubits when they are involved in a joint quantum operation.

[0050] The invention relates to a method for the functional partitioning of a qubit array of a quantum computer, a method for the parallel execution of at least one first quantum circuit and one second quantum circuit on a quantum computer, and a use of the method for determining a ground state of a quantum system. In particular, the invention enables the efficient use of the available physical qubits of available quantum computers. Specifically, it enables the physical qubits of the qubit array to be partitioned into qubit groups, each of which can be used to execute a quantum circuit. In particular, this may (but not necessarily) allow identical quantum circuits to be executed in parallel. This reduces the number of runs of the quantum computer while maintaining the same R. 409940

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[0052] Number of measurement results. In this way, one can parallelize over shots (individual measurements), observables or gradient descent directions for a VQE, which leads to a significant advantage when using a large number of physical qubits on NISQ computers.

[0053] The invention, including the features of the independent claims, which enables the advantages described above, is disclosed below.

[0054] This is achieved by a method according to claim 1 for the functional partitioning of a qubit arrangement of a quantum computer. Quantum computers programmable using quantum circuits can, in principle, be constructed from any quantum technology capable of performing single- and multi-qubit gate operations. Currently, architectures based, for example, on superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are being actively developed.

[0055] A quantum computer comprises a qubit arrangement, wherein the qubit arrangement comprises several physical qubits, which are preferably initialized by means of devices or units adapted to the technology with which the qubits are realized (e.g.

[0056] They may be equipped for initializing the qubit in a basic state), for manipulating (e.g., applying 1-qubit and / or 2-qubit gates) and / or for reading out the physical qubits.

[0057] Functional partitioning can be understood in particular as combining a subset of physical qubits of the quantum computer's qubit arrangement into a qubit group suitable for implementing a quantum circuit, and characterized, in particular, by its spatial arrangement, but especially by its R. 409940

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[0059] functional separation from the physical qubits of the qubit arrangement that do not belong to this qubit group, as a quantum computer independent of these qubits can be considered.

[0060] The method according to claim 1 includes in particular the following steps:

[0061] • Providing information on at least one first quantum circuit and one second quantum circuit intended for parallel execution on the quantum computer; in particular, this information may include the number of required physical qubits and their connectivity for each quantum circuit. If more than two quantum circuits are to be executed in parallel, this step provides the corresponding information for all quantum circuits intended for parallel execution on the qubit array. If the quantum circuits are at least partially identical, providing the information for one of the identical quantum circuits also provides the information for the other identical quantum circuits.

[0062] In this step, it is possible, but not necessary, to provide the quantum circuits themselves. The provision of information can be achieved, in particular, by input, by data transmission (wireless or wired data transmission), or by retrieval, for example, from a database. Providing the information makes available, in particular, the information required for the process regarding the number of physical qubits needed per quantum circuit and their respective required connectivity. Specifically, providing information on at least one first quantum circuit and one second quantum circuit can also include providing information on more than two quantum circuits. R. 409940

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[0064] • Providing a number of qubit groups, which determines the number of quantum circuits that can be executed in parallel; this number can be provided, in particular, by input, by data transmission (wireless or wired data transmission), or by retrieval, for example, from a database. It should be noted that the larger the number of qubit groups, the higher the risk of crosstalk. The method described above makes it possible to find a solution with reduced risk under the given constraints (requirements of the quantum circuits, number of qubit groups). An upper limit for the number of qubit groups is given by the number of physical qubits provided by the qubit array for the execution of quantum circuits.In particular, the sum of the physical qubits of all qubit groups should be smaller than the number of physical qubits provided by the qubit array for the execution of quantum circuits.

[0065] In particular, the sum of the physical qubits of all qubit groups should be significantly smaller than the number of physical qubits provided by the qubit array for the execution of quantum circuits. For example, the sum of the physical qubits of all qubit groups should be less than or equal to 50% of the number of physical qubits provided by the qubit array for the execution of quantum circuits.

[0066] • A qubit group is understood to be, in particular, a group or cluster of physical qubits of a qubit arrangement that are functionally, and especially also spatially, separated from the other physical qubits of the qubit arrangement. In particular, the physical qubits of the qubit group are suitable for executing a quantum circuit; in particular, the connectivity of the physical qubits of the qubit group is such that they are configured to execute the quantum circuit. Furthermore, the interaction of the qubits with the other qubits of the qubit arrangement is considered to be separate from the interaction with the R. 409940

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[0068] The optimization function S, which depends on a distance measure between the qubit groups on the qubit array, is provided. The optimization function is based on the following assumption: reducing or minimizing the interaction between the qubit groups reduces crosstalk. In other words, the optimization function takes into account the idea that the crosstalk error decreases with decreasing connectivity or increasing spatial distance between the qubit groups. The distance measure can, in particular, be a function of the spatial distance, for example, the position of the geometric center or the center of mass of the physical qubits of the qubit group. Alternatively or additionally, the distance measure can comprise the sum of the minimum distances between the clusters, taking qubit connectivity into account.

[0069] Alternatively or additionally, the fidelity of the qubits used in the individual qubit groups and their connectivity can be taken into account as additional parameters in the optimization function, especially to avoid highly noisy qubits.

[0070] • Optimizing the optimization function with respect to the distance measure between the qubit groups; in particular, this step may include maximizing the optimization function, especially maximizing the distance measure.

[0071] • Providing the positions of the qubit groups on the qubit array. In particular, the geometric centers or centers of mass of the qubit groups can be provided, specifically output, stored, transmitted, or displayed. For example, the positions of the qubit groups and / or the arrangements of the qubit groups can be provided by means of a connectivity graph of the physical qubits of the quantum computer. In particular, the quantum computer has only limited connectivity of the physical qubits, i.e., not every physical qubit can be connected to every other physical qubit. R. 409940

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[0073] Qubits interact. In other words, quantum operations, especially 2-qubit gates, can only be performed between specific physical qubits. This connectivity of physical qubits can be represented by a hardware-specific connectivity graph. The connectivity graph includes, in particular, information about the number of physical qubits and their connectivity. Specifically, the physical qubits are represented as nodes in the hardware-specific connectivity graph, and nodes of physical qubits configured to interact with each other are connected by edges. In other words, the connectivity graph provides information about which physical qubits can interact with which other physical qubits.For example, each physical qubit of the quantum computer in the connectivity graph can be assigned an identifier, such as a number uniquely attributable to the respective physical qubit. In this case, providing the positions of the qubit groups on the qubit array can be accomplished by specifying the identifiers of the physical qubits belonging to a common qubit group.

[0074] One advantage of the method is that it uses only generic basic information, in particular the number of physical qubits required per quantum circuit, and thus applies to an entire class of quantum circuits and is not problem-specific. In other words, the method can be run once for a quantum computer hardware and a desired number of qubit groups and predefined cluster sizes, for example, identical qubit groups of n qubits each, to find a partition that reduces cross-talk errors, after which each circuit with n qubits can be executed in parallel with the same trivial parallel acceleration on this quantum computer hardware. Similarly, R. 409940

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[0076] Of course, different sizes of qubit groups must also be taken into account.

[0077] If no result can be obtained during optimization, for example because the number of qubit groups was chosen too large (for example, if the sum of all physical qubits of the qubit groups is greater than the total number of physical qubits of the qubit arrangement), the number of qubit groups can be reduced according to one embodiment, in particular by the value one, and optimization can be carried out again with the reduced number.

[0078] According to one embodiment, the optimization function can take into account the structure of the quantum circuits of the different qubit groups that are to be executed in parallel. In particular, differing quantum circuits can have different optimal (local) assignments to the physical qubits of the qubit array. For example, the spatial arrangement to which a cluster c iIdeally, the coded format can vary. To account for this, the following approaches are possible, for example:

[0079] 1) Cluster-first strategy: Here, the individual clusters are selected first. i The qubit array is mapped to a local arrangement of physical qubits and symmetrically equivalent locations are sought by restricting the spatial optimization of S accordingly.

[0080] 2) Partition-First Strategy: Here, the spatial distribution, in other words the positions of the cluster locations, on the arrangement of physical qubits of the qubit array is first identified by optimizing the optimization function. Subsequently, the clusters c i Locally optimally mapped. R. 409940

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[0082] 3) Hybrid-optimal strategy: Optimization of the local structure of the clusters c i together with the optimization function.

[0083] One advantage of this method is that it allows mapping multiple quantum circuits to an array of physical qubits on a quantum computer, taking into account hardware-specific characteristics such as the connectivity of the physical qubits in the array, their respective quality, and the requirements imposed by the quantum circuits. Advantageously, this method reduces the total number of quantum computer calls by a factor of n, where n is the number of identical qubit groups assigned to the QPU, because the same statistical noise level is achieved for each individual measurement compared to implementing only one cluster at a time on the quantum computer. In other words, one quantum computer call yields n individual measurements of n identical qubit groups, whereas without the described parallelization approach, n calls would be required.

[0084] According to one embodiment, the optimization function comprises a sum of distance measures between qubit groups. The ideal mapping of the qubit groups to the physical qubits of the quantum computer should minimize the interactions between the qubit groups in order to reduce crosstalk errors, thereby improving the reliability of the quantum computer's measurement results.

[0085] According to one embodiment, the distance measure comprises a Euclidean distance between the geometric centroids of qubit groups. R. 409940

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[0087] The underlying idea of ​​this embodiment is based on maximizing the distance between all clusters c. k The distance to the clusters for the qubit group c j This refers in particular to the Euclidean distances of qubit group c. jand all other qubit groups. This is achieved in particular by maximizing the sum of the Euclidean L2 distances between the center of mass or geometric center of mass.

[0088]

[0089] , i.e. by maximizing the following optimization function:

[0090] c l mc k)

[0091]

[0092] where the second sum runs over all elements i of dimension n of the qubit topology. In particular, n = 2 for a two-dimensional quantum chip architecture.

[0093] One advantage is that this allows for spatial separation of the qubit groups, thus reducing crosstalk errors. The core assumption of this idea is that the crosstalk error decreases with increasing spatial distance between the qubit groups.

[0094] According to one embodiment, the distance measure, which is based on the Euclidean distance between the geometric centroids of the qubit groups, can be modified such that only distances within a cutoff radius r are considered. cut The variable factor is incorporated into the optimization function, while otherwise remaining constant. Here, the distance d is used. kj between all clusters c k and c j defined as

[0095] \

[0096] m r cl k, — m r

[0097]

[0098] cl 2

[0099] j J]

[0100] • d k j = r cut = const otherwise

[0101] And thus the optimization function R. 409940

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[0103] s — d-lk

[0104]

[0105] Ik

[0106] short-range, and the gradient contributions for all terms with d kj > r cut disappear. This is advantageous when a gradient-based optimization method is used to maximize S.

[0107] According to one embodiment, S is not maximized using gradient-based optimization, so the optimization can be reduced to a MaxMin diversity problem. In this embodiment, assume that all possible positions p of a cluster c are uniquely numbered by an index i. From these sets of possible cluster positions, m clusters are to be selected and optimally distributed. Let x i = 1 if this cluster positioning is selected, and x i = 0 otherwise. Then the following applies:

[0108] Maximize S

[0109]

[0110] = d tk x t x k

[0111] Under the boundary condition = m

[0112] Methods for solving the MaxMin diversity problem include:

[0113] • GRASP-based methods

[0114] • Local search methods, such as Iterated Tabu Search (ITS) or Variable Neighborhood Search (A_VNS)

[0115] • Population-based methods, such as the Scatter Search method (G_SS) or Memetic Algorithms (MA)

[0116] According to one embodiment, the optimization function comprises a sum of distance measures between the qubit groups, where the distance measure is the minimum shortest-path distance between qubit groups. The shortest-path distance reduces to the Manhattan distance or Manhattan metric in a square or cubic lattice and is a metric in which the distance between two points is defined as the sum of the absolute differences of their individual coordinates. This embodiment is based on maximizing the R. 409940

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[0118] Sum of the minimum distances between qubit groups with respect to qubit connectivity. Mathematically, this means, for example, with quadratic or cubic lattice connectivity, the sum of the minimum Manhattan norms between the clusters c. k to maximize the Manhattan norm dq,p) between two points q = (c^, q2,.., q n ~) and Q = (Pi> Pn) is defined as

[0119] n

[0120] d(q,p) = ^\qt - Pi\

[0121]

[0122] i=l

[0123] Therefore, the optimization problem to be solved is the maximization of the optimization function:

[0124] 'd min (cc k )

[0125] lk

[0126] of the minimum distance d min (c b c k~) between all qubit groups. If the qubit connectivity for some qubits deviates from a square / cubic lattice, the Manhattan norm may need to be adjusted accordingly by taking the missing connectivitys into account, either using a breadth-first search or depth-first search algorithm. From this, the minimum distances d can be determined. min (c b c k ~) with reduced connectivity as the sum of the individual path lengths. For the most general case, physical qubits and their connectivity can be represented as an undirected graph, which in principle allows for various shortest-path algorithms to determine d. min (c b c k ~) can be used, such as the Dijkstra algorithm.

[0127] According to one embodiment, the optimization function for at least some of the qubit groups includes a contribution resulting from a weighted sum of the qubit quality factors of the respective qubit group. In addition to maximizing the distance norms S as an optimization function, a further term can be added to the optimization function that takes the qubit quality factor into account. The physical R. 409940

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[0129] The qubits of a quantum computer often differ in their quality, particularly in their noise immunity. In other words, the qubit array of the quantum computer may include physical qubits with varying fidelities. This can be taken into account, for example, when arranging the qubit groups by adding a weighted term F to the optimization function. An example of such a weighted term is:

[0130] F = ak ifCqi).

[0131] where a is a scalar weighting factor and

[0132]

[0133] the sum of the qualities of all physical qubits

[0134]

[0135] of the qubit group k. This makes it advantageously possible to favor optimization solutions that use physical qubits with better accuracy, thus enabling more reliable and, in particular, less noise-prone single measurements. The accuracy of a physical qubit can be evaluated based on one or more of the following parameters: single-qubit accuracy, two-qubit gate accuracy, or the inverse of the readout error.

[0136] According to one embodiment, the optimization of the optimization function with respect to the distance measure between the qubit groups is terminated after a predefinable number of iterations, and the solution obtained in the last iteration step is used to provide the positions of the qubit groups on the qubit array. In general, maximizing the optimization function is non-trivial (potentially NP-hard, also called a minmax problem). The advantage, however, is that to achieve the aforementioned benefits, it is not necessary to find the best, i.e., in this case, the maximum, solution of the optimization function. The advantages arise even for a sufficiently good solution, in other words, a solution that can be found within a few, for example, 10 to 10,000 iterations of a gradient-free R. 409940

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[0138] Optimization schemes can be found. Even with such a solution, crosstalk errors can be reduced to such an extent that the reduction in noise in individual measurements exceeds the cost of maximizing the aforementioned optimization function.

[0139] According to one embodiment, a starting point for optimization is chosen as follows. It is performed

[0140] • Dividing the qubit array of the quantum computer into blocks. In particular, the qubit array is divided into as many blocks as the number of qubit groups. Specifically, a block comprises at least a number of physical qubits required to execute the quantum circuits, according to the provided information on the quantum circuits. In particular, at least some of the blocks comprise more physical qubits than the qubit group to be placed within the block; and • placing one qubit group per block.

[0141] One advantage is that a spatial separation of the qubit groups can be chosen as a starting point, in the form of an educated guess, thus simplifying or accelerating the optimization of the optimization function.

[0142] As a starting point for optimization, one can choose a trivial solution by dividing the physical qubits of the quantum computer into a regular grid and placing the clusters into separate blocks.

[0143] Alternatively, any arrangement of the qubit groups can be chosen as the starting point.

[0144] According to one embodiment, all qubit groups can each comprise the same number of physical qubits. This is particularly advantageous when the same quantum circuit is used with the different R. 409940

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[0146] The goal is to execute qubit groups in parallel, thus enabling a number of individual measurements corresponding to the number of qubit groups to be achieved by controlling the quantum computer in parallel, and thus enabling an expected value to be determined with fewer calls to the quantum computer compared to performing the individual measurements serially.

[0147] According to one embodiment, at least two of the qubit groups can have different numbers of physical qubits. This is particularly advantageous when different quantum circuits, especially quantum circuits requiring different numbers of physical qubits for execution on the quantum computer, are to be executed in parallel on the quantum computer.

[0148] A method according to claim 10 for executing at least one first quantum circuit and one second quantum circuit in parallel on a quantum computer, wherein the quantum computer comprises a qubit array. In this case, "parallel" means, in particular, that when the quantum computer is called up, several quantum circuits are executed simultaneously, and at the end, a result of an individual measurement can be determined for each qubit group. This contrasts with non-parallel execution, in which only one quantum circuit is executed. The method specifically comprises the following steps.

[0149] • Providing a mapping of the first quantum circuit to the first qubit group and providing a mapping of the second quantum circuit to the second qubit group;

[0150] • Initializing the quantum computer, including:

[0151] o initializing the first qubit group and

[0152] o initializing the second qubit group; R. 409940

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[0154] wherein the qubit groups are arranged at the positions of the qubit groups on the qubit arrangement provided by the method according to one of the preceding claims;

[0155] • Parallel execution of the first quantum circuit with the first qubit group and the second quantum circuit with the second qubit group; In other words, the gates provided in the quantum circuit are executed on the qubits occupied with the initial quantum state and measurements are carried out at the end of each, so that the execution of the quantum circuit serves to generate measurement results;

[0156] • Generating first measurement results of the first qubit group and generating second measurement results of the second qubit group

[0157] • Providing the first measurement results and the second measurement results.

[0158] Providing a mapping of the quantum circuit to the qubit group includes, in particular, assigning the qubits listed in the quantum circuit to the physical qubits of the quantum computer. Furthermore, providing this mapping can include an initial quantum state for the respective qubit group as well as gate parameters for controlling the quantum gates to execute the quantum circuit. This mapping can be achieved, in particular, through input, data transmission (wireless or wired data transmission), or retrieval, for example, from a database.

[0159] Initializing the qubit groups means, in particular, preparing an initial state for each qubit group and providing control signals based on the gate parameters to control the quantum gates depending on the gate parameters when executing the quantum circuit on the quantum computer; Are the quantum gates R. 409940

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[0161] For example, if a rotation gate is configured to cause a single-qubit rotation by an angle around one of the X, Y, or Z axes of the Bloch sphere, the angle is equal to the gate parameter. The control signals depend on the technology of the quantum computer used. For qubits based on superconducting circuits, manipulation of the qubits can be achieved, for example, via the applied voltage, the magnetic field, or coupling to microwave resonators. In such cases, the control signals are configured, for example, to adjust the magnetic field and / or the frequency of the microwave resonators. In particular, the control signals can include electrical signals.

[0162] The advantages of this method arise particularly from the advantages mentioned above. One advantage is that multiple individual measurements of the same quantum circuit can be performed in parallel with high reliability, and / or different quantum circuits or measurements of different observables can be carried out in parallel.

[0163] According to one embodiment, the first quantum circuit and the second quantum circuit are identical.

[0164] According to one embodiment, the first quantum circuit and the second quantum circuit differ from each other.

[0165] According to one embodiment, the first and / or the second quantum circuit are part of a hybrid variational quantum algorithm, in particular a quantum eigensolver (VQE). Measuring the expected values ​​of observables is an essential component of variational quantum algorithms. This requires a large number of measurements for statistical convergence to meet precision requirements. R. 409940

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[0167] fulfilling requirements such as chemical accuracy when applied to quantum chemistry calculations. In other words, this requires a large number of measurements from individual measurements to determine the expected values. To give an order of magnitude to the number of individual measurements: It is generally accepted that a measurement with precision E requires a number of individual measurements of order 1 / e 2This requires, for example, that if one selects five qubit groups which execute the same quantum circuit with the same quantum gates and the same gate parameters with one call to the quantum computer, the number of calls required is reduced by a factor of five to obtain the same number of individual measurements.

[0168] One advantageous application is in determining a ground state of a quantum system, where a Hamiltonian operator of the quantum system is measured when generating the measurement results.

[0169] According to one embodiment, the method is used for material simulation, where the Hamiltonian is a many-particle Hamiltonian describing the material. In another embodiment, the quantum system is described by a Hubbard Hamiltonian. The Hubbard model is an approximate model of a solid. It describes the behavior of electrons in a lattice assumed to be rigid. The repulsive Coulomb forces are considered only for those electrons occupying the same lattice site. The contribution of the electrons' kinetic energy is modeled by an overlap integral derived from the tight-binding model. Some examples of quantum systems that can be described by a Hubbard Hamiltonian are strongly correlated fermion systems, transition metals, and mobile electron systems (e.g., ferromagnetism, antiferromagnetism, R. 409940).

[0170] - 25 -

[0171] Ferrimagnetism), iT electron systems in quantum chemistry. One advantage is that the method accelerates the development and investigation of new materials and, in some cases, makes it possible in the first place. Furthermore, the properties of these new materials can thus be better adapted to the respective application.

[0172] Brief description of the drawings

[0173] Exemplary embodiments of the invention are shown in the drawings and are explained in more detail in the following description. Identical reference numerals in the figures denote identical or equivalently acting elements.

[0174] They show

[0175] Fig. 1 shows a sketch of a connectivity graph for the physical qubits of a quantum computer with 127 qubits, with qubit groups shown according to a first embodiment;

[0176] Fig. 2 shows a sketch of a connectivity graph for the physical qubits of a quantum computer with 127 qubits, with qubit groups shown according to a second embodiment;

[0177] Fig. 3 shows a flowchart of a method for the functional partitioning of a qubit array of a quantum computer; and

[0178] Fig. 4 shows a flowchart of a method for the parallel execution of at least one first quantum circuit and one second quantum circuit on a quantum computer.

[0179] Exemplary embodiments of the invention R. 409940

[0180] - 26 -

[0181] Fig. 1 shows a sketch 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, such as that used on IBM's superconducting quantum computer (127-qubit Eagle processor). The physical qubits are numbered row by row. The connectivity graph associated with a qubit array of the quantum computer comprises a total of 127 physical qubits 0,..., 126, which are represented in the sketch as circles or nodes and numbered for identification. Nodes of physical qubits 0,..., 126 that are configured to interact with each other are connected in the hardware-specific connectivity graph, in particular by edges. In other words, the connectivity graph 200 provides information about the number and connectivity of the physical qubits 0,..., 126.

[0182] In this embodiment, a quantum circuit with twelve physical qubits is to be implemented. In principle, such a quantum circuit could be implemented ten times in parallel on the 127-qubit Eagle processor, which would result in n = 10 clusters (qubit groups) on the connectivity graph. However, such a dense collection of qubit groups would lead to significant crosstalk errors. Therefore, a smaller number of qubit groups to be implemented is chosen, namely five. This reduces crosstalk between qubit groups 201, 202, 203, 204, and 205. There are many different ways in which these clusters can be mapped onto the connectivity graph.

[0183] In this embodiment, five qubit groups 201, 202, 203, 204, 205 were positioned on the physical qubits 0,..., 126 of the quantum computer as follows:

[0184] • The first qubit group 201 comprises the physical qubits numbered 0, 1, 2, 3, 4, 14, 15, 18, 19, 20, 21, 22. The second qubit group 202 R. 409940

[0185] - 27 -

[0186] includes the physical qubits numbered 8, 9, 10, 11, 12, 16, 17, 26, 27, 28, 29, 30.

[0187] • The third qubit group 203 comprises the physical qubits with the numbers 62,62,64,65,66,72,73,1,82,83,84,85.

[0188] • The fourth qubit group 205 comprises the physical qubits with the numbers 96, 97, 98, 99, 100, 109, 110, 114, 115, 116, 117, 118.

[0189] • The fifth qubit group 205 comprises the physical qubits numbered 104, 105, 106, 107, 108, 111, 112, 122, 123, 124, 125, 126.

[0190] Each qubit group 201, 202, 203, 204, 205 comprises the same number of physical qubits 0, ..., 126, namely twelve. Furthermore, the independent qubit groups exhibit the same structure with respect to the connectivity of the physical qubits they contain. The geometric centroid 2002 of each qubit group 201, 202, 203, 204, 205 is marked at its center, with some example distance measures 2001 between the geometric centroids, here the Euclidean distances (=L² distances), shown. The position of the qubit groups 201, 202, 203, 204, 205 was determined by maximizing the Euclidean distances between the geometric centroids of the qubit groups.

[0191] 201,202,203,204,205 found.

[0192] The underlying idea of ​​this embodiment is based on maximizing the distance between all qubit groups 201, 202, 203, 204, and 205. The distance to the clusters of the third qubit group 203 c3 refers in particular to the Euclidean distances 2001 between qubit group c3 and all other qubit groups 201, 202, 204, and 205. This is achieved, in particular, by maximizing the sum of the Euclidean L2 distances between the center of mass or geometric center of mass.

[0193] m^ k , i.e., by maximizing the following optimization function: R. 409940

[0194] - 28 -

[0195]

[0196] lk L2

[0197] The result of the approximate optimization is shown in Fig. 1.

[0198] Fig. 2 differs from Fig. 1 only in the optimization procedure used to determine the positions of the qubit groups 201, 202, 203, 204, and 205. In this embodiment, the shortest-path metrics 2000 between the qubit groups 201, 202, 203, 204, and 205 are shown. To determine the positions of the qubit groups 201, 202, 203, 204, and 205, a sum of distance measures between the qubit groups was used as the optimization function, where the distance measure is a, in particular, minimum shortest-path distance between two qubit groups. The shortest-path distance metric is a metric in which the distance between two nodes in a graph d min (, c k) is given as the edge lengths along the shortest path. For example, Dijkstra's algorithm can be used to determine the shortest-path distance. This implementation is based on maximizing the sum of the minimum distances between qubit groups 201, 202, 203, 204, 205 with respect to qubit connectivity. For a square lattice, this corresponds to the Manhattan metric. Mathematically, this means the sum of the minimum Manhattan norms between the clusters c k to maximize. The Manhattan norm d(q,p) between two points q = (q 17 q2,.., q n ) and q = (p 1; p2,..., p n ) is defined as n

[0199] d( q,p) = ^\qt ~ Pt\

[0200]

[0201] i=l

[0202] Therefore, the optimization problem to be solved is the maximization of the optimization function:

[0203] 'd min (ci, c k )

[0204] lk

[0205] of the minimum distance d min (, c k ) between all qubit groups. One possible solution is shown in Fig. 2. R. 409940

[0206] - 29 -

[0207] In general, the physical qubits that are not part of a qubit group 201, 202, 203, 204, 205 can be used to further reduce crosstalk between qubit groups 201, 202, 203, 204, 205 (similar to dynamic decoupling, which is used to reduce errors in conventional circuits). For this purpose, one or more of these physical qubits (for example, physical qubits 5 to 7, 13, 23 to 25, 31 to 61, etc., in Fig. 1 and Fig. 2) can reduce, completely prevent, or change the sign of the crosstalk through special gate operations, so that the crosstalk at least partially cancels out during the propagation time of the quantum circuit. Candidates for such special gate operations are 1-qubit gates such as the X-Gate or operations that lead to a collapse of the quantum state (such as simple measurements).

[0208] Fig. 3 shows a flowchart of a method 300 for the functional partitioning of a qubit arrangement of a quantum computer with the steps:

[0209] • Provide 301 of information about at least one first quantum circuit and one second quantum circuit intended for parallel execution on the quantum computer;

[0210] In particular, the first and / or the second quantum circuit can be part of a hybrid variational quantum algorithm, especially for determining a ground state of a quantum system.

[0211] • Provide 302 a number of qubit groups 201,202,203,204,205, which specifies the number of quantum circuits that can be executed in parallel;

[0212] • Provide 303 an optimization function which depends on a distance measure between the qubit groups on the qubit array; in particular, the distance measures sketched in Fig. 1 and Fig. 2 can be used for this purpose. R. 409940

[0213] - 30 -

[0214] • Optimizing 304 of the optimization function with respect to the distance measure between the qubit groups 201,202,203,204,205;

[0215] • Provide 305 of the positions of qubit groups 201, 202, 203, 204, 205 on the qubit array.

[0216] Fig. 4 shows a flowchart of a method 400 for parallel execution of at least one first quantum circuit and one second quantum circuit on a quantum computer, wherein the quantum computer comprises a qubit array, and wherein the method comprises the following steps:

[0217] • Providing 401 a mapping of the first quantum circuit onto the first qubit group 201 and providing a mapping of the second quantum circuit onto the second qubit group 202; In the case of the embodiment shown in Fig. 1 and Fig. 2, mappings for the third, fourth and fifth qubit groups are also provided.

[0218] • Initializing the quantum computer, including:

[0219] o initializing the first qubit group and

[0220] o initializing the second qubit group;

[0221] wherein the qubit groups 201,202,203,204,205 are arranged at the positions of the qubit groups 201,202,203,204,205 on the qubit arrangement provided according to, for example, the method shown in Fig. 3;

[0222] • Parallel execution 403 of the first quantum circuit with the first qubit group 201 and the second quantum circuit with the second qubit group 202; In the case of the embodiment shown in Fig. 1 and Fig. 2, the quantum circuits of the third, fourth and fifth qubit groups 203, 204, 205 are also executed.

[0223] • Generating 404 first measurement results of the first qubit group 201 and generating second measurement results of the second qubit group 202; In the case of the embodiment shown in Fig. 1 and Fig. 2, the measurement results of the third, fourth and fifth qubit groups 203, 204, 205 are also generated. In particular, in this step an observable is generated within the framework of an R. 409940

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[0225] A single measurement is used. To determine an expected value, a large number, for example several thousand individual measurements, is needed to then determine the expected value.

[0226] • Provide the first measurement results and the second measurement results. In particular, the measurement results can be provided, especially output, stored, transmitted or displayed.

Claims

R. 409940 - 32 - Claims 1. Method (300) for the functional partitioning of a qubit arrangement of a quantum computer: • Providing (301) information about at least one first quantum circuit and one second quantum circuit intended for parallel execution on the quantum computer; • Providing (302) a number of qubit groups (201,202,203,204,205) which specifies the number of quantum circuits that can be executed in parallel; • Providing (303) an optimization function which depends on a distance measure between the qubit groups on the qubit array; • Optimizing (304) the optimization function with respect to the distance measure between the qubit groups (201,202,203,204,205); • Providing the positions of the qubit groups (201,202,203,204,205) on the qubit array.

2. Method (300) according to claim 1, wherein the optimization function is a sum of distance measures of the qubit groups to each other (201,202,203,204,205).

3. Method (300) according to one of the preceding claims, wherein the distance measure comprises a Euclidean distance (2001) between the geometric centroids (2002) of the qubit groups (201,202,203,204,205).

4. Method (300) according to any one of the preceding claims, wherein the optimization function is a sum of distance measures of the qubit groups R. 409940 - 33 - (201,202,203,204,205) to each other, and where the distance measure is a shortest path distance (2000) of the qubit groups (201,202,203,204,205).

5. Method (300) according to one of the preceding claims, wherein the optimization function for at least a part of the qubit groups (201,202,203,204,205) includes a contribution which results from a weighted sum of the quality of the qubits of the respective qubit group (201,202,203,204,205).

6. Method (300) according to one of the preceding claims, wherein the optimization of the optimization function with respect to the distance measure between the qubit groups (201,202,203,204,205) is terminated after a predefinable number of iterations and the solution determined in the last iteration step is used for providing the positions of the qubit groups on the qubit arrangement.

7. Method (300) according to one of the preceding claims, wherein a starting point for the optimization is chosen as follows: • Dividing the qubit arrangement into blocks; • Arrange one qubit group (201,202,203,204,205) per block.

8. Method (300) according to any of the preceding claims, wherein all qubit groups (201,202,203,204,205) each comprise an equal number of physical qubits (0,...,126).

9. Method (300) according to any one of claims 1 to 7, wherein at least two of the qubit groups (201,202,203,204,205) have different numbers of physical qubits (0,..., 126).

10. Method (400) for parallel execution of at least one first quantum circuit and one second quantum circuit on a R. 409940 - 34 - Quantum computer, wherein the quantum computer comprises a qubit array, wherein the method comprises the following steps: • Providing (401) a mapping of the first quantum circuit to the first qubit group (201) and providing a mapping of the second quantum circuit to the second qubit group (202); • Initializing (402) the quantum computer, including: o initializing the first qubit group (201) and o initializing the second qubit group (202); wherein the qubit groups (201,202,203,204,205) are arranged at the positions of the qubit groups (201,202,203,204,205) on the qubit arrangement provided according to the method according to one of the preceding claims; • Parallel execution (403) of the first quantum circuit with the first qubit group (201) and the second quantum circuit with the second qubit group (202); • Generating (404) first measurement results of the first qubit group and generating second measurement results of the second qubit group • Provide (405) the first measurement results and the second measurement results.

11. Method (400) according to claim 10, wherein the first quantum circuit and the second quantum circuit are identical.

12. Method (400) according to claim 11, wherein the first quantum circuit and the second quantum circuit differ from each other.

13. Method (400) according to any one of claims 10 to 12, wherein the first and / or the second quantum circuit is part of a hybrid variational quantum algorithm, in particular a quantum eigensolver (VQE).

14. Use of the method (400) according to any one of claims 10 to 13 for determining a ground state of a quantum system, wherein during R. 409940 - 35 - Generating the measurement results involves measuring a Hamiltonian operator of the quantum system.

15. Use according to claim 14 for a material simulation, wherein the Hamiltonian is a many-particle Hamiltonian describing the material.

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