Method for parallel running of two quantum circuits and method for determining an expected value of an observable

Abstract

The invention relates to a method (400) for parallel running of at least one first quantum circuit and a second quantum circuit on a quantum computer, wherein the quantum computer comprises a qubit arrangement which is functionally divided into at least a first qubit group and a second qubit group, wherein the first qubit group and the second qubit group are spatially separated from one another, wherein the method (400) comprises the following steps: providing (406) a list including temporal delay values; Providing (401) 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, wherein the first quantum circuit and / or the second quantum circuit are time-delayed by means of the delay values; initializing (402) the quantum computer, comprising: initialising the first qubit group and initialising the second qubit group; parallel running (403) of the first quantum circuit with the first qubit group to generate (404) first measurement results of the first qubit group and parallel operation of the second quantum circuit with the second qubit group to generate (404) second measurement results of the second qubit group; providing (405) the first measurement results and the second measurement results.

Description

[0001] R. 413816

[0002] - 1 -

[0003] Description

[0004] title

[0005] Methods for the parallel execution of two quantum circuits and methods for determining an expectation value of an observable

[0006] State of the art

[0007] In “Variational approach-based quantum simulation of imaginary time evolution” (McArdle et al., npj Quantum Information (2019) 5:75; https: / / doi.org / 10.1038 / s41534-019-0187-2) a method for determining a ground state of a many-body system is described, which uses a hybrid system of quantum computer and classical computer.

[0008] 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.

[0009] Currently available quantum computers are not fully error-corrected, resulting in errors and noise inherent in any quantum circuits run on them. Therefore, such quantum computers are often referred to as NISQ (noisy intermediate-scale quantum) technology. The capabilities of currently available NISQ quantum computers are limited. Due to their restricted 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 computer is R.413816.

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[0011] Low-depth quantum circuits include hybrid quantum-classical algorithms, such as the variational quantum eigensolver (VQE).

[0012] However, variation algorithms such as VQE cannot efficiently exploit this large number of qubits due to challenges arising from the inherent noise.

[0013] Core and advantages of the invention

[0014] 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.

[0015] 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, to determine a ground state of a quantum system (i.e., the state with the lowest energy). In this approach, a quantum state (e.g., a wavefunction) is encoded with a variational approach in a quantum circuit with variation parameter 0^, and the corresponding expectation value E is calculated. 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.

[0016] 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 will then be connected to the quantum computer R. 413816

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[0018] returned to E k+1 to 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 6^ and corresponds to the ground state of the quantum system, or rather, describes an approximation of the ground state.

[0019] 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 employed 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 resulting 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 optimizer can get stuck in local minima or in so-called bar plateaus with vanishing gradients.Using a quantum computer, the expectation value of the system with respect to an observable, for example the Hamiltonian operator of the system, is determined, and a classical optimizer is used to improve the parameters of the approach.

[0020] 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 core of VQE lies in its hybrid methodology. A classical optimizer is used to adjust or optimize the parameters of the approach. The goal is to determine the expected value of the energy of the system.

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[0022] to minimize the Hamiltonian operator, which is crucial for reaching the ground state of the system.

[0023] 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).

[0024] 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.

[0025]

[0026] ^, generally denoted by a set of angles 0, which scale unfavorably with the complexity of the approach and the number of qubits. Furthermore, the inherent noise of NISQ devices leads to inefficient gradient-based optimizations.

[0027] 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 realized for shallow quantum circuits, for a few dozen variational parameters, and for a small number of physical qubits. Therefore, the power of quantum computers with 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.

[0028] Furthermore, another problem arises from the large number of shots (individual measurements) required to obtain statistically well-converged measurement results when designing quantum circuits. R. 413816

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[0030] The aforementioned problems essentially limit the potential for using an ever-increasing number of physical qubits on quantum computers, particularly 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, allowing this solution to parallelize over the number of individual measurements. In this way, many individual measurements can be performed in parallel by executing these identical quantum circuits in a single pass of the quantum computer.

[0031] 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.

[0032] Although this does not necessarily enable the processing of larger systems, it reduces the overall computing time for smaller systems.

[0033] The main problem with the parallelization solutions mentioned above is the occurrence of crosstalk between qubit groups, which leads to correlated and nonlocal errors, even between different, independent clusters of quantum circuits encoded on a quantum computer. To address this source of error, it is proposed to spatially separate 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 computing time.

[0034] The present invention proposes a method which

[0035] • making use of a larger number of physical qubits of a quantum computer and

[0036] • 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; R. 413816

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[0038] (hereinafter also referred to as qubit groups) are determined, each of which is suitable for executing a quantum circuit.

[0039] 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.

[0040] In particular, the invention makes it possible to reduce crosstalk between qubit groups and thus increase the reliability of the measurement results.

[0041] 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 reduced in particular by spatially separating clusters of parallel-encoded small quantum circuits on the same quantum computer and by further methods reducing crosstalk between the qubits.

[0042] 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 produced by the quantum circuit's computation. Operations on qubits are represented by boxes placed on these wires. Quantum gates are the elementary operations that a quantum computer can perform on its qubits. They are comparable to electronic gates, which perform the elementary operations of a classical computer. For quantum gates operating on two qubits (2-qubit gates), an interaction between the qubits in question is described in R. 413816.

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[0044] Physical qubits are required. For spin qubits, this can occur, among other things, via exchange interactions. Atoms in an ion trap, for example, can exchange photons. For qubits based on superconducting circuits, these qubits can be manipulated, for example, via an applied voltage, a magnetic field, or coupling to microwave resonators. In the following, "quantum circuit" refers to a quantum circuit for physical qubits. These quantum circuits can have additional quantum gates, especially swaps, compared to quantum circuits for logic qubits. These swaps may be added to the quantum circuit for physical qubits to transport logic qubits to interacting physical qubits when they are involved in a joint quantum operation.

[0045] The invention relates to a method for the parallel execution of at least one first quantum circuit and one second quantum circuit on a quantum computer and a method for determining an expectation value of an observable of a quantum system.

[0046] The method is based on a parallelization scheme in which the physical qubits of a quantum computer are divided into independent qubit groups (=clusters), with the qubit groups being spaced apart from each other and encoding independent quantum circuits. This spatial separation, achieved, for example, by placing physical qubits between the qubit groups that are not needed to execute the encoded quantum circuits, reduces the interaction between qubit groups.

[0047] Crosstalk errors can typically occur even between different, independent qubit groups, or the quantum circuits encoded therein. An advantage of the method presented below is that it avoids or at least reduces the occurrence of crosstalk between qubit groups, which leads to correlated and non-local errors. Therefore, the method according to claim 1 enables, in particular, the generation of more reliable measurement results. R. 413816

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[0049] In particular, the invention enables the efficient use of the available physical qubits of available quantum computers. Specifically, it allows the physical qubits of the qubit array to be divided into qubit groups, each of which can be used to execute a quantum circuit. In particular, this potentially (but not necessarily) allows identical quantum circuits to be executed in parallel. This reduces the number of runs of the quantum computer while maintaining the same number of measurement results. In this way, it is possible to parallelize VQE over shots (individual measurements), observables, or gradient descent directions, leading to a significant advantage when using a large number of physical qubits on NISQ computers.

[0050] This is achieved by the method according to claim 1, for the 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 arrangement which is functionally divided into at least one first qubit group and one second qubit group, wherein the first qubit group and the second qubit group are spatially separated from each other.

[0051] The parallel execution of the quantum circuits means, in particular, that by calling / activating the quantum computer, a number of individual measurements can be performed in parallel. This number corresponds, for example, to the number of qubit groups in the case of identical quantum circuits. Thus, an expected value can be determined with fewer calls to the quantum computer compared to performing the individual measurements serially. In other words, during parallel execution, at least one quantum gate of the first quantum circuit is executed in a time-overlapping manner with a quantum gate of the second quantum circuit.

[0052] Functional partitioning can be understood in particular as combining a subset of physical qubits of the quantum computer's qubit array into a qubit group suitable for implementing a quantum circuit and which, in particular by its spatial but especially by its functional separation, is distinct from the physical qubits of the qubit array not belonging to this qubit group, as described in R. 413816

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[0054] These qubits can be considered independent quantum computers. A functional division into qubit groups means, in particular, that the physical qubits of different qubit groups preferably do not interact with each other. This is achieved, in particular, by spatially separating the qubits. Specifically, the qubit groups can be spaced apart from each other on the qubit array. Spatial separation can be achieved, for example, by placing physical qubits that are not assigned to any of the qubit groups (i.e., are not needed for executing the quantum circuits) between the qubit groups, thus forming a kind of boundary for the qubit groups. "Functional" here means, in particular, that the physical qubits of a qubit group are needed specifically for executing the quantum circuit encoded on the qubit group and have therefore been grouped to fulfill a common function.The claim proposes a division into at least two qubit groups, and in particular also more than two or more than three qubit groups.

[0055] Quantum computers, programmable via quantum circuits, can in principle be constructed from any quantum technology capable of single- and multi-qubit gate operations. Currently, architectures based on, for example, superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are being actively developed.

[0056] A quantum computer comprises a qubit array, wherein the qubit array includes several physical qubits, which may preferably be equipped with devices or units adapted to the technology with which the qubits are realized for initializing (e.g. initializing the qubit in a basis state), manipulating (e.g. applying 1-qubit and / or 2-qubit gates) and / or reading the physical qubits.

[0057] The method according to claim 1 comprises in particular the following steps:

[0058] - Providing a list comprising time delay values; in particular, the list may include at least one time delay value. R. 413816

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[0060] For example, the time delay value can be an idle time between quantum gate k and quantum gate k + 1, and / or an idle time between the start time and the first quantum gate. This advantageously allows the quantum circuits of the different qubit groups to start at different times and / or to run asynchronously. This, in turn, can advantageously reduce or eliminate crosstalk during the execution of the quantum circuits.

[0061] In particular, the list can include a subset of the qubit groups, at least two, but especially one for each qubit group, time delay values. Specifically, the list can include delay values ​​for each qubit group for each quantum gate, with a delay value in an interval from 0 inclusive up to a maximum time value. According to one embodiment, the time delay value can be an integer multiple of the clock rate of the quantum chip (i.e., n / clock frequency).

[0062] - 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, wherein the first quantum circuit and / or the second quantum circuit are time-delayed by means of the delay values; - Initializing the quantum computer, comprising:

[0063] Initializing the first qubit group and

[0064] Initializing the second qubit group;

[0065] - Parallel execution of the first quantum circuit with the first qubit group to generate first measurement results of the first qubit group and the second quantum circuit with the second qubit group to generate second measurement results of the second qubit group;

[0066] - Providing the first and second measurement results. In particular, this step involves measuring an observable in a single measurement. To determine an expected value, a large number, for example, several thousand individual measurements, are needed to calculate the expected value.

[0067] 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 an initial quantum state for the respective qubit group R. 413816

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[0069] as well as gate parameters for controlling the quantum gates to execute the quantum circuit. Provision can be achieved in particular through input, data transmission (wireless or wired data transmission), or retrieval, for example from a database.

[0070] Initializing the qubit groups specifically involves preparing an initial state for each qubit group and providing control signals based on the gate parameters to activate the quantum gates when the quantum circuit is executed on the quantum computer. For example, if the quantum gates are configured as rotation gates, which, for instance, cause a one-qubit rotation by an angle around one of the X, Y, or Z axes of the Bloch sphere, then the angle is equal to the gate parameter. The control signals depend on the technology of the quantum computer used.In qubits based on superconducting circuits, manipulation can be achieved, for example, via the applied voltage, the magnetic field, or coupling to microwave resonators. The control signals are configured, for instance, to adjust the magnetic field and / or the frequency of the microwave resonators. In particular, the control signals can be electrical signals. The resulting expectation value benefits from the fact that crosstalk occurs at different positions within the quantum circuits and can therefore cancel each other out. Thus, an improved result, especially one with reduced crosstalk errors, can be achieved compared to expectation values ​​determined from measurements without the insertion of delay values.

[0071] One advantage of this method, which systematically reduces crosstalk errors, is that it allows consideration to arrange the qubit groups closer together spatially, since the increasing influence of crosstalk errors due to the denser arrangement can be reduced again by this method.

[0072] In particular, different lists of delay values ​​can be used, so that R. 413816 depends on the choice of delay values.

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[0074] Measurement results are obtained. In particular, random delay values ​​and delay values ​​determined according to a specific scheme can be compiled for the list and used as a basis for delaying the quantum circuits. When determining the expected value, the results are then averaged over the various lists of delay values.

[0075] The first variant of the "temporal offset" strategy involves calculating only the average across different sets. The resulting expected value will benefit from the fact that crosstalk occurs at different positions in the circuit and can therefore cancel each other out.

[0076] According to one embodiment, the list includes at least one delay value that shifts the start time of the second quantum circuit relative to the start time of the first quantum circuit, so that the quantum circuits are executed at least partially staggered in time. In particular, this reduces crosstalk between the qubit groups. By staggering the execution, especially when executing identical quantum circuits, the initial delay ensures that all quantum gates are executed at different times than the quantum gates of the other qubit groups. In particular, it is not necessary, but still possible, to include idle times in the quantum circuit.

[0077] According to one embodiment, the list of time delay values ​​includes at least one idle time, which is inserted at one or more points in the first quantum circuit and / or second quantum circuit.

[0078] According to one embodiment, the delay values ​​are not chosen arbitrarily, but are the result of optimizing a correlator of a first observable of the first qubit group, which depends on a first list of delay values, and a second observable of the second qubit group, which depends on a second list of delay values. In other words, the delay values ​​are optimized here such that the interaction of the qubit groups is reduced or minimized. R. 413816

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[0080] An observable, particularly a quantum mechanical observable, can be understood as a measurable quantity and its associated operator, acting in state space, a Hilbert space. Examples of observables are the energy (the associated operator being the Hamiltonian of the quantum system), the position coordinates, the momentum coordinates, and the spin components of a particle. Observables assign values ​​to the results of certain measurements, corresponding to the eigenvalue of the operator. A crucial difference between classical quantities and quantum mechanical observables is that some pairs of quantum mechanical observables are not simultaneously measurable. If the operators of two quantum mechanical observables do not commute, then a measurement of the first operator changes the quantum state in a way that is incompatible with the subsequent measurement of the second observable, and vice versa.Quantities that can be precisely determined simultaneously are called commuting observables; they have the property that the order of their operators in the product can be interchanged without changing the result. Observables that cannot be measured with arbitrary precision simultaneously are also called complementary observables.

[0081] This embodiment is based on considering the measurements of the observables Ô₁ on the first qubit group and Ô₂ on the second qubit group (the observables Ô₁ and Ô₂ can be the same or different). The correlation function (=correlator) is determined as follows:

[0082] c_kl({τ_m^k},{τ_n^l}) = ⟨Ô_k(τ_m)Ô_l(τ_n)⟩ where the following definition applies:

[0083] ⟨Ô_k(τ_m)Ô_l(τ_n)⟩ = ⟨ψ_kψ_l|Ô_k(τ_m)Ô_l(τ_n)|ψ_kψ_l⟩ with |ψ₁ψ₂⟩ ​​= |ψ₁⟩ ⊗ |ψ₂⟩ Here, |ψᵢ⟩ (here with i = 1,2) describes a quantum state of the i-th quantum group, {τ_m} the list of delay values ​​for the k-th quantum group, and {τ_n} the list of delay values ​​for the l-th quantum group. By optimizing, in particular minimizing, the correlation function c ki (i.e., by determining the delay values ​​that result in the lowest correlation between the observables) can then be used for R. 413816

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[0085] For each quantum circuit, a list of delay values ​​is identified that reduces crosstalk between the clusters (=qubit groups).

[0086] According to one embodiment, the first observable and the second observable can be identical.

[0087] According to one embodiment, the first observable and the second observable can differ from each other.

[0088] According to one embodiment, the functional partitioning of the qubit arrangement is determined by the following steps:

[0089] • 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 can 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. In particular, providing the quantum circuits themselves is possible in this step, but not required.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. The provision of information specifically makes available the information required for the process regarding the number of physical qubits needed per quantum circuit and their respective required connectivity. In particular, the provision of information on at least one first quantum circuit and one second quantum circuit can also include the provision of information on more than two quantum circuits. R. 413816.

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[0091] • 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. Specifically, 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.

[0092] For example, the sum of the physical qubits of all qubit groups should be less than or equal to 50% of the physical qubits provided by the qubit array for the execution of quantum circuits.

[0093] A qubit group is understood to be, in particular, a group or cluster of physical qubits in a qubit array that are functionally, and especially spatially, separated from the other physical qubits in the array. Specifically, 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 in the array is reduced compared to the interaction with the members of their own qubit group.

[0094] • Providing an optimization function S that depends on a distance measure between the qubit groups on the qubit array; The optimization function is based on the following assumption: By reducing or minimizing the interaction between the qubit groups, the R. 413816

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[0096] Crosstalk is reduced. In other words, the optimization function takes into account the idea that the crosstalk error decreases with decreasing connectivity or increasing spatial distance between qubit groups. The distance measure can, in particular, be a function of the spatial distance, for example, the position of the geometric center or center of mass of the physical qubits in 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. Alternatively or additionally, the quality can be

[0097] The fidelities of the qubits used in the individual qubit groups and their connectivity are taken into account as additional parameters in the optimization function, especially to avoid highly noisy qubits.

[0098] • 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.

[0099] • 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 and / or arrangements of the qubit groups can be provided by means of a connectivity graph of the physical qubits of the quantum computer. Specifically, the quantum computer has only limited physical qubit connectivity; that is, not every physical qubit can interact with every other physical qubit. In other words, quantum operations, especially 2-qubit gates, can only be performed between certain physical qubits. This physical qubit connectivity 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, in the hardware-specific connectivity graph, the physical qubits are represented as nodes, and nodes of physical qubits configured to interact with each other are connected by edges. In other words, the connectivity graph includes information about which physical qubits are connected. R. 413816.

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[0101] Qubits with which other physical qubits can interact. 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, the positions of the qubit groups on the qubit array can be provided by specifying the identifiers of the physical qubits belonging to a common qubit group.

[0102] One advantage of this method is that it uses only generic basic information, specifically the number of physical qubits required per quantum circuit, and is therefore applicable 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 crosstalk errors. Afterward, each circuit with n qubits can be executed in parallel on this quantum computer hardware with the same trivial parallel acceleration. Analogously, different sizes of qubit groups can, of course, also be taken into account.

[0103] 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.

[0104] 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.

[0105] For example, the spatial arrangement in which a cluster c t R. 413816

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[0107] Ideally, the coded format can vary. To account for this, the following approaches are possible, for example:

[0108] 1) Cluster-first strategy: Here, the individual clusters are selected first. t 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.

[0109] 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 t Locally optimally mapped.

[0110] 3) Hybrid-optimal strategy: Optimization of the local structure of the clusters c t together with the optimization function.

[0111] One advantage of the method is that it allows for the mapping of multiple quantum circuits onto an arrangement of physical qubits of a quantum computer, taking into account hardware-specific characteristics such as the connectivity of the physical qubits of the qubit arrangement, their respective quality, requirements arising from the quantum circuits, etc.

[0112] Advantageously, this method allows the total number of quantum computer calls to be reduced by a factor of n, where n is the number of identical qubit groups assigned to the QPU, since the same statistical noise of the individual measurements is achieved 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.

[0113] 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 achieve R. 413816

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[0115] To reduce crosstalk errors, thereby improving the reliability of the quantum computer's measurement results.

[0116] According to one embodiment, the distance measure comprises a Euclidean distance between the geometric centroids of qubit groups. The underlying idea of ​​this embodiment is based on maximizing the distance c between all clusters. k The distance to neighboring clusters of the qubit group refers specifically to the Euclidean distances between the qubit group and 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, i.e., by maximizing the following optimization function:

[0117] S =

[0118]

[0119] Ik L2

[0120] 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.

[0121] According to one embodiment, the optimization function comprises a sum of distance measures between the qubit groups, where the distance measure is a, in particular 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. The Manhattan distance 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 sum of the minimum distances between the qubit groups with respect to the qubit connectivity. Mathematically, this means, for example, in a square or cubic lattice connectivity, maximizing 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 q2,.., q n ) and q = (p n p2,..., p n ) is defined as

[0122] n

[0123] d(q,p) = Σ|qᵢ - pᵢ|

[0124]

[0125] i = l R. 413816

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[0127] Therefore, the optimization problem to be solved is the maximization of the optimization function:

[0128] F ' dmin(. Cl> Ck)

[0129]

[0130] Ik

[0131] of the minimum distance d min (c h 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 h 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 h c k ) can be used, such as the Dijkstra algorithm.

[0132] 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 fidelities of the qubits in the respective qubit group. In addition to maximizing the spacing norms S as an optimization function, another term can be added to the optimization function that takes the qubit fidelity into account. The physical qubits of a quantum computer often differ in their fidelity, particularly with respect to their noise susceptibility. In other words, the qubit array of the quantum computer includes, for example, physical qubits with different fidelities. This can be taken into account, for instance, when arranging the qubit groups by adding a weighted term F to the optimization function. An example of such a weighted term is:

[0133] F = a ' k ' ifCqi').

[0134] where a is a scalar weighting factor and

[0135]

[0136] (q ( ) the sum of the goodnesses of all physical qubits q t of the qubit group k. This makes it advantageously possible to favor solutions in the optimization that use physical qubits with better quality factors and thus enable more reliable and, in particular, less noise-prone individual measurements. The quality factor of an R. 413816

[0137] - 21 -

[0138] Physical qubits can be evaluated based on one or more of the following metrics: single-qubit accuracy, two-qubit gate accuracy, or the inverse of read errors.

[0139] According to one embodiment, the optimization of the optimization function with respect to the distance measure between the qubit groups is terminated after a predefined 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). However, the advantage 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 with a sufficiently good solution; in other words, a solution that can be found within a few iterations, for example, 10 to 10,000, of a gradient-free optimization scheme.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 optimization function mentioned above.

[0140] According to one embodiment, a starting point for optimization is chosen as follows. It is performed

[0141] • Dividing the qubit arrangement of the quantum computer into blocks.

[0142] In particular, the qubit array is divided into as many blocks as corresponds to 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 information provided. In particular, at least some of the blocks comprise more physical qubits than the qubit group to be arranged within the block; and

[0143] • an arrangement of one qubit group per block.

[0144] 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 and accelerating the optimization of the optimization function. R. 413816

[0145] - 22 -

[0146] 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.

[0147] Alternatively, any arrangement of the qubit groups can be chosen as the starting point.

[0148] According to one embodiment, each qubit group can comprise the same number of physical qubits. This is particularly advantageous if the same quantum circuit is to be executed in parallel with the different qubit groups, and thus a number of individual measurements corresponding to the number of qubit groups can be achieved by controlling the quantum computer in parallel, and therefore an expected value can be determined with fewer calls to the quantum computer compared to performing the individual measurements serially.

[0149] 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.

[0150] According to one embodiment, one of the methods described above can be used to determine an expected value of an observable of a quantum system. Such a method comprises the steps of: 1) Executing one of the methods described above, in particular on the quantum computer;

[0151] 2) Adjusting the list of time delay values ​​and / or repeating steps 1) and 2) until a termination criterion is reached; a termination criterion can be derived, for example, from the target number of individual measurements to determine the expected value. In particular, several individual measurements can be carried out with the same list of delay values. R. 413816

[0152] - 23 -

[0153] The list can be adjusted after a single measurement.

[0154] In particular, individual measurements that depend on the delay values ​​can be generated.

[0155] 3) Determining the expected value from the measurement results provided in step 1). This step can be performed on a classical computer. Specifically, averaging the individual measurements, which may depend on the delay values, is used to obtain the expected value. The expected value can be provided by input, data transmission (wireless or wired), or retrieval, for example, from a database. The term "quantum system" refers to physical systems in which phenomena of quantum mechanics are visible. Examples of such phenomena include the quantization of energy or other observables, particle-wave interference, non-locality, and quantum mechanical tunneling.Quantum systems encompass the entire microscopic world, including elementary particles and atoms, but also electrical conductors with nanometer dimensions, semiconductors, large molecules, and certain materials whose macroscopic properties are determined by quantum mechanical interactions on microscopic scales. A quantum system can be described, in particular, by a Hamiltonian operator. In quantum mechanics, the Hamiltonian operator of a system is an operator that describes the total energy of that system, including kinetic and potential energy. Its spectrum, the energy spectrum of the system, includes the eigenvalues ​​of the Hamiltonian operator, i.e., the energy eigenvalues. This is the set of possible results that can be obtained by measuring the total energy of the system.

[0156] In particular, a Hamiltonian operator of a quantum system can be used as an observable. Specifically, this can be a many-body Hamiltonian operator that describes or approximates a material.

[0157] In particular, the method can be used to determine a ground state energy of a material system. According to one embodiment, the method is carried out for a material simulation, wherein the Hamiltonian is a many-particle Hamiltonian, which is R. 413816

[0158] - 24 -

[0159] The material is described. According to one embodiment, the quantum system is described by a Hubbard-Hamilton operator. The Hubbard model is an approximate model of a solid. It describes the behavior of electrons in a lattice assumed to be rigid. The repulsive Coulomb forces are only considered for those electrons 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-Hamilton operator are strongly correlated fermion systems, transition metals, and mobile electron systems (e.g.,...).

[0160] Ferromagnetism, anti-ferromagnetism, ferrimagnetism), π-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.

[0161] 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.

[0162] According to one embodiment, the first quantum circuit and the second quantum circuit are identical.

[0163] According to one embodiment, the first quantum circuit and the second quantum circuit differ from each other.

[0164] 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, such as chemical accuracy when applied to quantum chemistry. (R. 413816)

[0165] - 25 -

[0166] Calculations. In other words, a large number of measurements from individual measurements are required 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 s requires a number of individual measurements of order 1 / ε². If, for example, one chooses five qubit groups that execute the same quantum circuit with the same quantum gates and the same gate parameters with a single 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.

[0167] Brief description of the drawings

[0168] 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.

[0169] They show

[0170] 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;

[0171] 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;

[0172] Fig. 3 shows a flowchart of a method for the functional partitioning of a qubit arrangement of a quantum computer;

[0173] 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 and

[0174] Fig. 5 shows a flowchart of a procedure for determining an expected value of an observable of a quantum system.

[0175] Exemplary embodiments of the invention R. 413816

[0176] - 26 -

[0177] 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.

[0178] In this embodiment, a quantum circuit with twelve physical qubits will be implemented. In principle, such a quantum circuit could be implemented ten times in parallel on the 127-qubit Eagle processor, resulting 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.

[0179] 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:

[0180] • 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 comprises the physical qubits numbered 8, 9, 10, 11, 12, 16, 17, 26, 27, 28, 29, 30.

[0181] • The third qubit group 203 comprises the physical qubits numbered 62, 62, 64, 65, 66, 72, 73, 1, 82, 83, 84, 85. R. 413816

[0182] - 27 -

[0183] • The fourth qubit group 205 comprises the physical qubits with the numbers 96, 97, 98, 99, 100, 109, 110, 114, 115, 116, 117, 118.

[0184] • The fifth qubit group 205 comprises the physical qubits numbered 104, 105, 106, 107, 108, 111, 112, 122, 123, 124, 125, 126.

[0185] In total, 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 included physical qubits 0,..., 126. At the center of each qubit group 201, 202, 203, 204, 205, the geometric centroid 2002 of the respective qubit group is marked, with some example distance measures 2001 of the geometric centroids, here the Euclidean distances (=L2 distances), shown. The position of the qubit groups 201,202,203,204,205 was found by maximizing the Euclidean distances between the geometric centroids of the qubit groups 201,202,203,204,205.

[0186] 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 specifically 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.

[0187] , i.e. by maximizing the following optimization function:

[0188] S =

[0189]

[0190] Ik L2

[0191] The result of the approximate optimization is shown in Fig. 1.

[0192] Fig. 2 differs from Fig. 1 only in the optimization procedure used to determine the positions of the qubit groups 201, 202, 203, 204, 205. In this embodiment, the shortest path metrics 2000 between the qubit groups 201, 202, 203, 204, 205 are shown. To determine the positions of the qubit groups 201, 202, 203, 204, 205, a sum of distance measures between the R. 413816 was used as the optimization function.

[0193] - 28 -

[0194] Qubit groups are used, 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 h 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 =

[0195] (q₁, q₂,..., qₙ) and q

[0196]

[0197] = (Pi, Pn) is defined as

[0198] n

[0199] d(q,p) = Σ|qᵢ - pᵢ|

[0200]

[0201] i = l

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

[0203] S = Σ d min (cₗ, c k )

[0204] lk

[0205] of the minimum distance d min (c h c k ) between all qubit groups. One possible solution is

[0206] shown in Fig. 2.

[0207] Fig. 3 shows a flowchart of a method 300 for the functional partitioning of a qubit arrangement of a quantum computer with the steps:

[0208] • Provide 301 information on at least one first quantum circuit and one second quantum circuit intended for parallel execution on the quantum computer; in particular, the first and / or the second quantum circuit may be part of a hybrid variational quantum algorithm, especially for determining a ground state of a quantum system. • Provide 302 a number of qubit groups 201, 202, 203, 204, 205, which specifies the number of quantum circuits executable in parallel;

[0209] • 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. 413816

[0210] - 29 -

[0211] • Optimizing 304 of the optimization function with respect to the distance measure between the qubit groups 201,202,203,204,205;

[0212] • Provide 305 of the positions of qubit groups 201, 202, 203, 204, 205 on the qubit array.

[0213] Fig. 4 shows a flowchart of a method 400 for the parallel execution of at least one first quantum circuit and one second quantum circuit on a quantum computer. In particular, the qubit arrangement of the quantum computer is functionally divided into at least one first qubit group 201 and one second qubit group 202, for example as in Fig.

[0214] The method is outlined in Figures 1 and 2. It includes, in particular, the following steps:

[0215] • Provide 406 a list containing time delay values;

[0216] • 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, wherein the first quantum circuit and / or the second quantum circuit are time-delayed by means of the delay values; 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.

[0217] • Initializing the quantum computer, including:

[0218] Initializing the first qubit group 201 and

[0219] Initializing the second qubit group 202;

[0220] • Parallel execution 403 of the first quantum circuit with the first qubit group 201 to generate 404 first measurement results of the first qubit group and of the second quantum circuit with the second qubit group 202 to generate 404 second measurement results of the second qubit group; 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.

[0221] • Provide 405 of the first measurement results and the second measurement results.

[0222] In this step, an observable is measured in a single measurement. To determine an expected value, a large number, for example several thousand individual measurements, is needed to then calculate the R. 413816

[0223] - 30 -

[0224] To determine the expected value. In particular, the measurement results can be output, stored, transmitted, or displayed.

[0225] Fig. 5 shows a flowchart of a procedure 500 for determining an expectation value of an observable of a quantum system with the steps:

[0226] 1) Provide 501 a list containing time delay values;

[0227] 2) Inserting 502 of the delay values ​​into at least one first quantum circuit and / or a second quantum circuit; in other words, idle times between the quantum gates of the quantum circuits can be introduced by means of the delay values, so that the time intervals between the quantum gates can be influenced. In particular, they can be influenced in such a way that the quantum circuits operate asynchronously in time. In other words, the quantum circuits are separated not only spatially by the position of the qubit groups but also in time.

[0228] 3) Execution 503 of the method 400 for the parallel execution of quantum circuits, in particular on the quantum computer, for example according to the embodiment shown in Fig. 4;

[0229] 4) Adjusting 504 of the list of time delay values ​​and / or re-executing steps 1) to 3) until a termination criterion is reached;

[0230] 5) Determine 505 of the expected value from the measurement results provided in step 3).

Claims

R. 413816 - 31 - Claims 1. 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 arrangement functionally divided into at least one first qubit group (201) and one second qubit group (202), wherein the first qubit group (201) and the second qubit group (202) are spatially separated from each other, wherein the method (400) comprises the following steps: • Provide (406) a list including time delay values; • 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), wherein the first quantum circuit and / or the second quantum circuit are time-delayed by means of the delay values; • Initializing (402) the quantum computer, including: oein Initializing the first qubit group (201) and Initializing the second qubit group (202); • Parallel execution (403) of the first quantum circuit with the first qubit group (201) to generate (404) first measurement results of the first qubit group and of the second quantum circuit with the second qubit group (202) to generate (404) second measurement results of the second qubit group; • Provide (405) the first measurement results and the second measurement results.

2. Method (400) according to claim 1, wherein the list includes at least one delay value which shifts a start time of the second quantum circuit relative to a start time of the first quantum circuit, such that the quantum circuits are executed at least partially staggered in time.

3. Method (400) according to any one of the preceding claims, wherein the list of time delay values ​​includes at least one idle time, which is connected to a R. 413816 - 32 - or is inserted at several locations in the first quantum circuit and / or second quantum circuit.

4. Method (400) according to any of the preceding claims, wherein the delay values ​​are the result of an optimization of a correlator of a first observable of the first qubit group, which depends on a first list of delay values ​​and of a second observable of the second qubit group, which depends on a second list of delay values.

5. Method (400) according to claim 4, wherein the first observable and the second observable are identical.

6. Method (400) according to claim 4, wherein the first observable and the second observable differ from each other.

7. Method (400) according to one of the preceding claims, wherein the functional partitioning (300) of the qubit arrangement is determined by the following steps: • 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); • Provide (305) the positions of the qubit groups (201,202,203,204,205) on the qubit array.

8. Method (400) according to claim 7, wherein the optimization function is a sum of distance measures of the qubit groups (201,202,203,204,205) to each other. R. 413816 - 33 - 9. Method (400) according to one of claims 7 or 8, wherein the distance measure comprises a Euclidean distance (2001) between the geometric centroids (2002) of the qubit groups (201,202,203,204,205).

10. Method (400) according to any one of claims 7 to 9, wherein the optimization function comprises a sum of distance measures of the qubit groups (201,202,203,204,205) to each other, and wherein the distance measure is the shortest path distance (2000) of the qubit groups (201,202,203,204,205).

11. Method (400) according to one of claims 7 to 10, 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 factors of the qubits of the respective qubit group (201,202,203,204,205).

12. Method (400) according to one of claims 7 to 11, 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.

13. Method (400) 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.

14. Method (400) 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) or wherein at least two of the qubit groups (201,202,203,204,205) have different numbers of physical qubits (0,...,126).

15. Method (500) for determining an expectation value of an observable of a quantum system with the steps: R. 413816 - 34 - 1) Performing (503) the method according to one of the preceding claims, in particular on the quantum computer; 2) Adjusting (504) the list of time delay values ​​and / or re-executing steps 1) and 2) until a termination criterion is reached; 3) Determine (505) the expected value from the measurement results provided in step 1).

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