Method for reducing error in the determination of the expected value of an observable, method for determining a ground state energy, hybrid computer platform, error reduction algorithm

Abstract

The invention describes a method (100) for reducing error in the determination of an observable of a quantum system in a quantum state on a noisy quantum computer platform, comprising the following steps: • dividing (101) an observable operator into series of base operators; • providing (102) calibration operators; • measuring (103) the calibration operators, comprising determining (104) noisy measurement results (1040) and gate-error-reduced measurement results (1050) of the calibration operators; • ascertaining (106) a calibration function (1060) for mapping the noisy measurement results (1040) to the gate-error-reduced measurement results (1050) of the calibration operators; • determining (107) noisy measurement results (1070) of the series of base operators; • determining (108) error-reduced measurement results (1080) of the series of base operators by mapping (109) the noisy measurement results (1070) by means of the calibration function (1060); • determining (110) the observable by combining the error-reduced measurement results (1080) of the series of base operators.

Description

[0001] Description

[0002] title

[0003] Method for error reduction in the determination of the expectation value of an observable, method for determining a ground state energy, hybrid computer platform, error reduction algorithm

[0004] State of the art

[0005] When errors occur during the measurement of expectation values ​​of quantum mechanical observables for a given quantum state on currently available quantum computer platforms, a distinction can be made between errors during the calculation (gate errors) and errors during the measurement process (readout errors).

[0006] In “Scalable mitigation of measurement errors on quantum computers” (Nation et al.; PRX Quantum, 2:040326, Nov 2021) a method for reducing readout errors is described.

[0007] In “Error mitigation for short-depth guantum circuits” (Temme et al.; Phys. Rev. Lett., 119:180509, Nov 2017) an extrapolation method for reducing gate errors, “zero noise extrapolation” (ZNE), and an alternative method, “probabilistic error cancellation” (PEC), are described.

[0008] Core and advantages of the invention

[0009] Currently available quantum computing platforms, so-called noisy intermediate-scale quantum (NISQ) devices, can be used to calculate expectation values ​​of quantum mechanical observables for a given quantum state. The quantum state can be prepared by a quantum circuit that performs operations on a series of qubits using quantum gates (hereinafter referred to as gates). NISQ computers typically comprise a small number of qubits, for example, less than or equal to 1000 qubits, in particular 50 to several hundred qubits. The term "intermediate-scale" reflects the low computing power of these low-qubit, noise-sensitive quantum computers.

[0010] When measuring expectation values ​​of quantum mechanical observables for a given quantum state on currently available quantum computer platforms, errors can occur during the calculation (gate errors) and during the measurement process (readout errors).

[0011] Many strategies have been developed to correct both types of errors. Readout errors can be corrected, for example, using matrix-free measurement error mitigation, as described in "Scalable mitigation of measurement errors on quantum computers" (Paul D. Nation, Hwajung Kang, Neereja Sundaresan, and Jay M. Gambetta. PRX Quantum, 2:040326, Nov 2021), or the method described in "Model-free readout-error mitigation for quantum expectation values" (Ewout van den Berg, Zlatko K. Minev, and Kristan Temme. Phys. Rev. A, 105:032620, Mar 2022) (TREX, [2]).

[0012] Gate errors can be corrected, for example, using extrapolation methods such as the zero noise extrapolation (ZNE) method described in “Error mitigation for short-depth quantum circuits” (Temme et al.; Phys. Rev. Lett., 119:180509, Nov 2017) or probabilistic error amplification (PEA, see, for example, in “Evidence for the utility of quantum computing before fault tolerance.” (Kim et al. Nature 618(7965): 500- 505, Jun 2023)). An alternative is the use of probabilistic error cancellation (PEC, see, for example, “Error mitigation for short-depth quantum circuits” (Temme et al.; Phys. Rev. Lett., 119:180509, Nov 2017). These methods typically increase the variance and can lead to expected values ​​with extremely large errors (so-called outliers).

[0013] The present invention relates to a method for error reduction in the determination of an observable, in particular an expectation value of an observable, a method for determining a ground state energy of a quantum system, a hybrid computer platform, and an error reduction algorithm.

[0014] In the method described below, a calibration function is determined using a method for reducing gate errors. Thus, using the methods proposed in the claims, it is possible to determine the expected values ​​of quantum mechanical observables with greater accuracy and also reduce the variance and the number of outliers compared to known methods.

[0015] Further advantages include the fact that the number of measurements required to determine the expected value of a quantum mechanical observable in a quantum state does not need to be increased, or can even be reduced, compared to the determination using ZNE, PEC, PEA, or similar methods. This advantageously saves resources and increases the reliability of the measurement results.

[0016] In particular, the invention enables the provision of error-reduced, particularly gate-error-reduced, and thus more accurate measurement results for expected values ​​of observables using a noisy quantum computer platform. A further advantage of providing an improved method for error reduction is that it enables more reliable and efficient material simulations, thus enabling, supporting, and especially accelerating the development of new materials.

[0017] This is achieved with a method for error reduction in the determination of an observable of a quantum system in a quantum state on a noisy quantum computer platform according to claim 1.

[0018] A noisy quantum computer platform can be understood, in particular, as a hybrid computer platform comprising a classical computer and a quantum computer, in particular a NISQ computer. The classical computer can also be understood as a network of several classical computers, whereby the tasks of the classical computer can be distributed among one or more classical computers within the framework of the process, for example, depending on resource requirements. Quantum computers programmable using quantum circuits can, in principle, be constructed from any quantum technology capable of implementing single- and multi-qubit gate operations. Architectures based on, for example, superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are currently being actively developed.A quantum computer comprises a qubit arrangement, wherein the qubit arrangement comprises a plurality of physical qubits, which can preferably be provided with devices or units adapted to the technology with which the qubits are realized for initializing (e.g., initializing the qubit in a basic quantum state), manipulating (e.g., applying 1-qubit and / or 2-qubit gates), and / or reading the physical qubits. Initializing a quantum computer comprises preparing an initial quantum state and providing gate-parameter-based control signals for controlling quantum gates of the quantum circuit to prepare a quantum state, for example, for measuring an expected value of an observable. The control signals depend on the technology of the quantum computer used.In the case of qubits based on superconducting circuits, the manipulation of the qubits can be achieved, for example, via the applied voltage, the magnetic field, or via coupling to microwave resonators, so that the control signals are configured, for example, to adjust the magnetic field and / or the frequency of the microwave resonators. In particular, the control signals can comprise electrical signals. The initial quantum state specifies, in particular, which quantum states the qubits of the quantum computer are assigned at the beginning of the process.

[0019] Algorithms and applications that utilize quantum mechanical resources can be easily and efficiently written in the language of quantum logic circuits. A quantum circuit is a computational routine constructed 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 representing the final quantum data generated by the quantum circuit's computation. Operations on qubits are represented by boxes placed on these wires. Quantum gates are the elementary operations that a quantum computer can perform on its qubits. They are comparable to electronic gates, which perform the elementary operations of a classical computer. However, a quantum gate operates on quantum mechanical systems such as spin.Quantum operations are mathematically realized by matrix multiplication with unitary matrices. Unitary matrices are always invertible, and thus the input values ​​of a circuit can be reconstructed from the output values. For quantum gates that operate on two qubits (2-qubit gates), an interaction between the physical qubits in question is required. For spin qubits, this can occur, among other things, through 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 the applied voltage, the magnetic field, or via coupling to microwave resonators.

[0020] For example, an important task of currently available quantum computers is calculating the ground state of a quantum mechanical system. Currently available noisy intermediate-scale quantum (NISQ) computers are limited in their capabilities. Due to their limited size (small number of qubits) and inherent gate errors, such noisy NISQ computers allow the execution of short quantum circuits, i.e., quantum circuits of small depth, and the results typically exhibit large error bars.

[0021] Errors can occur both during the execution of the quantum circuit—in other words, during the application of the quantum gates (gates for short)—and during the readout—in other words, during the measurement after the application of the quantum circuit. The method according to claim 1 enables a reduction of these errors, in particular gate errors. In other words, the method enables a significant reduction in the variance of the expected values, a reduction in outliers, and it does not require an increase in the number of measurements. This makes it possible to implement longer quantum circuits, i.e., quantum circuits with greater depth, and thus also to reduce errors in the results of more complex algorithms, which require longer quantum circuits, and thus potentially enable their application in the first place.

[0022] An observable, especially a quantum observable, can be understood as a measurable quantity and its associated operator acting in a state space, a Hilbert space. Examples of observables are energy (the associated operator is the Hamiltonian of the quantum system), spatial coordinates, momentum coordinates, and the components of a particle's spin. Observables assign values ​​to the results of certain measurements that correspond to the eigenvalue of the operator. A crucial difference between classical quantities and quantum observables is that some pairs of quantum observables cannot be measured simultaneously. If the operators of two quantum 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 its result. Observables that cannot be measured simultaneously with arbitrary precision are also called complementary observables. The term "quantum system" refers to physical systems in which manifestations of quantum mechanics are visible. Examples of such manifestations are the quantization of energy or other observables, the interference of particle waves, non-locality, or quantum mechanical tunneling. Quantum systems encompass the entire microscopic world, such as elementary particles and atoms, but also electrical conductors with dimensions in the nanometer range, semiconductors, large and small molecules, and certain materials whose macroscopic properties are determined by quantum mechanical interactions on microscopic scales.In particular, a quantum system can be described by a Hamiltonian. In quantum mechanics, the Hamiltonian of a system is an operator that describes the total energy of that system, including kinetic energy and potential energy. Its spectrum, the energy spectrum of the system, includes the eigenvalues ​​of the Hamiltonian, i.e., the energy eigenvalues. This is the set of possible results that can be obtained by measuring the total energy of the system. Due to its close relationship to the energy spectrum and the time evolution of a system, it is of fundamental importance for most formulations of quantum theory and quantum chemistry. Measurement errors in the results can be reduced using the method proposed here. This can improve the reliability of the measurement results or, in some cases, even enable the usability of the results.

[0023] The method according to claim 1 comprises dividing an observable operator into series of basis operators. The division can preferably be carried out by a user, by executing an at least partially computer-implemented method on the quantum computer platform, in particular on a classical computer. Alternatively or additionally, the division can be carried out by data transmission or wireless or wired data transmission or by retrieving information about the division of the observable operator, for example from a database. In particular, during the division, the observable operator can be divided into a sum of Pauli series, where a Pauli series is an example of a series of basis operators. A Pauli series here comprises a sequence, in other words a product ora sequence of Pauli matrices on several qubits, where the Pauli matrices are examples of basis operators and can be denoted by I,X,Y,Z, respectively, where:.

[0024] Mi !) • ! ;) • ” ?) • ; - In other words, the Pauli series is not a product of the matrices of the same qubit (i.e. again a 2x2 matrix) but in particular a tensor product of the matrices of several qubits (i.e. for two qubits the Pauli series is then a 4x4 matrix). An example of a series of basis operators in the form of a Pauli series is: XYXZ. The Pauli matrices are special complex Hermitian 2x2 matrices. Together with the 2x2 identity matrix, which in this context, they form both a basis of the 4-dimensional real vector space of all complex Hermitian 2x2 matrices and a basis of the 4-dimensional complex vector space of all complex 2x2 matrices. Alternatively, sets of other matrices can be used as basis operators and a sequence of these basis operators as a series for the partition of the Observable operator. An example of partitioning the observable operator Ö into a sum of Pauli series Ä( is: where l numbers the Pauli series and a t corresponds to a weighting of the Zth Pauli series. In other words, the observable operator Ö in this example is expressed as a sum of weighted Pauli series. More generally, the observable operator Ö is expressed as a sum of weighted series of basis operators. A series can be understood as a basis operator or a product, especially a tensor product, of several basis operators of different qubits.

[0025] The method according to claim 1 further comprises providing calibration operators, wherein these comprise at least one of the series of basic operators. "At least one of the series" can be understood in particular to mean that at least one of the series is used as a calibration operator, that two or more of the series of basic operators are each used as a calibration operator, in other words, that each of the two or more series of basic operators each forms a calibration operator, or that all of the series of basic operators are each used as calibration operators, in other words, that each of the series of basic operators each forms a calibration operator.

[0026] In the above example, where Pauli series are used as series of basis operators (Pauli matrices), the calibration operators would therefore be at least one Pauli series, preferably two or more, in particular all Pauli series.

[0027] Preferably, the provision can be performed by user input, by data transmission or wireless or wired data transmission, or by retrieval, for example, from a database. In particular, the information required for the method regarding the choice of calibration operators and / or the division of the observable operator into the series of basis operators is made available by the provision. This step is preferably performed by the classical computer.

[0028] The method further comprises measuring the calibration operators. The measurement may, in particular, include determining noisy measurement results of the calibration operators.

[0029] Executing a quantum circuit on a quantum computer specifically means that gates provided in the quantum circuit are executed on the qubits assigned to the initial quantum state, and measurements of observables / operators, i.e., expected values, are performed, so that executing the quantum circuit serves to generate measurement results. In other words, measuring the calibration operators involves executing the quantum circuit to prepare the quantum state for which the calibration operators are to be measured. Executing the quantum circuit preferably occurs on the quantum computer. In particular, it involves executing the quantum circuit multiple times and measuring the calibration operators on the quantum computer. Measuring expected values ​​of observables is an essential component of quantum algorithms.This requires a large number of individual measurements for statistical convergence to meet precision requirements, such as chemical accuracy in applications to quantum chemistry calculations. To give an order of magnitude for the number of measurements: it is usually true that a measurement of an observable with precision s requires a number of measurements of order 1 / s. 2 requires.

[0030] Commuting operators can be measured simultaneously, meaning that a single measurement can measure multiple commuting calibration operators simultaneously. The results for each calibration operator can be stored individually or transferred to a storage unit, particularly a cloud. This advantageously allows for reducing the number of individual measurements required to evaluate the quantum system's observables, saving resources, and reducing potential errors.

[0031] The term “determining noisy measurement results of the calibration operators” can be understood in particular as determining the expected value of the observables from the results of the individual measurements. In other words, the noisy measurement results can in particular be measurement results determined using the noisy quantum computer platform, in particular the quantum computer, to which no error reduction methods were applied, i.e. for which neither readout errors nor gate errors were corrected. Alternatively or additionally, the noisy measurement results can in particular be measurement results that were obtained using a method for reducing readout errors, for example with matrix-free measurement error reduction, as described, for example, in “Scalable mitigation of measurement errors on quantum computers” (Paul D. Nation, Hwajung Kang, Neereja Sundaresan, and Jay M. Gambetta.PRX Quantum, 2:040326, Nov 2021) or the method described in “Model-free readout-error mitigation for quantum expectation values” (Ewout van den Berg, Zlatko K. Minev, and Kristan Temme. Phys. Rev. A, 105:032620, Mar 2022.” (TREX, [2]). In this case, where the noisy measurement results are readout-error reduced, the “determination of noisy measurement results of the calibration operators” then also includes applying a method for reducing readout errors to the measurement results determined using the noisy quantum computer platform, in particular the quantum computer. In this case, the noisy measurement results are then, in particular, readout-error reduced measurement results.

[0032] The matrix-free measurement error reduction method calibrates a readout error matrix for each qubit. This means that the probabilities are determined at which a quantum state (|1) of the respective qubit is incorrectly measured as 0 and a quantum state (|0) is incorrectly measured as 1. Using this error matrix, the method is able to correct or reduce readout errors and thus provide measurement results with reduced readout errors. The only overhead is the calibration measurements for determining the error matrices, which require a large number (on the order of 10,000) individual measurements, but need only be performed once and can then be used for multiple observables and quantum circuits.

[0033] Furthermore, the measurement includes performing gate error reduction to generate gate-error-reduced measurement results of the calibration operators. This step is preferably performed on the quantum computer. In particular, one of the gate error reduction methods described above, such as extrapolation methods such as the "zero noise extrapolation" (ZNE) method described in "Error mitigation for short-depth quantum circuits" (Temme et al.; Phys. Rev. Lett., 119:180509, Nov 2017) or probabilistic error amplification (PEA, see, for example, in "Evidence for the utility of quantum computing before fault tolerance." (Kim et al.; Nature 618(7965): 500-505, Jun 2023)), can be used. An alternative is the use of probabilistic error cancellation (PEC, see for example “Error mitigation for short-depth quantum circuits” (Temme et al.; Phys. Rev. Lett., 119:180509, Nov 2017) to generate gate-error-reduced measurement results of the calibration operators.

[0034] Specifically, zero-noise extrapolation is a strategy for reducing gate errors. It involves finding a way to vary the size of the error, measuring the result at different values ​​of the error, and then extrapolating to the zero-error result (zero-noise extrapolation). In other words, zero-noise extrapolation (ZNE) can amplify gate errors by adding additional gates, such as Pauli gates, to the quantum circuit, which would cancel out on a noise-free quantum computer. On a noisy quantum computing platform, the noise of the gates adds up instead of canceling them out. For example, when applying ZNE, CNOT gates are replaced by three CNOT gates that perform exactly the same operations but amplify the noise by a factor of three.After measuring the expected value at different noise factors (noise gain levels), a function is fitted to the measurement results and evaluated at a noise factor of zero. This extrapolation leads to a gate-error-reduced result. The variance of the expected values ​​at the different noise factors due to the statistical measurement process or readout errors leads to an even higher variance of the extrapolated result. In some cases, this can lead to expected values ​​with large errors; for example, the expected values ​​of Pauli series lie in the interval [-1; 1], and their error can be greater than 0.5.

[0035] Probabilistic error amplification (PEA) works similarly to ZNE, but differs in the method used to amplify the noise. Probabilistic error cancellation (PEC) is capable of determining expected values ​​with noise factors less than 1, but at the cost of an increased number of measurements.

[0036] The measuring step can, in particular, comprise storing the noisy and gate-error-reduced measurement results. In particular, a plurality of individual measurements are performed using the quantum computer. The noisy and gate-error-reduced measurement results can then be determined from the results of these individual measurements. A plurality of individual measurements is required to determine the noisy and gate-error-reduced measurement results. In particular, the results of each individual measurement can be transmitted from the quantum computer to the classical computer and / or a storage unit, and the results of the individual measurements can be stored there.Transmission can be understood in particular to mean that the results of each of the individual measurements are transmitted to the classical computer and / or to a storage unit, and the measurement results are determined from the results of the individual measurements on the classical computer. The transmission of the results of the individual measurements from the quantum computer to the classical computer can be carried out wirelessly or wired, both via the Internet and locally to the classical computer, using a communication unit. Alternatively or additionally, the results of the individual measurements can be transferred from the quantum computer to a buffer, in particular to a storage device external to the quantum computer and the classical computer, such as a cloud, from which the data can be retrieved using the classical computer.For example, the quantum computer can be part of a cloud computing platform to which the classical computer has access, allowing it to retrieve the results of the quantum computer's individual measurements from there. These are just a few examples of how the quantum computer platform can determine measurement results. Other variants for determining measurement results are also possible.

[0037] In particular, it is possible to use measurement results from the gate error reduction step to generate gate-error-reduced measurement results of the calibration operators to determine the noisy measurement results of the calibration operators. For example, if ZNE is used for gate error reduction, the noisy measurement results correspond to the measurement results from the ZNE measurement for a noise factor of 1.

[0038] Furthermore, the method according to claim 1 comprises determining a calibration function for mapping the noisy measurement results to the gate-error-reduced measurement results. For this purpose, a parameterized function f(x) is defined on an interval [-1, 1], the parameters of which are adjusted such that this function f(x) maps the noisy measurement results to the gate-error-reduced measurement results. Preferably, a monotonic function is chosen as the function f(x), for which the following applies: f(-1) = -1 and f(1) = 1.

[0039] An example that satisfies the properties and can therefore be used as such a parameterized function f(x) is: p(x) denotes a polynomial whose coefficients are determined by fitting the function to map the noisy measurement results to the gate-error-reduced measurement results. In other words, a fit is performed with the parameterized function f(x), and the parameters are determined such that the function provides a mapping rule from the noisy measurement results to the gate-error-reduced measurement results. The function can be adapted to the measurement results by fitting, i.e., by adjusting the parameters.

[0040] Another example of a suitable parameterized function f (x) is: (x) = max (— ​​1, min (1, p(x))) Here, too, p denotes a polynomial whose coefficients are determined by adapting the function to map the noisy measurement results to the gate-error-reduced measurement results.

[0041] The function provided with the parameters determined by fitting the function is then the calibration function. Fitting the function to determine the parameters is part of the step of determining a calibration function to map the noisy measurement results to the gate-error-reduced measurement results. Specifically, this step is performed on the classical computer. In other words, the calibration function is preferably determined on the classical computer.

[0042] Furthermore, the method comprises determining noisy measurement results of the series of basis operators. This step is preferably carried out at least partially on the quantum computer. For this purpose, the expected values ​​for the series of basis operators are determined as described above for the calibration operators, in particular by

[0043] • the quantum computer is initialized with an initial quantum state,

[0044] • a quantum circuit is applied to the initial quantum state to prepare a quantum state for which the expectation value of the operator is to be measured, and

[0045] • the operator for this quantum state is measured.

[0046] It should also be noted here that the aforementioned steps serve to perform a single measurement. To determine the expected value of the operator, a large number of individual measurements are performed, and the expected value of the operator is then determined from these individual measurements, preferably on a conventional computer.

[0047] Since the calibration operators at least partially coincide with the series of basis operators, the expected values ​​for these coincident operators can be adopted or copied directly from the measurement results of the calibration operators. In particular, the measurement results can be retrieved from the storage unit and copied or saved for the corresponding series of basis operators. As already explained at the beginning for the calibration operators, the noisy measurement results can be directly the measurement results determined from the individual measurements of the quantum computer. Alternatively or additionally, the measurement results determined by the quantum computer can be readout-error reduced, preferably by the classical computer, and these readout-error-reduced measurement results can be used as noisy measurement results of the series of basis operators.

[0048] The method further comprises determining error-reduced measurement results of the groups of basic operators by mapping the noisy measurement results of the series of basic operators using the calibration function. This step is preferably performed on a conventional computer. In other words, the noisy measurement results of the series of basic operators are mapped to the error-reduced measurement results of the series of basic operators using the calibration function. This means that the error-reduced measurement results of the series of basic operators are calculated from the noisy measurement results of the series of basic operators using the calibration function.

[0049] From the thus determined, error-reduced measurement results of the series of basic operators, the expected value of the observable can now be determined by combining the error-reduced measurement results of the series of basic operators according to the initially selected division of the observable operator, for example, into a sum of weighted series of basic operators. "Determining an observable" can, in particular, involve determining an expected value of the observable. In other words, the observable is determined by combining the error-reduced measurement results of the series of basic operators. This step is preferably performed on a classical computer.In particular, the determination may comprise calculating the expected value of the observables by summing the error-reduced measurement results of the series of basis operators, which are weighted by means of the weights used when dividing the observable operator into the series of basis operators.

[0050] Preferably, the expected value of the observable determined in this way can be output. The output can be transmitted, for example, to a hybrid computer platform, a classical computer, a quantum computer, a cloud, and / or a display device. Alternatively or additionally, the output can be used as the input of another algorithm, in particular a material simulation algorithm in which the quantum system is analyzed for certain material properties.

[0051] The method advantageously enables the determination of an error-reduced expectation value of an observable of a quantum system in a quantum state using a noisy quantum computer platform, in particular comprising a NISQ computer. In particular, the method enables the reliability of the measurement results of the quantum computer platform to be improved, thus opening up the use of quantum computers for new classes of problems that require higher accuracy and improved reliability of measurement results.

[0052] Determining an observable, in particular the expected value of an observable, can be understood as determining at least one, two, or more than two observables. In the case of determining the expected values ​​of multiple observables, the series of basis operators, in particular Pauli series, comprise all series of basis operators required to obtain the expected values ​​of all observables.

[0053] According to one embodiment, measuring the calibration operators can involve simultaneously measuring commuting calibration operators. One advantage is that the number of individual measurements can be reduced compared to the number of individual measurements required when each calibration operator is measured in a separate individual measurement, since each individual measurement provides measured values ​​for multiple (commuting) calibration operators, from which the expected values ​​of the calibration operators can then be subsequently calculated. This consequently enables resource and time savings.

[0054] According to one embodiment, the method may comprise preparing the quantum state on the noisy quantum computing platform. In particular, this may be done by applying a quantum circuit to an initial quantum state. When initializing the quantum computer, in other words setting the initial quantum state, the qubits of the quantum computer are brought into the initial quantum state. For example, all qubits are brought into the state |0) or all qubits are brought into the state |1). Alternatively, the qubits are brought into an initial qubit state sequence, i.e., a first number of qubits are brought into the state |0) and a second number of qubits are brought into the state |1). An example of a qubit state sequence of five qubits in which all qubits are in the state |0) is: 00000.Initialization further includes providing control signals, which are specified in particular by the quantum circuit for preparing the quantum state for which the observable expectation value is to be determined. During the preparation of the quantum state, the quantum gates manipulate the states of the qubits of the quantum computer according to the quantum gates provided in the quantum circuit, so that the prepared quantum state is present after the quantum circuit is executed.

[0055] According to one embodiment, performing a gate error reduction to generate gate error-reduced measurement results of the calibration operators comprises performing one or more of the following gate error reduction methods: Zero-Noise Extrapolation (ZNE), Probabilistic Error Amplification (PEA), Probabilistic Error Cancellation (PEC).

[0056] Furthermore, a method for determining a ground state energy of a quantum system is proposed, wherein the method is applied for error reduction in the determination of a Hamiltonian, in particular the expectation value of the Hamiltonian of the quantum system, in a ground state of the quantum system in order to obtain an error-reduced ground state energy.

[0057] In particular, the corresponding ground-state energy can be output and / or stored. The output can be transmitted, for example, to a hybrid computing platform, a classical computer, a quantum computer, a cloud, and / or a display device. Alternatively or additionally, the output can be used as the input of another algorithm, in particular a material simulation algorithm, in which the quantum system is analyzed for certain material properties.

[0058] Finding the ground state of a quantum mechanical system is an important task in the context of atomistic material simulations and in the field of quantum chemistry.

[0059] According to one embodiment, the quantum system whose ground-state energy is to be determined is a many-body system, which can be described, for example, by a Hubbard Hamiltonian. The Hubbard model is an approximate model of a solid. It describes the behavior of electrons in a lattice assumed to be rigid. The repulsive Coulomb forces are only considered for those electrons that are located at the same lattice site. The kinetic energy contribution of the electrons is modeled by an overlap integral derived from the tight-binding model. Some examples of quantum systems that can be described by a Hubbard Hamiltonian are strongly correlated fermion systems, transition metals, mobile electron systems (e.g., ferromagnetism, antiferromagnetism, ferrimagnetism), and TT electron systems in quantum chemistry.One advantage is that the process accelerates and, in some cases, even enables the development and investigation of new materials. Furthermore, the properties of these new materials can be better adapted to the respective application.

[0060] The advantages listed above also apply to the use of the aforementioned methods for a material simulation, where a Hamiltonian of the quantum system is a many-body Hamiltonian that describes the material to be simulated.

[0061] A hybrid computer platform, which is an example of a noisy quantum computer platform, comprises a classical computer and a quantum computer for executing a quantum circuit, which are adapted such that the steps of the methods described above are executable and / or that the use of the method for a material simulation is executable, has the advantage that it can be used particularly efficiently for material simulation.Adaptation can be understood in particular as meaning that, for example, the hardware of the quantum computer can be tuned to the quantum circuit so that the mapping of the logical qubits of the quantum circuit to the physical qubits of the quantum computer is possible and preferably requires the addition of as few as possible to a negligible number of additional SWAP operations, which ensure the interaction of the physical qubits occupied by the logical qubits when executing, for example, 2-gate operations. In particular, the hardware of the quantum computer can be selected based on the provided quantum circuit, thus enabling an even more efficient and less noise-susceptible execution of the method. The hybrid computer platform can be understood in particular as comprising at least one classical computer and at least one quantum computer.Preferably, it includes a cloud computing platform to which the classical computer has access, so that it can retrieve, in particular, the measurement results of the quantum computer from there.

[0062] According to one embodiment, a NISQ computer is used as a quantum computer.

[0063] Furthermore, an error reduction algorithm is proposed which causes a hybrid computer platform, comprising a classical computer and a quantum computer for executing a quantum circuit, to execute at least one of the methods described above.

[0064] A computer-readable storage medium on which the error reduction algorithm, which is an implementation of the method described above, is stored can be controlled in particular by the quantum computer and / or the classical computer to provide the error reduction algorithm.

[0065] Short description of the drawings

[0066] Embodiments of the invention are illustrated in the drawings and explained in more detail in the following description. Identical reference numerals in the figures denote identical or equivalent elements.

[0067] It shows

[0068] Fig. 1 is a flowchart of a procedure for determining an observable of a quantum system in a quantum state on a noisy quantum computer platform;

[0069] Fig. 2 is a sketch of a diagram showing an example of a calibration function;

[0070] Fig. 3 is a sketch of a diagram comparing the measurement results adjusted using different error reduction methods; and

[0071] Fig. 4 is a schematic representation of a hybrid computer platform.

[0072] Embodiments of the invention

[0073] Fig. 1 shows a flowchart of a method 100 for reducing errors when determining an observable 1100 of a quantum system in a quantum state on a noisy quantum computer platform. The method 100 determines expectation values ​​of an observable operator Ö, in other words, an operator Ö describing the observable. For this purpose, the quantum state of the quantum system for which the expectation value is to be determined is prepared using a quantum circuit on the quantum computer. The following notation is used below: an observable operator Ä describes a quantum mechanical operator;

[0074] Ä describes a noisy expected value (in other words a noisy measurement result) of the observable A;

[0075] Ä describes a gate-error-reduced expected value (in other words a gate-error-reduced measurement result) of the observable A, where, for example, zero-noise extrapolation, probabilistic error amplification and / or probabilistic error cancellation are applied for gate error reduction;

[0076] Ä describes the expected value of the observable A 1100 determined by the error reduction method 100 shown in Fig. 1. The embodiment of the method 100 shown in Fig. 1 comprises the following steps:

[0077] Splitting 101 an observable operator into series of basis operators; In this embodiment, the Pauli matrices are used as basis operators. During splitting, the observable operator is split into a sum of Pauli series, where a Pauli series P tis an example of a series of basis operators. A Pauli series comprises a sequence of several Pauli matrices, where the Pauli matrices are denoted by I, X, Y, Z, respectively, where:

[0078] In this example, the observable operator A is divided / decomposed into a weighted sum of Pauli series as follows: where a t the weighting of the Z-th Pauli series P t designated.

[0079] Providing 102 calibration operators, wherein the calibration operators comprise at least one of the series of basis operators; In this step, a set of calibration Pauli series C k as calibration operators. This set of calibration Pauli series C k can be at least one of the Pauli series P t Alternatively or additionally, the set of calibration Pauli series C ktwo or more, in particular all of the Pauli series P t In general, the set of calibration Pauli series C k be constructed in such a way that it can be measured in as few measurements as possible, which means that as many of the Pauli series as possible should commute. One way to construct such commuting Pauli series is to use some Pauli series P t from the series of basis operators and to construct Pauli series from them, which are related to the Pauli series P selected from the series of basis operators tcommute and can thus be measured in a single common measurement. Constructing commuting operators can, for example, involve replacing some Pauli operators (Pauli matrices) of the Pauli series with the identity operator. An example of this is the Pauli series: XYXZ. Commuting Pauli series can be achieved by replacing one or more Pauli operators with the identity operator / . Four of the 15 possible examples of commuting Pauli series constructed from this Pauli series are: XYXI, XYIZ, XYII, IYII.

[0080] Measuring 103 the calibration operators, comprising: o Determining 104 noisy measurement results C k 1040 of the calibration operators and o performing 105 a gate error reduction, for example ZNE, to generate gate error-reduced measurement results C k1050 of the calibration operators; In particular, this step, alternatively or additionally steps 103 and / or 105, is performed simultaneously for all commuting Pauli series. If ZNE is used as a method for gate error reduction, the noisy measurement results correspond to C k 1040 of the calibration operators to the measurement results for ZNE at a noise factor of 1. In other words, the determination 104 of noisy measurement results C k 1040 of the calibration operators in this case are done by using / determining / copying the ZNE measurement results at a noise factor of 1.

[0081] Determining 106 a calibration function 1060 for mapping the noisy measurement results 1040 to the gate-error-reduced measurement results 1050. For this purpose, in particular, a parameterized function (%) is defined on an interval [-1, 1], the parameters of which are adjusted such that this function f(x) maps the noisy measurement results 1040 to the gate-error-reduced measurement results 1050. Preferably, a monotonic function is selected as the function f(x), for which the following applies: f(-l) = -1 and f(l) = 1.

[0082] An example that satisfies the properties and can therefore be used as such a parameterized function f(x) is: p(x) denotes a polynomial whose coefficients are determined by fitting the function for mapping the noisy measurement results 1040 to the gate-error-reduced measurement results 1050. In other words, a fit is performed with the parameterized function f(x), and the parameters are determined such that the function provides a mapping rule from the noisy measurement results 1040 to the gate-error-reduced measurement results 1050.

[0083] Another example of the parameterized function f (x) is: (%) = max (— ​​1, min (1, p(x))) p(x) also denotes a polynomial whose coefficients are determined by adapting the function to map the noisy measurement results 1040 to the gate-error-reduced measurement results 1050.

[0084] Determining 107 noisy measurement results 1070 of the series of basis operators, in this embodiment all Pauli series P (; This gives the noisy expected values ​​P t . All Pauli series Pj, which are already in the calibration operators C k did not need to be measured again, but their expected value can be copied from to Pj.

[0085] Determining 108 error-reduced measurement results 1080 of the series of basic operators by mapping 109 the noisy measurement results 1070 of the series of basic operators with the calibration function 1060; In other words, in this embodiment, the P t using the calibration function 1060 with the parameters determined by the fit, to obtain the error-reduced expectation values ​​of the Pauli operators: Pi = fW-

[0086] Determining 110 the observable 1100 by combining the error-reduced measurement results 1080 of the series of basis operators. In this step, the error-reduced Pauli operators are recombined using the weights used in the splitting to form the observable operator A, whose expectation value should be determined using the method 100 shown in Fig. 1:

[0087] This method 100 is preferably executed on a noisy quantum computer platform, in particular a hybrid computer platform comprising a classical computer and a quantum computer. The method 100 can be implemented using an error reduction algorithm that causes a noisy quantum computer platform, in particular a hybrid computer platform comprising a classical computer and a quantum computer for executing a quantum circuit, to execute the method 100 shown in Fig. 1.

[0088] In a variant of the embodiment shown here, the method 100 is used to determine two or more observables Ä n used. In this case, the theorem of Pauli series P t chosen in such a way that the expected values ​​of all of the two or more observables can be determined.

[0089] The diagrams shown in Fig. 2 and Fig. 3 show results for an embodiment in which ten-qubit Pauli series (tensor product of ten Pauli matrices) were measured using IBM's superconducting-qubit quantum computer "ibm_torino." In this embodiment, the calibration operators C k chosen to include all Pauli series used for the partition of the observable operator: C k = P k . The noisy measurement results 1040 were each readout error reduced using the matrix-free error reduction method as described in "Scalable mitigation of measurement errors on quantum computers" (Paul D. Nation, Hwajung Kang, Neereja Sundaresan, and Jay M. Gambetta. PRX Quantum, 2:040326, Nov 2021).

[0090] Fig. 2 shows a sketch of a diagram showing an example of a calibration function. In particular, Fig. 2 shows the adaptation of the calibration function 1060 to the measurement results 1040, 1050, with the measurement points 1061 also shown, as well as the calibration function 1060 adapted to the measurement results 1040, 1050. The gate-error-reduced measurement results 1040 are plotted against the noisy measurement results 1050 (here: measurement results processed using matrix-free measurement error reduction). The measurement results are entered as points, each marked with a cross. The noisy measurement results 1040 (expected values) are plotted on the x-axis, and the gate-error-reduced measurement results 1050 (expected values), here the ZNE results, are plotted on the y-axis. A straight line and another curved curve are entered in the diagram.The curved curve, which moves on both axes in the interval [-1,1], is the calibration function 1060 (with a free parameter a) of the form:.

[0091] 2 TCX f(x) = — tan -1 (a ■ (tan— )) n 2

[0092] By adjusting the free parameter a, the function was adapted to the measurement results. Shown is the calibration function 1060, which maps the noisy measurement results 1040 to the gate-error-reduced measurement results 1050. The calibration function 1060 can be determined, for example, by a fit on a conventional computer.

[0093] Fig. 3 shows a sketch of a diagram which compares the measurement results which were adjusted using various error reduction methods. The measurement results are plotted on the y-axis: the noisy measurement results 1040, the gate error-reduced measurement results 1050 and those obtained using the error reduction method 100 shown in Fig. 1, in which the error reduction is achieved by applying the calibration function 1060. The exact, theoretical expected values, i.e. the expected values ​​precisely determined by simulation on a conventional computer, are plotted on the x-axis. If all the expected values ​​plotted here had an error of zero, all the points would lie on the straight line 300 shown. The diagram therefore clearly shows that the expected values ​​which were calculated using the method shown in Fig.1 proposed method 100 for error reduction, have a smaller error than the expected values ​​which were corrected only with ZNE 1050 or only with the matrix-free error reduction 1040.

[0094] Fig. 5 is a schematic representation of a hybrid computing platform 200, which is an example of a noisy quantum computing platform on which the method 100, as exemplified in Fig. 1, can be executed. The hybrid computing platform 200 includes the classical

[0095] Computer 201 and the quantum computer 202, which are adapted to carry out the method 100 as described by way of example in Fig. 1.

Claims

Claims 1. A method (100) for reducing errors in the determination of an observable of a quantum system in a quantum state on a noisy quantum computer platform, comprising the following steps: • Splitting (101) an observable operator into series of basic operators; • Providing (102) calibration operators, wherein the calibration operators comprise at least one of the series of basic operators; • Measuring (103) the calibration operators, comprising o determining (104) noisy measurement results (1040) of the calibration operators and o performing (105) a gate error reduction to generate gate error-reduced measurement results (1050) of the calibration operators; • Determining (106) a calibration function (1060) for mapping the noisy measurement results (1040) of the calibration operators to the gate-error-reduced measurement results (1050) of the calibration operators; • Determining (107) noisy measurement results (1070) of the series of basis operators; • Determining (108) error-reduced measurement results (1080) of the series of basic operators by mapping (109) the noisy measurement results (1070) of the series of basic operators with the calibration function (1060); • Determine (110) the observables by combining the error-reduced measurement results (1080) of the series of basis operators.

2. Method (100) according to one of the preceding claims, wherein the measuring (103) of the calibration operators and / or the determining (107) of noisy measurement results (1070) of the series of basis operators comprises a simultaneous measurement of commuting calibration operators.

3. Method (100) according to one of the preceding claims, wherein the basis operators are Pauli matrices.

4. The method (100) of any preceding claim, wherein the method comprises preparing the quantum state on the noisy quantum computer platform.

5. The method (100) of claim 4, wherein preparing comprises applying a quantum circuit to an initial quantum state.

6. The method (100) according to any one of the preceding claims, wherein performing (105) a gate error reduction to generate gate error-reduced measurement results (1050) of the calibration operators comprises performing one or more of the following gate error reduction methods: zero-noise extrapolation, probabilistic error amplification, probabilistic error cancellation.

7. The method according to any one of the preceding claims, wherein determining (106) the calibration function (1060) comprises adapting a parameterized function, wherein the parameters of the parameterized function are adapted such that this function maps the noisy measurement results (1040) to the gate-error-reduced measurement results (1050).

8. A method for determining a ground state energy of a quantum system, wherein the method (100) according to one of the preceding claims is applied for error reduction in the determination of a Hamiltonian of the quantum system in a ground state of the quantum system in order to obtain an error-reduced ground state energy.

9. Method (100) according to one of the preceding claims, wherein the quantum system is described by a many-body Hamiltonian of a material.

10. The method (100) according to any one of claims 8 or 9, wherein the Hamiltonian of the quantum system is a Hamiltonian for the Hubbard model.

11. Use of the method (100) according to one of the preceding claims for a material simulation, wherein a Hamiltonian of the quantum system is a many-body Hamiltonian which describes the material to be simulated.

12. A hybrid computer platform (200) comprising a classical computer (201) and a quantum computer (202) for executing a quantum circuit, which are adapted so that the steps of the method (100) according to any one of claims 1 to 10 can be carried out and / or the use of the method (100) according to claim 11 can be carried out.

13. The hybrid computer platform (200) of claim 12, wherein the quantum computer (202) is a NISQ computer.

14. An error reduction algorithm which causes a hybrid computer platform (200) comprising a classical computer (201) and a quantum computer (202) for executing a quantum circuit, in particular a hybrid computer platform (200) according to one of claims 12 or 13, to execute the method (100) according to one of claims 1 to 10.

15. A computer-readable storage medium on which the error reduction algorithm (1012) according to claim 14 is stored.

Images