Methods for error reduction in determining an expected value of an observable, methods for determining ground state energy, hybrid computer platform, error reduction algorithm
Patent Information
- Application Number
- CN202580021301.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Priority Date
- 2024-03-14
- Filing Date
- 2025-03-05
- Publication Date
- 2026-10-20
AI Technical Summary
[0057]计算机可读存储介质尤其可以由量子计算机和/或经典计算机操控，用于提供误差减少算法，在该计算机可读存储介质上存储有误差减少算法，该误差减少算法是上述方法的一种实现。
Smart Images
Figure FT_1 
Figure FT_2 
Figure FT_3
Abstract
Description
Background Technology
[0001] Errors that occur when measuring the expected value of a quantum mechanical observable for a given quantum state on currently available quantum computer platforms can be distinguished between errors during computation (gate errors) and errors during the measurement process (readout errors).
[0002] A method for reducing readout errors is described in “Scalable mitigation of measurement errors on quantum computers” (Nation et al.; PRX Quantum, 2:040326, November 2021).
[0003] In “Error mitigation for short-depth quantum circuits” (Temme et al.; Phys. Rev. Lett., 119:180509, November 2017), an extrapolation method for reducing gate errors, “zero noise extrapolation” (ZNE), and an alternative method, “probabilistic error cancellation” (PEC), are described. Summary of the Invention
[0004] Currently available quantum computing platforms, so-called noisy-intermediate-scale-quantum (NISQ) structural elements, can be used to calculate the expected value of quantum mechanical observables for a given quantum state. In this case, 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 mostly consist of a small number of qubits, such as less than or equal to 1000 qubits, especially 50 to several hundred qubits. The name "intermediate-scale" should take into account the low computational power of quantum computers with few qubits and sensitivity to noise.
[0005] When measuring the expected value of a quantum mechanical observable for a given quantum state on a currently available quantum computer platform, errors may occur during computation (gate error) and during the measurement process (readout error).
[0006] Many strategies have been developed to correct for both types of errors. Readout errors can be corrected, for example, by matrix-free measurement error reduction 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, November 2021), or by methods 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, March 2022 (TREX, [2])).
[0007] Gate errors can be corrected, for example, by 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, November 2017), or by probabilistic error amplification (PEA, e.g., see "Evidence for the utility of quantum computing before fault tolerance." (Kim et al., Nature 618(7965):500–505, June 2023)). An alternative is to use probabilistic error cancellation (PEC, e.g., see "Error mitigation for short-depth quantum circuits" (Temme et al.; Phys. Rev. Lett., 119:180509, November 2017). These methods typically lead to increased variance and may result in expected values with extremely large errors (so-called outliers).
[0008] The present invention relates to a method for reducing errors in determining the expected value of observables, especially observable quantities, a method for determining the ground state energy of a quantum system, a hybrid computer platform, and an error reduction algorithm.
[0009] In the method described below, a calibration function is determined using methods for reducing gate errors. Therefore, the method proposed in the claims can achieve: determining the expected value of a quantum mechanical observable with higher accuracy, and further reducing variance and the number of outliers compared to known methods.
[0010] Another advantage is that, compared to determinations using ZNE, PEC, PEA, or similar methods, the proposed method does not require increasing, or can even reduce, the number of measurements needed to determine the expected values of quantum mechanical observables in a quantum state. This advantageously saves resources and improves the reliability of measurement results.
[0011] This invention particularly enables the provision of more accurate, and especially less gate-error-adjusted, measurement results of observable expectations using a noisy quantum computer platform. An additional advantage of providing an improved method for error reduction is that it allows for more reliable and efficient material simulations, and thus enables, supports, and, in particular, accelerates the development of new materials.
[0012] This is achieved using the method for reducing errors when determining observables of a quantum system in a quantum state on a noisy quantum computer platform, as described in claim 1.
[0013] Noisy quantum computer platforms can be understood in particular as hybrid computer platforms, which include classical computers and quantum computers, especially NISQ computers. Classical computers can also be understood as a network of multiple classical computers, where, for example, depending on resource requirements, tasks of the classical computers can be distributed across one or more classical computers within this framework. Quantum computers capable of being programmed using quantum circuits can, in principle, be constructed using any quantum technology, provided that the technology can implement both single-qubit and multi-qubit gate operations. Currently, architectures based on, for example, superconducting circuits, ion traps, semiconductor quantum dots, photons, and neutral atoms are under active development. A quantum computer includes a qubit arrangement comprising multiple physical qubits, which are preferably configured to be used, via devices or units correspondingly adapted to the technology for realizing the qubits, for initializing the physical qubits (e.g., initializing the qubits in a ground quantum state), for manipulating the physical qubits (e.g., applying single-qubit gates (1-Qubit-Gatter) and / or two-qubit gates (2-Qubit-Gatter)), and / or for reading out the physical qubits. Initializing a quantum computer includes preparing initial quantum states and providing control signals based on gate parameters. These control signals are used to manipulate the gates of the quantum circuit to prepare quantum states, for example, to measure desired values of observables. The control signals depend on the technology of the quantum computer used. In the case of qubits based on superconducting circuits, manipulation of the qubits can be performed, for example, via an applied voltage, magnetic field, or via coupling with a microwave resonator, such that the control signals are, in this case, set, for example, to set the magnetic field and / or to set the frequency of the microwave resonator. In particular, the control signals can include electrical signals. The initial quantum states specifically indicate which quantum states the quantum computer's qubits are in at the start of the method.
[0014] Algorithms and applications utilizing quantum mechanics resources can be written simply and efficiently in the language of logical quantum circuits. A quantum circuit is a computational routine constructed from coherent quantum operations. Each horizontal line or circuit in a quantum circuit represents a qubit, where the left end of the line represents the original quantum data, and the right end represents the final quantum data produced by the computation through the quantum circuit. Operations on a qubit are represented by boxes placed on these lines. A quantum gate is a fundamental operation that a quantum computer can perform on its qubits. The quantum gate is similar to the electronic gate that performs the fundamental operations of a classical computer. However, quantum gates work with quantum mechanical systems, such as spin. Quantum operations are mathematically implemented through matrix multiplication using unitary matrices. Unitary matrices are always invertible, so the input values can be reconstructed from the output values of the circuit. For a quantum gate operating on two qubits (a two-qubit gate), an interaction between the relevant physical qubits is required. In the case of spin qubits, this can additionally occur via exchange interactions. Atoms in an ion trap can, for example, exchange photons. In the case of qubits based on superconducting circuits, these qubits can be manipulated, for example, by an applied voltage, magnetic field, or by coupling with a microwave resonator.
[0015] One of the important tasks currently available for quantum computers is calculating the ground state of quantum mechanical systems. The capabilities of currently available noisy medium-scale quantum (NISQ) computers are limited. Due to their limited size (few qubits) and inherent gate errors, such noisy NISQ computers allow for the implementation of short quantum circuits, that is, shallow quantum circuits, and the results typically have large error bars (Fehlerbalken).
[0016] Errors can occur both during the implementation of quantum circuits, i.e., during the application of quantum gates (hereinafter referred to as gates), and during readout, i.e., during measurement after the application of quantum circuits. The method according to claim 1 enables a reduction in these errors, particularly gate errors. In other words, the method enables a significant reduction in the variance of the expected value, a reduction in outliers, and without increasing the number of measurements. This makes it possible to implement longer quantum circuits, i.e., quantum circuits with greater depth, and thus also reduces the errors resulting from the more complex algorithms requiring longer quantum circuits, and consequently enables their application when necessary.
[0017] Observables, especially quantum mechanical observables, can be understood as measurement parameters and operators associated with them, operating in state space, i.e., Hilbert space. Examples of observables are energy (the associated operator being the Hamiltonian operator of a quantum system), position coordinates, momentum coordinates, and the spin component of a particle. An observable assigns a value corresponding to an eigenvalue of an operator to the result of a particular measurement. A decisive difference between classical parameters and quantum mechanical observables is that several pairs of quantum mechanical observables cannot be measured simultaneously. If the operators of two quantum mechanical observables do not commute, then the following applies: a measurement of the first operator alters the quantum state in a way incompatible with a subsequent measurement of the second observable, and vice versa. Measurement parameters that can be determined simultaneously and accurately are called commutative observables; these parameters have the property that the order of their operators in a product can be interchanged without changing the result of the product. Observables that cannot be measured simultaneously and arbitrarily accurately are also called complementary observables.
[0018] The term "quantum system" refers to a physical system in which quantum mechanical phenomena can be observed. Examples of such phenomena include the quantization of energy or other observable quantities, interference of particle waves, nonlocality, or quantum mechanical tunneling. Quantum systems encompass the entire microscopic world, such as elementary particles and atoms, but also include electrical conductors, semiconductors, macromolecules, and small molecules with dimensions in the nanometer range, as well as specific materials whose macroscopic properties are determined by quantum mechanical interactions at the microscopic scale. Quantum systems can be described, in particular, by the Hamiltonian operator. In quantum mechanics, the Hamiltonian operator of a system is the operator that describes the total energy of the system, including both kinetic and potential energy. The spectrum of this Hamiltonian operator, i.e., 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 achieved by measuring the total energy of the system. Due to the close relationship between this Hamiltonian operator and the energy spectrum and the time evolution of the system, it is fundamental to most formulations of quantum theory or quantum chemistry. Measurement errors in the results can be reduced using the methods presented herein. Therefore, the reliability of the measurement results can be improved, or even partially made usable.
[0019] The method according to claim 1 includes splitting the observable operator into a string of base operators. Preferably, the splitting can be performed by a user, by implementing at least a partially computer-based method on a quantum computer platform, particularly on a classical computer. Alternatively or additionally, the splitting can be performed by data transmission, wireless data transmission, wired data transmission, or by retrieving information about the splitting of the observable operator, for example, from a database. In particular, the observable operator can be split into a sum of Pauli strings during the splitting, where a Pauli string is an example of a string of base operators. In this case, the Pauli string comprises a sequence, in other words, a product or juxtaposition of Pauli matrices on multiple qubits, where the Pauli matrices are examples of base operators and can be used to represent, where applicable: .
[0020] In other words, a Pauli string is not a product of matrices of the same qubit (i.e., again a 2x2 matrix), but rather a tensor product of matrices of multiple qubits (i.e., for two qubits, a Pauli string is a 4x4 matrix). An example of a base operator string in Pauli string form is... The Pauli matrix is a special complex Hermitian 2×2 matrix. Together with the 2×2 identity matrix, in this context, it forms both the basis of the 4-dimensional real vector space of all complex Hermitian 2×2 matrices and the basis of the 4-dimensional complex vector space of all complex 2×2 matrices. Alternatively, sets of other matrices can be used as basis operators, and sequences of these basis operators can be used as strings for splitting observable operators. Split into Pauli string An example of a sum is: , in, Number the Pauli string, and Corresponding to the The weights of each Pauli string. In other words, in this example, the observable operator. This is expressed using the sum of weighted Pauli sequences. More generally, observable operators... It is expressed as the sum of weighted base operator strings. In particular, the product of a single base operator or multiple base operators of different qubits, especially the tensor product, can be understood as a string.
[0021] Additionally, the method according to claim 1 includes providing calibration operators, wherein these calibration operators include at least one of the base operator strings. "At least one of the strings" can be particularly understood as follows: - So that at least one of the strings is used as a calibration operator. - Two or more of the base operator strings are used as calibration operators, in other words, each of the two or more base operator strings forms a calibration operator, or... - All the base operator strings in the base operator string are used as calibration operators. In other words, each base operator string in the base operator string forms a calibration operator.
[0022] In the example above where Pauli strings are used as base operator strings (Pauli matrices), the calibration operator will therefore be at least one Pauli string, preferably two or more, and especially all of them.
[0023] Preferably, the provision can be made via user input, data transmission, wireless data transmission, wired data transmission, or, for example, by retrieving information from a database. This provision makes, in particular, the information required by the method for selecting calibration operators and / or splitting observable operators into strings of base operators available. This step is preferably performed by a classical computer.
[0024] Additionally, the method includes measuring a calibration operator. This measurement may, in particular, include noisy measurements determining the calibration operator.
[0025] Implementing quantum circuits on a quantum computer specifically means implementing gates set in the quantum circuit on qubits in their initial quantum states, and performing measurements on observables / operators, i.e., on desired values, so that the implementation of the quantum circuit is used to produce the measurement results. In other words, measuring the calibration operator involves implementing the quantum circuit to prepare a quantum state for which the calibration operator should be measured. It is preferable to implement the quantum circuit on a quantum computer. This implementation specifically involves multiple implementations of the quantum circuit and measurements of the calibration operator on the quantum computer. Measuring the desired value of the observable is a crucial component of quantum algorithms. This requires a large number of single measurements for statistical convergence to meet accuracy requirements, such as chemical accuracy when applied to quantum chemical calculations. To give the order of magnitude of the number of measurements: generally applicable, in terms of accuracy... Measuring observables requires orders of magnitude The number of measurements.
[0026] Commuting operators can be measured simultaneously, meaning that multiple commuting calibration operators can be measured simultaneously in a single measurement. The result of each calibration operator can be stored separately or transmitted to a storage unit, particularly the cloud. This advantageously enables a reduction in the number of single measurements required to analyze and process observables in quantum systems, saving resources and reducing potential errors.
[0027] "Determining the noisy measurement results of the calibration operator" can be understood in particular as determining the expected value of the observable from the results of a single measurement. In other words, the noisy measurement results can be, in particular, measurements determined by a noisy quantum computing platform, especially a quantum computer, where error reduction methods have not been applied to the measurement results; that is, neither readout error nor gate error has been corrected in the measurement results. Alternatively or additionally, the noisy measurement result may be a measurement result corrected by means of error reduction methods for readout errors, such as matrix-free measurement error reduction 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, November 2021) or in “Model-free readout-errormitigation for quantum expectation values” (Ewout van den Berg, Zlatko K. Minev and Kristan Temme, Phys. Rev. A, 105:032620, March 2022 (TREX, [2])). For the case where the noisy measurement result is obtained with readout error reduction, “determining the noisy measurement result of the calibration operator” further includes applying the methods for reducing readout errors to the measurement result determined by means of a noisy quantum computer platform, especially a quantum computer. In this case, the measurement results containing noise are especially those with reduced readout error.
[0028] A matrix-free measurement error reduction method calibrates the readout error matrix for each qubit. This means determining the following probabilities: under these probabilities, the quantum state of the corresponding qubit... Measured incorrectly as ,and The state was incorrectly measured as Using this error matrix, the method can correct or reduce readout errors, and thus provide measurement results with reduced readout errors. The only overhead is the calibration measurements used to determine the error matrix, which require a large number (on the order of 10,000) of single measurements, but only need to be performed once, and can subsequently be used for multiple observables and quantum circuits.
[0029] Additionally, the measurement includes implementing gate error reduction to generate a calibration operator from the gate error-reduced measurement result. Preferably, this step is implemented on a quantum computer. In particular, one of the methods for gate error reduction described at the beginning can be used, such as an extrapolation method, for example, the "zero-noise extrapolation" (ZNE) method described in "Error mitigation for short-depth quantum circuits" (Temme et al.; Phys. Rev. Lett., 119:180509, November 2017), or a probabilistic error amplification (PEA, e.g., see "Evidence for the utility of quantum computing before fault tolerance." (Kim et al., Nature 618(7965):500–505, June 2023)). An alternative is to use probabilistic error cancellation (PEC, see, for example, “Error mitigation for short-depth quantum circuits” (Temme et al.; Phys. Rev. Lett., 119:180509, November 2017) to produce measurements with reduced gate errors for the calibrated operator.
[0030] Zero-noise extrapolation is a strategy for reducing gate errors. This strategy involves finding a way to vary the magnitude of the error, measure results under different error values, and then extrapolate to a 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. These additional gates would cancel each other out on a noiseless quantum computer. On a noisy quantum computer platform, the noise of the gates accumulates rather than compensates for it. For example, in applying ZNE, three CNOT gates are replaced with three CNOT gates that perform the exact same operation but amplify the noise by a factor of three. After measuring the expected value under different noise factors (the level of noise amplification), the function is fitted to the measurement results, i.e., analyzed when the noise factor is zero. This extrapolation results in a reduction in gate errors. The variance of the expected value due to statistical measurement procedures or readout errors under different noise factors leads to a further increase in the variance of the extrapolated result. In several cases, this can lead to expected values with large errors; for example, the expected value of a Pauli string may lie in the interval [missing information]. In, and its error can be greater than .
[0031] Probabilistic error amplification (PEA) works similarly to ZNE, but differs in its approach; it amplifies noise. Probabilistic error suppression (PEC) can determine noise levels when the noise factor is less than a certain threshold. The expected value under the given circumstances, but at the cost of an increased number of measurements.
[0032] The measurement process may, in particular, include storing noisy measurement results and gate-error-reduced measurement results. Specifically, a large number of single measurements are performed using a quantum computer. The noisy measurement results and gate-error-reduced measurement results can then be determined from the results of these single measurements. A large number of single measurements are required to determine the noisy measurement results and gate-error-reduced measurement results. Specifically, the results of each single measurement can be transferred from the quantum computer to a classical computer and / or storage unit, and the results of the single measurements are stored therein. The transfer can be understood in such a way that the results of each single measurement are transferred to the classical computer and / or storage unit, and the measurement result is determined from the results of the single measurements on the classical computer. The transfer of the results of the single measurements from the quantum computer to the classical computer can be done wirelessly or wired via a communication unit, both via the Internet and locally. Alternatively or supplementarily, the results of the single measurements can be transferred from the quantum computer to a buffer, especially to a storage device external to both the quantum computer and the classical computer, such as the cloud, from which the data can be retrieved by the classical computer. For example, a quantum computer could be part of a cloud computing platform, to which a classical computer has access, allowing the classical computer to retrieve the results of a single measurement from the quantum computer. This is just one example of how measurement results can be determined using a quantum computing platform. Other variations for determining measurement results are equally possible.
[0033] In particular, it is possible to use the measurement results from the step of implementing gate error reduction to generate the calibration operator, in order to determine the noisy measurement results of the calibration operator. For example, if ZNE is used for gate error reduction, the noisy measurement results correspond to those from the step of implementing gate error reduction to generate the calibration operator with a noise factor of The measurement results of ZNE measurement.
[0034] Additionally, the method according to claim 1 includes determining a calibration function for mapping noisy measurement results to measurement results with gated error reduction. For this purpose, in particular, a parameterized function is used. Defined in interval The parameters of the parameterized function are adapted to make the function... The noisy measurement results are mapped to the measurement results with reduced gate error. Preferably, a monotonic function is chosen as the function. This applies to the monotonic function: and .
[0035] It satisfies the characteristics and can therefore be used as an example of such a parameterized function. The example is: The polynomial is represented by a function that maps noisy measurements to measurements with reduced gate error to determine its coefficients. In other words, the coefficients of the polynomial are determined by performing a parameterization function. The function is fitted to the measurement results, and the parameters are determined so that it provides a mapping rule from noisy measurements to measurements with reduced gate error. By fitting, that is, by adapting the parameters, the function can be adapted to the measurement results.
[0036] Appropriate parameterized functions Another example is: Here, It also represents a polynomial, whose coefficients are determined by using an adaptation function to map noisy measurements to measurements with reduced gate error.
[0037] A function whose parameters are determined by an adaptation function is a calibration function. Adapting the function to determine the parameters is part of the following step: determining a calibration function to map noisy measurements to measurements reduced by gate error. In particular, this step is performed on a classical computer. In other words, it is preferable to determine the calibration function on a classical computer.
[0038] Additionally, the method includes determining noisy measurements of the base operator string. Preferably, this step is implemented, at least partially, on a quantum computer. For this purpose, the expected value of the base operator string is determined, as described above for the calibration operator, in particular: ● Initialize the quantum computer with the initial quantum state. ● Applying quantum circuits to an initial quantum state to prepare a quantum state in which the expected value of the operator should be measured, and ● Measurement operator for this quantum state.
[0039] It should also be noted that the above steps are for performing a single measurement. To determine the expected value of the operator, a large number of single measurements are performed, and the expected value of the operator is subsequently determined by these single measurements, preferably on a classical computer.
[0040] Since the calibration operator is at least partially consistent with the base operator string, the expected values of these consistent operators can be directly adopted or copied from the measurement results of the calibration operator. In particular, the measurement results can be retrieved from the storage unit and copied or stored for the relevant base operator string. As already explained at the beginning regarding the calibration operator, noisy measurement results can be directly determined by a single measurement of a quantum computer. Alternatively or supplementarily, readout error reduction can be performed on the measurement results determined by a quantum computer, preferably by a classical computer, and these readout error-reduced measurement results can be used as noisy measurement results of the base operator string.
[0041] Furthermore, the method includes determining the error-reduced measurement result of the base operator group by mapping the noisy measurement results of the base operator string to the error-reduced measurement result of the base operator string using a calibration function. Preferably, this step is implemented on a classical computer. In other words, the noisy measurement results of the base operator string are mapped to the error-reduced measurement results of the base operator string using a calibration function. That is, the error-reduced measurement result of the base operator string is calculated from the noisy measurement results of the base operator string using a calibration function.
[0042] The expected value of an observable can now be determined from the error-reduced measurement results of the base operator strings by combining the error-reduced measurement results of the base operator strings according to the initial split of the observable operator, for example, splitting it into a weighted sum of base operator strings. "Determining the observable" can particularly include determining the expected value of the observable. In other words, the observable is determined by combining the error-reduced measurement results of the base operator strings. Preferably, this step is performed on a classical computer. The determination can be particularly achieved by summing the error-reduced measurement results of the base operator strings, weighting the error-reduced measurement results using the weights used when splitting the observable operator into base operator strings.
[0043] Preferably, a definite expected value of the observable can be output. This output can be, for example, transmitted to a hybrid computing platform, classical computer, quantum computer, cloud, and / or display device. Alternatively or additionally, the output can be used as input to another algorithm, particularly an algorithm for materials simulation, in which the quantum system is studied with respect to certain material properties.
[0044] This method advantageously enables the determination of the expected values of observables of a quantum system in a quantum state with reduced errors using a noisy quantum computing platform, particularly one including NISQ computers. In particular, this method enables improvements in the reliability of measurements obtained from quantum computing platforms, and thus facilitates the use of quantum computers for new categories of problems requiring higher accuracy and improved reliability in measurement results.
[0045] Determining the expected value of an observable, especially an observable quantity, can be understood in particular as determining at least one, two, or more than two observables. In the case of determining the expected value of multiple observables, the basic operator string, especially the Pauli string, comprises all the basic operator strings needed to obtain the expected values of all observables.
[0046] According to one implementation, measuring calibration operators may include simultaneously measuring commuting calibration operators. An advantage is that the number of measurements per measurement can be reduced compared to measuring each calibration operator in a separate single measurement, because each single measurement provides measurements of multiple (commuting) calibration operators, from which the desired value of the calibration operator can then be calculated. Therefore, this enables both resource and time gains.
[0047] According to one implementation, the method may include preparing quantum states on a noisy quantum computer platform. In particular, this can be done by applying quantum circuitry to the initial quantum state. When initializing the quantum computer, or in other words, setting the initial quantum state, the qubits of the quantum computer are placed in the initial quantum state. For example, all qubits are placed in the initial state. Or place all qubits in a state Alternatively, the qubits are placed in the initial qubit state sequence; that is, the first number of qubits are placed in the state sequence. And place the second number of qubits in the state Five qubits, all qubits are in a state. An example of a sequence of qubit states is 00000. Additionally, this initialization includes providing control signals, which are pre-given, in particular, by a quantum circuit for preparing a quantum state for which an expected value of an observable should be determined. During the preparation of the quantum state, quantum gates manipulate the states of the quantum computer's qubits according to the gates set in the quantum circuit, such that the prepared quantum state exists after the implementation of the quantum circuit.
[0048] According to one implementation, the gate error reduction used to generate the calibration operator through gate error reduction measurement results includes one or more of the methods for gate error reduction mentioned below: zero-noise extrapolation (ZNE), probability error amplification (PEA), and probability error elimination (PEC).
[0049] Furthermore, a method for determining the ground state energy of a quantum system is proposed, wherein the method is applied to reduce the error when determining the expected value of the Hamiltonian operator, especially the Hamiltonian operator of the quantum system, in the ground state of the quantum system, so as to obtain the error-reduced ground state energy.
[0050] In particular, the associated ground-state energy can be output and / or stored. This output can be made, for example, by transmitting it 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 input to another algorithm, particularly an algorithm for materials simulation, in which the quantum system is studied with respect to certain material properties.
[0051] Finding the ground state of a quantum mechanical system is an important task in the context of atomic-level materials simulation and in the field of quantum chemistry.
[0052] According to one implementation, the quantum system whose ground-state energy should be determined is a many-body system, which can be described, for example, by the Hubbard-Hamilton operator. The Hubbard model is an approximate model of solids. This Hubbard model describes the behavior of electrons in a lattice assumed to be rigid. Here, repulsive Coulomb forces are considered only for electrons located at the same lattice points. The share of the electron's kinetic energy is modeled by the overlapping integral from the tight-binding model. Several examples of quantum systems that can be described by the Hubbard-Hamilton operator are strongly correlated fermionic systems, transition metals, mobile electron systems (e.g., ferromagnetic, antiferromagnetic, and subferrimagnetic), and π-electron systems in quantum chemistry. The advantage is that this method thus accelerates the development and research of new materials, and to some extent even enables the development and research of new materials. Furthermore, the properties of these new materials can therefore be better adapted to their respective applications.
[0053] The advantages listed above also apply to the use of the aforementioned method for materials simulation, where the Hamiltonian operator of the quantum system is a multi-body Hamiltonian operator that describes the material to be simulated.
[0054] A hybrid computing platform is an example of a noisy quantum computing platform comprising a classical computer and a quantum computer for implementing quantum circuits, the classical computer and the quantum computer being adapted to enable the implementation of the steps of the above-described method and / or to enable the use of the method for materials simulation. This hybrid computing platform has the advantage of being particularly efficient for materials simulation. Adaptation can be understood in particular as the quantum computer hardware being coordinated with the quantum circuits, enabling the mapping of logical qubits of the quantum circuits to physical qubits of the quantum computer, preferably requiring only a minimal to a very small number of additional SWAP operations, which, in the case of implementing, for example, two-gate operations, ensure the interaction of the physical qubits occupied by the logical qubits. In particular, the quantum computer hardware can be selected based on the provided quantum circuits, thus enabling further, more efficient, and noise-insensitive implementations of the method. The hybrid computing platform can be understood in particular as comprising at least one classical computer and at least one quantum computer. Preferably, the hybrid computing platform includes a cloud computing platform to which the classical computer has access, enabling the classical computer to access the measurement results from the quantum computer, in particular.
[0055] According to one implementation, the NISQ computer is used as a quantum computer.
[0056] In addition, an error reduction algorithm is proposed, which prompts a hybrid computer platform, including classical computers and quantum computers for implementing quantum circuits, to implement at least one of the above methods.
[0057] Computer-readable storage media, in particular, can be manipulated by quantum computers and / or classical computers to provide error reduction algorithms, on which such error reduction algorithms are stored, and which are implementations of the methods described above. Attached Figure Description
[0058] Embodiments of the invention are shown in the accompanying drawings and explained in more detail in the following description. The same reference numerals in the drawings denote the same or equivalent elements.
[0059] The attached diagram shows: Figure 1 A flowchart illustrating the process of determining observables of a quantum system in a quantum state on a noisy quantum computer platform; Figure 2 A sketch of the following line graph is shown: This line graph illustrates an example of a calibration function; Figure 3The following is a sketch of a line graph: this line graph compares measurement results that have been adapted using different error reduction methods; and Figure 4 A schematic diagram of a hybrid computing platform is shown. Detailed Implementation
[0060] Figure 1 A flowchart illustrating the process of a method 100 for reducing errors when determining observables 1100 of a quantum system in a quantum state on a noisy quantum computer platform. Method 100 determines the observable operator. The expected value, in other words, the operator that determines the observable. The expected value of the quantum state is determined. To this end, the following quantum state of the quantum system is prepared on a quantum computer using quantum circuits: the expected value of this quantum state should be determined. The following notation will be used: - Observable operators Describe quantum mechanical operators; - Describing observables The expected value including noise (in other words, the measurement result including noise); - Describing observables The expected value of the reduction in gate error (in other words, the measurement result of the reduction in gate error), wherein, for example, zero-noise extrapolation, probability error amplification and / or probability error elimination are applied for gate error reduction; - Describing observables by means of Figure 1 The expected value determined by the error reduction method 100 shown. 1100.
[0061] Method 100 Figure 1 The illustrated embodiments include the following steps: - The observable operator is split into 101 base operator strings; in this embodiment, the Pauli matrix is used as the base operator. During the split, the observable operator is divided into a sum of Pauli strings, where each Pauli string... This is an example of a base operator string. In this case, the Pauli string consists of a sequential arrangement (Aneinanderreihung) of multiple Pauli matrices, where the Pauli matrices are represented here by... This indicates that, among which, applicable to: .
[0062] In this embodiment, the observable operator Decompose / decompose into a weighted sum of Pauli strings as follows: , in, Indicates the first A string of Pauli The weight.
[0063] - Provide 102 calibration operators, wherein the calibration operators include at least one base operator string from the base operator strings; in this step, select the set of calibration Pauli strings. As a calibration operator. This set of calibration Pauli strings. It can include Pauli strings At least one Pauli string in the set. Alternatively or supplementarily, calibrate the set of Pauli strings. It can include Pauli strings Two or more, especially all, Pauli strings. Generally, the set of calibrated Pauli strings... The set of calibrated Pauli strings should be constructed such that it can be measured in as few measurements as possible, which means that as many Pauli strings as possible should commute. One possibility for constructing such commuting Pauli strings is to select several Pauli strings from the base operator strings. The following Pauli string is constructed from it: the Pauli string and the Pauli string selected from the base operator string. Commutation occurs, and therefore can be measured in a single, common measurement. Operators for constructing commutation can, for example, include replacing several Pauli operators (Pauli matrices) of a Pauli string with identity operators. An example of this is a Pauli string: A Pauli string that commutes with this Pauli string can be identified by one or more Pauli operators. This is achieved through substitution. Four of the 15 possible examples of commutative Pauli strings constructed from this Pauli string are: , , , .
[0064] - Measurement of calibration operator 103, including: ○ Determine the noisy measurement results of the 104 calibration operators. 1040, and ○ Implement 105 gate error reduction, for example, ZNE is used to generate measurement results with gate error reduction for calibration operators. 1050; In particular, this step, or alternatively or supplementarily, also steps 103 and / or 105 are performed simultaneously for all commutative Pauli strings. If ZNE is used as a method for reducing gate errors, the noisy measurement results of the calibration operator... 1040 corresponds to a noise factor of In this case, the measurement results for ZNE are determined. In other words, in this case, the noisy measurement results of the 104 calibration operators are determined. 1040 can be achieved at a noise factor of In this case, use / confirm / copy the ZNE measurement results.
[0065] - A calibration function 1060 is determined to map the noisy measurement result 1040 to the measurement result 1050 with gate error reduced; for this purpose, a parameterized function is specifically defined. Defined in interval Above, the parameters of the parameterized function are adapted so that the function... The noisy measurement result 1040 is mapped to the measurement result 1050 with reduced gate error. Preferably, a monotonic function is chosen as the function. This applies to the monotonic function: and 。
[0066] It satisfies the characteristics and can therefore be used as an example of such a parameterized function. One example is: This represents a polynomial whose coefficients are determined by an adaptation function used to map the noisy measurement result 1040 to the measurement result 1050 with gate error reduced. In other words, it uses a parameterized function. Perform a fit and determine the parameters such that the function provides a mapping rule from the noisy measurement result 1040 to the measurement result 1050 with gate error reduced.
[0067] For parameterized functions Another example is: Here, It also represents a polynomial whose coefficients are determined by an adaptation function used to map the noisy measurement result 1040 to the measurement result 1050 with gate error reduced.
[0068] - Determine the 107-base operator strings, i.e., all Pauli strings in this embodiment. The measurement result containing noise is 1070; thus, the expected value containing noise is obtained. It has already been included in the calibration operator. All Pauli strings It does not need to be remeasured, but its expected value can be changed from... Towards copy.
[0069] - By mapping the noisy measurement result 1070 of the 109-based operator string with the calibration function 1060, the error-reduced measurement result 1080 of the 108-based operator string is determined; in other words, in this embodiment, mapping is performed using the calibration function 1060 with parameters determined by fitting. In order to obtain the expected value of the Pauli operator after error reduction: .
[0070] - By combining the error-reduced measurement results 1080 of the base operator string, the observable 1100 is determined. In this step, the error-reduced Pauli operators are reassembled into observable operators using the weights used during the splitting. The expected value of this observable operator should be obtained by means of... Figure 1 The method 100 shown determines: , Preferably, the method 100 is implemented on a noisy quantum computer platform, particularly a hybrid computer platform comprising classical and quantum computers. Method 100 can be implemented using an error reduction algorithm that facilitates implementation on the noisy quantum computer platform, particularly a hybrid computer platform comprising classical computers and quantum computers for implementing quantum circuits. Figure 1 Method 100 shown.
[0071] In a variation of the embodiment shown herein, method 100 is used to determine two or more observables. In this case, choose the Pauli set. This makes it possible to determine the expected value of all observables in the two or more observables.
[0072] Figure 2 and Figure 3 The line drawing sketch shown illustrates the results for an embodiment in which a ten-qubit Pauli string (the tensor product of ten Pauli matrices) has been measured using IBM's superconducting qubit-based quantum computer "ibm_torino". In this embodiment, the calibration operator... This selection ensures that the calibration operator includes all Pauli strings used for splitting observable operators: Readout errors of the noisy measurement result 1040 were reduced using matrix-free error reduction methods, 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, November 2021).
[0073] Figure 2 The following line graph is shown as a sketch: This line graph shows an example of a calibration function. Figure 2 Specifically, the calibration function 1060 is adapted to the measurement results 1040 and 1050, along with the measurement point 1061 plotted, and the calibration function 1060 adapted to the measurement results 1040 and 1050. The gate-error-reduced measurement result 1040 is plotted in relation to the noisy measurement result 1050 (here: the measurement result processed by matrix-free measurement error reduction). The measurement results are recorded as points marked with crosshairs. The noisy measurement result 1040 (expected value) is plotted on the x-axis, and the gate-error-reduced measurement result 1050 (expected value), here the ZNE result, is plotted on the y-axis. A straight line and another curved curve are plotted in the line graph. On both axes... The bend curve that varies within the interval is of the following form (with free parameters). Calibration function 1060: By adapting free parameters The function has been adapted to the measurement results. The calibration function 1060 is shown, which maps the noisy measurement result 1040 to the gate-error-reduced measurement result 1050. The calibration function 1060 can be determined, for example, through fitting on a classical computer.
[0074] Figure 3 The following is a sketch of a line graph: This line graph compares measurement results adapted using different error reduction methods. The measurement results are plotted on the y-axis: measurement result 1040 with noise, measurement result 1050 with gate error reduction, and measurement result 1050 with... Figure 1The method 100 for reducing errors shown in the diagram involves applying a calibration function 1060. Precise, theoretical expected values are plotted on the x-axis, that is, expected values precisely determined by simulation on a classical computer. If all expected values plotted here have zero error, then all points lie on the plotted line 300. Therefore, it can be clearly seen from this line graph that, by means of… Figure 1 The expected value determined by the method 100 for reducing errors proposed in the paper has a smaller error than the expected value corrected by ZNE 1050 alone or by only no matrix error reduction 1040.
[0075] Figure 5 is a schematic diagram of a hybrid computing platform 200, which is an example of a noisy quantum computing platform on which implementations such as... Figure 1 The exemplary method 100 is shown in the figure. The hybrid computing platform 200 includes a classical computer 201 and a quantum computer 202, which are adapted for implementing methods such as those described in the figure. Figure 1 Method 100 is described exemplarily in the example.
Claims
1. A method (100) for reducing errors when determining observables of a quantum system in a quantum state on a noisy quantum computer platform, the method comprising the steps of: ● Split the observable operator into a string of base operators (101); ● Provides (102) calibration operators, where, The calibration operator includes at least one of the base operator strings; ● Measurement (103) of the calibration operator, including ○ Determine the noisy measurement results (1040) of the calibration operator described in (104), and ○ Implement (105) gate error reduction to generate the gate error reduction measurement results (1050) of the calibration operator; ● Determine (106) a calibration function (1060) for mapping the noisy measurement result (1040) of the calibration operator to the gate error reduction measurement result (1050) of the calibration operator; ● Determine the noisy measurement results (1070) of the base operator string (107); ● By mapping (109) the noisy measurement result (1070) of the base operator string with the calibration function (1060), the error-reduced measurement result (1080) of the base operator string is determined (108). ● The observable measurement (110) is determined by combining the error-reduced measurement results (1080) of the base operator string.
2. The method (100) according to any one of the preceding claims, wherein, Measuring (103) the calibration operator and / or determining (107) the noisy measurement results of the base operator string (1070) includes simultaneously measuring the commutative calibration operator.
3. The method (100) according to any one of the preceding claims, wherein, The base operator is a Pauli matrix.
4. The method (100) according to any one of the preceding claims, wherein, The method includes preparing the quantum state on the noisy quantum computer platform.
5. The method (100) according to claim 4, wherein, The preparation involves applying a quantum circuit to an initial quantum state.
6. The method (100) according to any one of the preceding claims, wherein, The implementation of (105) gate error reduction for generating the gate error-reduced measurement result (1050) of the calibration operator includes implementing one or more of the methods mentioned below for gate error reduction: zero-noise extrapolation, probability error amplification, probability error elimination.
7. The method according to any one of the preceding claims, wherein, The calibration function (1060) is determined to include an adaptation parameterization function, wherein the parameters of the adaptation parameterization function are such that the function maps the noisy measurement result (1040) to the gate error-reduced measurement result (1050).
8. A method for determining the ground state energy of a quantum system, wherein, The method (100) according to any one of the preceding claims is used to reduce the error when determining the Hamiltonian operator of the quantum system in the ground state of the quantum system, so as to obtain the error-reduced ground state energy.
9. The method (100) according to any one of the preceding claims, wherein, The quantum system is described by the many-body Hamiltonian operator of the material.
10. The method (100) according to any one of claims 8 or 9, wherein, The Hamiltonian operator of the quantum system is the Hamiltonian operator used in the Hubbard model.
11. Use of the method (100) according to any one of the preceding claims for material simulation, wherein, The Hamiltonian operator of the quantum system is a multi-body Hamiltonian operator, which describes the material to be simulated.
12. A hybrid computer platform (200) comprising a classical computer (201) and a quantum computer (202) for implementing quantum circuits, the classical computer and the quantum computer being adapted to enable the implementation of the steps of the method (100) according to any one of claims 1 to 10 and / or the implementation of the use of the method (100) according to claim 11.
13. The hybrid computing platform (200) according to claim 12, wherein, The quantum computer (202) is an NISQ computer.
14. An error reduction algorithm that causes a hybrid computer platform (200) to implement the method (100) according to any one of claims 1 to 10, the hybrid computer platform comprising a classical computer (201) and a quantum computer (202) for implementing quantum circuits, the hybrid computer platform being particularly a hybrid computer platform (200) according to any one of claims 12 or 13.
15. A computer-readable storage medium storing the error reduction algorithm (1012) according to claim 14.