Method and apparatus for measuring energy of quantum chemical system

Abstract

A method and apparatus for measuring energy of a quantum chemistry system. In one aspect, a method includes: for a Hamiltonian that describes a chemical system and includes a sum of multiple terms; wherein each term includes i) a respective operator that affects a respective single particle-based rotation and ii) one or more particle density operators, for each group including terms having the same operator that affects the respective single particle-based rotation, repeatedly measuring expected values for terms included in the group, including: calculating, by quantum computing, the desired values of terms included in the group; performing a corresponding single particle based rotation on a qubit system encoding a state of the chemical system; measuring a Jordan-Wigner transform of one or more particle density operators in the set under a calculation basis to obtain respective measurements of the set; calculating a total number of particles or a spin component using the obtained measurements; determining whether the calculated total particle number or spin component is equal to a corresponding target value; and in response to determining that the calculated total population or spin component is equal to the corresponding target value, retaining the measurement for determining the energy of the chemical system by a classical calculation; and determining the energy of the chemical system by classical calculations using the obtained measurements.

Description

[0001] This application is a divisional application of the invention patent application with application date of July 28, 2020, application number 202080055624.7, and invention name “Efficient and noise-resistant quantum chemical measurement”. Technical Field

[0002] This specification relates to quantum computing. Background Art

[0003] Quantum measurement is a core component of experimental quantum computing. Performing a measurement requires preparing a quantum system in a specific state and applying quantum operations and measurements to that system with high-precision control. Therefore, performing a measurement is costly. Summary of the Invention

[0004] This specification describes measurement strategies for quantum chemistry.

[0005] Generally speaking, one innovative aspect of the subject matter described in this specification can be implemented in a method for measuring the energy of a chemical system, the method comprising: obtaining a Hamiltonian describing the chemical system, wherein the Hamiltonian is expressed in an orthonormal basis; decomposing the Hamiltonian describing the chemical system into a sum of terms by classical computation, wherein each term includes i) a corresponding operator that affects a corresponding single-particle basis rotation and ii) one or more particle density operators; for each group of terms including the same operator that affects the corresponding single-particle basis rotation, repeatedly measuring expectation values ​​of the terms included in the group, comprising: performing the corresponding single-particle basis rotation on a qubit system encoding a state of the chemical system by quantum computation; and measuring a Jordan-Wigner transform of the one or more particle density operators in the group in a computational basis to obtain corresponding measurement results for the group; and determining the energy of the chemical system by classical computation using the obtained measurement results.

[0006] Other embodiments of this aspect include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. One or more systems of classical computers and quantum computers can be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which software, firmware, hardware, or a combination thereof causes the system to perform the actions during operation. One or more computer programs can be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform the actions.

[0007] The foregoing and other embodiments may each optionally include one or more of the following features, alone or in combination. In some embodiments, measuring the expected value of an item included in the group further includes performing error mitigation by post selection. Performing error mitigation by post selection may include: calculating the total number of particles or spin components using the obtained measurement results; determining whether the calculated total number of particles or spin components is equal to the corresponding target value; in response to determining that the calculated total number of particles or spin components is equal to the corresponding target value, providing the measurement results for determining the energy of the chemical system by classical calculation; and in response to determining that the calculated total number of particles or spin components is not equal to the corresponding target value, discarding the measurement results.

[0008] In some embodiments, a Hamiltonian describing a chemical system includes a one-electron component and a two-electron component, and wherein decomposing the Hamiltonian describing the chemical system into a sum of multiple terms by classical calculation includes: diagonalizing each scalar coefficient in the two-electron component, including: expressing each scalar coefficient in the two-electron component as a second sum of multiple terms on a single-particle basis, each term in the second sum of multiple terms including a Hermitian coefficient matrix of a monomer operator formed by a first pair of spin-orbits, a matrix, and a product of a Hermitian coefficient matrix of a monomer operator formed by a second pair of spin-orbits; for each term in the sum of multiple terms, determining a matrix of the monomer operator in the diagonalized corresponding Hermitian coefficient matrix; and determining a corresponding operator that affects a rotation of a corresponding basis using the determined matrix of the diagonalized monomer operator.

[0009] In some embodiments, the method further comprises discarding finite feature values ​​that are less than a predetermined threshold.

[0010] In some embodiments, the method further includes grouping terms of the decomposed Hamiltonian that are diagonalized under the same single-particle basis, including: for each term in the decomposed Hamiltonian, determining which single-particle basis the term diagonalizes; and assigning the term to the group corresponding to the determined single-particle basis.

[0011] In some embodiments, measuring the expected values ​​of the items included in the group includes simultaneously measuring the expected values ​​of the items included in the group.

[0012] In some embodiments, performing the corresponding basis rotation includes applying a corresponding Givens rotation circuit to the qubit system.

[0013] In some embodiments, determining the energy of the chemical system using the obtained measurements by classical calculation includes: determining an average measurement corresponding to each group; and summing the determined averages.

[0014] In some embodiments, the arbitrary orthonormal basis comprises a Gaussian or molecular orbital basis.

[0015] In some embodiments, the Hamiltonian describing a chemical system comprises a plurality of terms, each term comprising a product of one or more of: i) an annihilation operator for the corresponding spin-orbit, ii) a creation operator for the corresponding spin-orbit, and ii) a scalar coefficient given by a one-electron integral or a two-electron integral over basis functions in an orthonormal basis.

[0016] In some embodiments, the chemical system comprises a symmetrically stretched hydrogen chain, a symmetrically stretched water molecule, or a stretched nitrogen dimer.

[0017] The subject matter described in this specification can be implemented in a specific manner to achieve one or more of the following advantages. The presently described disclosure represents a significant and broadly applicable improvement to the state of the art in the field of quantum measurement. The presently described measurement strategy reduces the time required to measure the energy of a quantum system (e.g., in quantum chemistry) to within a fixed accuracy. Furthermore, the presently described measurement strategy can be implemented to perform measurements that are less sensitive to readout errors than measurements performed using a naive strategy. Furthermore, the presently described measurement strategy achieves a powerful form of error mitigation at minimal cost.

[0018] The details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 An example quantum computing system is depicted.

[0020] Figure 2 is a flow chart of an example process for measuring the energy of a chemical system.

[0021] Figure 3A-3C Shown are example measurement times versus number of qubits / spin orbitals for different measurement strategies and different chemical systems.

[0022] Figure 4 is a flow chart of an example process for performing error mitigation using post-selection.

[0023] Figure 5 Shown is a plot of the single-qubit depolarization probability versus the readout bit flip probability for ground-state measurements of a stretched chain of six-qubit hydrogen atoms.

[0024] The same reference numbers and names in different drawings indicate the same elements. DETAILED DESCRIPTION

[0025] The variational quantum eigensolver (VQE) framework is an example of a promising approach for efficiently utilizing small and noisy quantum devices to simulate quantum chemistry. VQE methods use quantum devices as coprocessors that prepare parameterized quantum wave functions and measure the expectation values ​​of observables. Combined with classical optimization algorithms, it is possible to minimize the expectation value of the Hamiltonian as a function of the parameters, thereby approximating the wave function, energy, and other properties of the ground state.

[0026] One difficulty associated with applying VQE to non-trivial systems is the large number of circuit repetitions required to perform accurate measurements. In the VQE framework, the expectation value is computed by Hamiltonian averaging, where the Hamiltonian is decomposed into a sum of easily measurable operators (e.g., Pauli strings) whose expectation values ​​are independently sampled by repeated measurements. When the measurements are optimally distributed over these easily sampled operators, the expected value is averaged. When the number of measurements required is between ,in Its expected value is estimated to be The Hamiltonian of represents a scalar, and Denotes the target accuracy. Techniques used to assess the feasibility of VQE often apply this upper limit, leading to the conclusion that chemical applications require a considerable number of measurements.

[0027] This specification describes an improved measurement strategy that is based on the decomposition of a two-electron integral tensor and does not rely on the properties of easily measurable operators (such as Pauli strings). The decomposition of the two-electron integral tensor results in fewer sets of terms to be measured separately and requires fewer repeated measurements of the ground state to obtain a fixed accuracy. For example, the measurement strategy can reduce the total number of measurements required by up to four orders of magnitude. In addition, compared to existing measurement strategies, the measurement strategy is less sensitive to readout errors caused by long Jordan-Wigner strings. In addition, the measurement strategy provides a powerful form of error mitigation at minimal cost by allowing post-selection of either the total number of particles or the spin operator to be measured simultaneously with each measurement.

[0028] Example Hardware

[0029] Figure 1 An example quantum computing system 100 suitable for implementing the presently described measurement strategies is depicted. The example system 100 is an example of a system implemented as a classical or quantum computer program on one or more classical computers or quantum computing devices in one or more locations, in which the systems, components, and techniques described below may be implemented.

[0030] System 100 may include quantum hardware 102 in data communication with a classical processor 104. For convenience, classical processor 104 and quantum hardware 102 are shown as separate entities, however, in some embodiments, classical processor 104 may be included in quantum hardware 102, e.g., quantum hardware 102 may include one or more components for performing classical computing operations.

[0031] System 100 may receive input data, such as input data 106, which may include data representing a physical system of interest. The received data representing the physical system of interest may include data representing a physical system to be probed or simulated. For example, it may be desirable to determine the energy of a physical system. In some embodiments, the received data may represent a physical system described by an electronic structure Hamiltonian, such as a single atom or molecule, a material, or a chemical substance.

[0032] The system may generate output data, such as output data 108, representing the results of simulating a physical system of interest. Output data 108 may include data representing determined properties of the physical system (e.g., measured energy of the physical system) or data that can be used to determine properties of the physical system (e.g., data representing raw measurements). For example, as described above, in some embodiments, the physical system may be a chemical substance, such as atoms or molecules. In these cases, the data representing the simulation results can be used to determine properties of the chemical substance, such as the rate of a chemical reaction, its electronic structure, and / or its optical / thermal properties.

[0033] System 100 is configured to perform classical computations using a classical processor 104 and quantum hardware 102 in conjunction with quantum computing. Quantum hardware 102 includes components for performing quantum computations using quantum circuits. For example, quantum hardware 102 includes a quantum system 120 and a control device 122. Quantum system 120 includes one or more multi-level (e.g., two-level) quantum subsystems, such as qubits, for performing algorithmic operations or quantum computations. The specific implementation of the multi-level quantum subsystems included in quantum hardware 102 and how they interact with each other depend on various factors, including the type of quantum computation being performed by quantum hardware 102. For example, the multi-level quantum subsystems may include qubits implemented via atoms, molecules, or solid-state quantum systems. In other examples, qubits may include, but are not limited to, superconducting qubits or semiconductor qubits.

[0034] The multi-level quantum subsystem can be frequency-tunable. For example, each qubit can have an associated operating frequency that can be adjusted, for example, using one or more control devices 122 by applying voltage pulses via one or more drive lines coupled to the qubits. Example operating frequencies include a qubit idle frequency, a qubit interaction frequency, and a qubit readout frequency. Different frequencies correspond to different operations that a qubit can perform. For example, setting the operating frequency to a corresponding idle frequency can cause a qubit to enter a state in which it does not strongly interact with other qubits and can be used to perform a single-qubit gate (e.g., as part of a quantum circuit). As another example, where qubits interact via a coupler with fixed coupling, the qubits can be configured to interact with each other by setting their respective operating frequencies to a gate-related frequency that is detuned from their common interaction frequency. In other cases, such as where qubits interact via a tunable coupler, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to enable interaction between the qubits, and then by setting their respective operating frequencies to a gate-related frequency that is detuned from their common interaction frequency. Such interactions can be performed in order to execute multi-qubit gates (e.g., as part of a quantum circuit).

[0035] The control device 122 may also include a measurement device, such as a readout resonator. The measurement results (measurement data) obtained via the measurement device may be provided to a classical processor included in the quantum hardware 102 or the classical processor 104 for processing and analysis.

[0036] The classical processor 104 may include a Hamiltonian decomposition module 114 configured to process received input data, e.g., data representing a Hamiltonian expressed in an orthonormal basis (such as a Gaussian or general molecular orbital basis), to generate data representing a decomposed version of the Hamiltonian that is identical to or approximates the Hamiltonian expressed in an orthonormal basis.

[0037] The decomposed version of the Hamiltonian consists of a sum of multiple terms, where the sum of multiple terms includes diagonal operators in the corresponding single-particle basis. For example, as shown below with reference to Figure 2 In more detail, each term in the sum of the multiple terms may include i) a corresponding operator affecting the rotation of the corresponding basis and ii) one or more particle density operators. Figure 2 An example procedure for decomposing the Hamiltonian describing a chemical system into a sum of such polynomials is described in detail.

[0038] The classical processor 104 may also include a basis rotation grouper 116 configured to group terms of the decomposed Hamiltonian (consisting of sums and products of particle density operators) that are diagonal under the same single-particle basis. The basis rotation grouper 116 is configured to provide data representing groups 126 of terms that are diagonal under the same single-particle basis to the quantum hardware 102. The quantum hardware 102 may then measure the expectation value of the terms associated with each single-particle basis by applying a Givens rotation circuit that performs a change of basis before measuring the Jordan-Wigner transformed number operator in the computation basis. Figure 2 An example process for grouping terms of the decomposed Hamiltonian and measuring expected values ​​of terms in a group is described in more detail.

[0039] The classical processor 104 may further include a post-processing module 118 configured to process the measurement results 128 received from the quantum hardware 102. For example, the post-processing module 118 may be configured to calculate a statistical average of the measurement results. In some embodiments, the post-processing module 118 may further include a post-selection module 124 for performing error mitigation, as described below with reference to Figure 4 Described in more detail.

[0040] Basis rotation grouping measurement strategy

[0041] Figure 2 is a flow chart of an example process 200 for measuring the energy of a quantum chemical system. For convenience, process 200 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system (e.g., Figure 1 The system 100 may perform the process 200 .

[0042] The system obtains a Hamiltonian that describes the quantum chemical system (step 202). The Hamiltonian can be obtained by a classical computing device. The Hamiltonian can be input by a user or can be extracted from another source (e.g., a memory). The Hamiltonian that describes the physical system can be expressed in any standard orthogonal basis (e.g., a Gaussian or a general molecular orbital basis). The Hamiltonian can include a one-electron component and a two-electron component, where each of the one-electron component and the two-electron component includes a corresponding scalar coefficient that can be given by a known integral over a basis function that depends on a particular discretization scheme. An example Hamiltonian is given in Equation (1) below.

[0043]

[0044] In equation (1), and represents the annihilation operator and creation operator of spin-orbit p, 、 represents a scalar coefficient that can be given by a known integral over basis functions that depends on the particular discretization scheme. In some embodiments, the Hamiltonian can describe a periodic physical system. In other embodiments, the Hamiltonian can describe a non-periodic physical system, such as a single molecule.

[0045] The Hamiltonian describing a quantum chemical system can be mapped to a qubit Hilbert space using various methods for simulating indistinguishable fermions with discernible qubits. For example, the Hamiltonian describing a quantum chemical system can be mapped to a qubit Hilbert space using the Jordan-Wigner transformation or the Bravyi-Kitaev transformation.

[0046] The Jordan-Wigner transformation is attractive because it is simple and allows many useful circuit primitives to be constructed explicitly. However, one drawback of using the Jordan-Wigner transformation is that it maps operators that act on a constant number of fermion modes to qubit operators, supporting at most Under simple readout error models, such as symmetric bit-flip channels, support The Pauli word for a qubit is This causes the expected value measurement to be driven to zero by the multiplication factor that is in the As described below, a system implementing the example process 200 can avoid this challenge without leaving the Jordan-Wigner framework, thereby allowing estimation of single- and two-particle fermion operator expectation values ​​by measuring 1- and 2-local qubit operators, respectively.

[0047] The system performs classical computation to decompose the Hamiltonian describing the chemical system into a sum of multiple terms, each of which includes i) a corresponding operator that affects the rotation of the corresponding single-particle basis and ii) one or more particle density operators (also called number operators) (step 204). That is, each term in the sum of multiple terms corresponds to a corresponding single-particle basis, and each term in the sum is a diagonal operator. For example, the system can decompose the Hamiltonian given by equation (1) above into the following equation (2)

[0048]

[0049] Among them, the operator represents the operator that affects the corresponding single-particle basis rotation, for example, the single-particle basis rotation given by the following equation (3)

[0050]

[0051] In some cases, the Hamiltonian describing a chemical system (e.g., a Hamiltonian of the form given in Equation (1)) can be difficult to simulate. For example, the Hamiltonian may describe a non-periodic system. The matrix representing the Hamiltonian describing a periodic system (expressed in a given basis) often includes an underlying structure that can be used to simplify the computations associated with simulating the periodic system, e.g., repetition in the underlying structure can be used to reduce the number of multiplication operations required to simulate the periodic system. However, the matrix representing the Hamiltonian describing a non-periodic system (expressed in a given basis) may not include this underlying structure. Therefore, efficiently simulating such systems can be challenging. However, by decomposing the Hamiltonian (expressed in a standard orthonormal basis) into a sum of multiple terms as described above, the Hamiltonian is mapped to a combination of sub-Hamiltonians with specific structures and expressed in different corresponding bases, to which simulation techniques can be effectively applied.

[0052] To decompose the Hamiltonian describing a chemical system into a sum of multiple terms, the system first diagonalizes the scalar coefficients of the terms in the two-electron component of the Hamiltonian by expressing each scalar coefficient as the sum of the corresponding multiple terms on a single-particle basis. Each term in the sum of multiple terms consists of the Hermitian coefficient matrix of the monomer operator formed by the first pair of spin-orbits ,matrix and the Hermitian coefficient matrix of the single operator formed by the second pair of spin-orbits For example, the system can use the following equation (4) to diagonalize the scalar coefficients of the two-electron component of the Hamiltonian .

[0053]

[0054] The two-electron component of the Hamiltonian can then be rewritten to obtain the Hamiltonian given in equation (5) below.

[0055]

[0056] in, and each (For fixed ) is the Hermitian coefficient matrix of the singleton operator.

[0057] Then, the system determines the matrix ,matrix The corresponding Hermitian coefficient matrices of the diagonalized single-particle operators in their corresponding single-particle Hilbert spaces. The system uses the determined matrix To determine the operator that affects the corresponding single-particle basis rotation required to obtain Equation (2) .

[0058] In some embodiments, the system may discard finite eigenvalues ​​less than a predetermined threshold during the process of decomposing the Hamiltonian describing the chemical system into a sum of multiple terms, e.g., to obtain a controllable approximation to the original Hamiltonian. The predetermined threshold may be preselected based on the specific quantum chemical system (or equivalently, the Hamiltonian describing the quantum chemical system) and / or the capabilities of the hardware used to perform step 206. For example, to produce a more controllable approximation, a larger threshold may be used, i.e., more eigenvalues ​​may be discarded.

[0059] The system groups the terms of the decomposed Hamiltonian (step 206) that are on the same single particle basis. For example, for each term in the decomposed Hamiltonian given by equation (2) above, the system can determine under which single-particle basis the term is diagonalized and assign the term to the group corresponding to the single-particle basis.

[0060] The system then includes, for each set of measurements (step 206), the expected values ​​of the items in the set. Within a set, the measurements of the items can be performed approximately simultaneously (eg, within the limits of hardware precision).

[0061] In order to measure the expected value of a term included in a group, the system performs a basis rotation corresponding to the group on a qubit system encoding the state of the chemical system to be measured by quantum computing. For example, the system can apply a corresponding Givens rotation circuit to the qubit system. Then, the system measures the qubit system. This includes measuring the Jordan-Wigner transformation of one or more particle density operators included in the term in the computational basis to obtain the corresponding measurement result. That is, by performing a basis rotation, for example, applying the corresponding Givens rotation circuit directly on the quantum state of the qubit system before measurement. circuit, the system can then simultaneously sample the rotated basis and Expected value, to estimate the energy as , where the subscript on the expected value is Indicates that they are applying a basis transformation Then sampled.

[0062] and The reason why they can be sampled simultaneously is because under the Jordan-Wigner transformation, , which is the diagonal qubit operator. Therefore, the system is able to use only different groups of terms sample all the terms in the Hamiltonian. Fortunately, Extremely easy to implement, even on hardware with minimal connectivity. For example, any change to the single-particle basis can be done using Two-qubit gates can be performed with a gate depth of exactly N, even with only linearly connected qubit arrays.

[0063] The system may repeatedly (e.g., a predetermined number of times) measure the expected values ​​of the items included in the group to obtain a plurality of measurement results, and post-process the plurality of measurement results to obtain a final measurement result. For example, the system may post-select a measurement result from the plurality of measurement results and / or determine a statistical average of the plurality of measurement results. Figure 4 An example process for performing error mitigation using post-selection is described.

[0064] The system uses the obtained measurement results for each group through classical calculations, such as by summing the average values ​​of the measurements corresponding to each group, to determine the energy of the chemical system (step 208). In some embodiments, the system can use the determined energy of the chemical system to perform further calculations to determine properties of the physical system. For example, the example process 200 can be used to determine properties of individual molecules and simulate catalysts or drugs for performing materials science simulations, or to determine the rate of a chemical reaction.

[0065] Figure 3A-3C Shown are the number of qubits / spin orbitals plotted against an example measurement time (the time required to obtain an estimate of the ground state energy) for different chemical systems and different measurement strategies / methods, namely, the currently described measurement strategy (called "basis spin grouping"), three existing measurement strategies ("split measurement," "Pauli grouping," and "Pauli grouping, RDM confinement," and two upper limits from the fermion L1 norm and the qubit L1 norm).

[0066] Figure 3A-3C The data plotted are, for example, based on known rules (such as those requiring each operator via Measured as time The fraction of ), assuming 10,000 circuit repetitions per second and optimally distributed measurements, is generated using the calculation of the expected value variance to determine the required measurement time. Because in practice the variance of each operator is not known in advance, it is assumed that an adaptive measurement scheme that schedules additional measurements based on the observed sample variance can approximate the ideal partitioning of measurement time, and for simplicity, Figure 3A-3C Only quantities based on ideal partitioning are presented. Assume that the target accuracy corresponds to 1.0 milli-Hartree Error bars.

[0067] The calculations performed to generate the data plotted in Figure 3 were performed on symmetrically stretched hydrogen chains with varying bond lengths and numbers of atoms ( Figure 3A ), symmetrical stretching of water molecules ( Figure 3B ) and stretched nitrogen dimers ( Figure 3C ), all calculations are performed in multiple basis sets. All calculations for systems up to 20 qubits use the configuration interaction singles and doubles (CISD) approximation to the ground state, and for systems above 20 qubits use Hartree-Fock states. Bounds based on the L1 norm are computed in fermionic Hilbert space and in terms of operators acting directly on the qubits.

[0068] Figure 3A The data generated for symmetric stretched hydrogen chains with different bond lengths and numbers of atoms under various basis sets are shown. The hydrogen chains contain symmetric interatomic spacings ranging from 0.6 to 1.3 Å under the STO-3G, 6-31G, or cc-pVDZ basis sets. Computations performed on systems requiring the same number of qubits (spin-orbits) are drawn together in columns and slightly spread out horizontally for easier visibility.

[0069] Figure 3B The data generated for a symmetric stretched water molecule under various basis sets are shown. The bonds in the water molecule are at 0.8 and 1.5 symmetric stretching between them. Water calculations were performed with the STO-3G basis set on nitrogen and oxygen molecules with and without frozen 1s orbitals, and with the 6-31G basis set with frozen nuclei. Calculations performed on systems requiring the same number of qubits (spin-orbits) are plotted together in columns and slightly expanded horizontally for easier visualization.

[0070] Figure 3C Data generated for a stretched nitrogen dimer using various basis sets are shown. The spacing between nitrogen atoms ranges from 0.9 to 1.6 Nitrogen calculations are performed with and without the 1s orbital frozen on nitrogen / oxygen molecules using the STO-3G basis set, and with the nucleus frozen using the 6-31G basis set. Calculations performed on systems requiring the same number of qubits (spin-orbits) are plotted together in columns and slightly expanded horizontally for easier visibility.

[0071] Figure 3A-3CEach of these demonstrates the effectiveness of the described measurement strategy compared to each of three existing measurement strategies and two L1-norm values. For example, for system sizes greater than or equal to 12 qubits for hydrogen chains, 24 qubits for water molecules, and 16 qubits for nitrogen dimers, the measurement time is reduced using the described basis rotation grouping method. Furthermore, the relative improvement in measurement time between the described method and other existing methods increases with increasing numbers of qubits, i.e., larger system sizes.

[0072] In addition to the reduction in measurement time, the measurement strategy described above with reference to the example process 200 has the additional benefit of reducing sensitivity to readout errors and enabling a powerful form of error mitigation through post-selection. These properties are a consequence of the fact that the Hamiltonian measured in the example process 200 is measured only in terms of the particle density operator under different basis sets. As described above, applying the Jordan-Wigner transformation to the terms of the quantum chemical Hamiltonian results in support for up to However, the particle density operator is more simply transformed Thus, the measurement strategy described in example process 200 avoids the extension of locality caused by long Jordan-Wigner strings, and individual terms from the one-particle and two-particle components of the Hamiltonian can be measured by measuring 1-local qubit operators and 2-local qubit operators, respectively.

[0073] Furthermore, the measurement strategy currently described provides opportunities for mitigating errors. For example, when determining eigenvalues ​​with symmetry operators (such as the total number of particles) or spin Quantity When the state of ρ is of interest, it is desirable to have a method to remove the components of some experimentally prepared states ρ that violate this constraint.

[0074] There are two strategies to achieve this goal. The first existing strategy consists in measuring the symmetry operator directly and non-destructively and discarding those results where undesirable eigenvalues ​​are observed, projecting them into the appropriate symmetry sector by post-selection. The difficulty of performing these measurements efficiently limits the application of this strategy. In addition, some implementations of this first existing strategy focus on using a depth of Circuit to measure and The parity of , which may induce further errors during their implementation. A second existing strategy builds on the first and uses additional measurements and classical post-processing to compute the expectation value of the projected state without requiring additional circuit depth.

[0075] As referenced below Figure 4As described, the presently described measurement strategy implements a new form of error mitigation based on post-selection and eliminates the challenges posed by the non-locality of the Jordan-Wigner transformed operator during measurement.

[0076] Error mitigation via postselection

[0077] Figure 4 is a flow chart of an example process 400 for performing error mitigation using post-selection. As described above, the example process 400 can be performed in conjunction with the example process 200 for measuring energy of a chemical system.

[0078] For convenience, process 400 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system (e.g., Figure 1 The system 100 may perform the process 400 .

[0079] The system obtains the measurement result (step 402) generated during step 206 of example process 200. That is, the system obtains the measurement result by applying the corresponding Givens rotation circuit to the qubit system to perform the corresponding basis rotation and measuring the Jordan-Wigner transformation of one or more numerical operators included in the term under the computational basis, the measurement result representing the result of measuring a group of terms having the same operator that affects the corresponding basis rotation.

[0080] The system uses the obtained measurement results to calculate the total number of particles or the value of a spin component (e.g., the z component of the spin) (step 404). Since the obtained measurement results consist of a bit string, the system can calculate the total number of particles by counting 1s or 0s, or calculate the value of a spin component by subtracting the sum of the particle density operators acting on the "spin-down" spin orbitals from the sum of the particle density operators acting on the "spin-up" spin orbitals (and dividing by 2).

[0081] The system determines whether the calculated total number of particles or the value of the spin component is equal to the corresponding target value. In response to determining that the calculated total number of particles or the spin number is equal to the corresponding target value, the system determines that no error has occurred and provides the measurement result for post-processing, as described above with reference to Figure 2 In response to determining that the calculated total number of particles or spins is not equal to the corresponding target value, the system determines that an error has occurred and discards the measurement result.

[0082] Figure 5 Shown is a plot of the single-qubit depolarization probability measured in the ground state versus the readout bit-flip probability for a stretched chain of six hydrogen atoms under an error model consisting of single-qubit dephasing noise applied after every two-qubit gate and a symmetric bit-flip channel during readout. Figure 5Four graphs AD corresponding to different measurement strategies are shown. In each graph, the square and the number represent the absolute error in millihartrees.

[0083] Panel A shows the errors caused by the "Pauli grouping" measurement strategy, which involves measuring compatible Pauli words simultaneously in the usual molecular orbital basis. Panel B shows the errors when using the currently described "basis rotation" or "basis rotation grouping" scheme, which performs a single-particle basis change before the measurement. Panel C shows the errors when using the "Pauli grouping strategy" along with additional measurements and post-processing, which effectively projects the measured state onto the total number of particles and Figure D shows that when the currently described basis rotation strategy is used and the correct number of particles and To approximate the real ansatz circuit, three random Givens rotation networks of identical composition were simulated to act on the ground state before measurement.

[0084] Figure 5 It is shown that the amount of error introduced using the presently described measurement strategy (with and without post-selection techniques) can be less than the amount of error introduced using the Pauli grouping measurement strategy (with and without post-selection techniques). Furthermore, both the Pauli grouping strategy and the basis rotation strategy benefit from implementing post-selection error mitigation techniques. Furthermore, although the presently claimed basis rotation strategy requires a circuit depth that is one-third of the circuit depth used by the Pauli grouping strategy, the error remaining after post-selection error mitigation is comparable in many cases and is lower when noise during measurement is the dominant error channel. Even without post-selection (where the presently described measurement strategy benefits from a strictly more powerful form of error mitigation), the locality of the presently described Jordan-Wigner transformed operators provides some benefits in suppressing the effects of readout errors.

[0085] The digital and / or quantum subject matter and implementations of digital functional operations and quantum operations described in this specification may be implemented in digital electronic circuits, suitable quantum circuits, or more generally, in quantum computing systems, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more thereof. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0086] The embodiments of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or for controlling the operation of the data processing device. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access storage device, one or more qubits, or a combination of one or more thereof. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, which is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing device.

[0087] The terms quantum information and quantum data refer to information or data carried, held, or stored by a quantum system, wherein the smallest non-trivial system is a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" includes all quantum systems that can be appropriately approximated as a two-level system in the relevant context. Such quantum systems can include multi-level systems, e.g., having two or more energy levels. For example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational base state is identified by the ground state and the first excited state, however, it should be understood that other arrangements in which the computational state is identified by higher-order excited states are also possible.

[0088] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all types of devices, equipment, and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The apparatus may also be or further include specialized logic circuitry, such as an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator—that is, a quantum data processing apparatus designed to simulate or generate information about a specific quantum system. Specifically, a quantum simulator is a specialized quantum computer that lacks the capability to perform general-purpose quantum computations. In addition to the hardware, the apparatus may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these.

[0089] A digital computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) may be written in any form of programming language (including compiled or interpreted languages ​​or declarative or procedural languages), and it may be deployed in any form (including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment). A quantum computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) may be written in any form of programming language (including compiled or interpreted languages ​​or declarative or procedural languages) and translated into a suitable quantum programming language, or may be written in a quantum programming language (e.g., QCL or Quipper).

[0090] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program may be stored as part of a file that stores other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files storing one or more modules, subroutines, or portions of code). A digital and / or quantum computer program may be deployed to execute on a single digital computer or a single quantum computer, or on multiple digital and / or quantum computers located at one location or distributed across multiple locations and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using quantum systems (e.g., qubits). Typically, digital data communication networks cannot transmit quantum data, but quantum data communication networks can transmit both quantum data and digital data.

[0091] The processes and logic flows described in this specification may be performed by one or more programmable digital and / or quantum computers, optionally operating in conjunction with one or more digital and / or quantum processors, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows may also be performed by a dedicated logic circuit (e.g., an FPGA or ASIC) or a quantum simulator, or a combination of a dedicated logic circuit or a quantum simulator and one or more programmable digital and / or quantum computers, and the apparatus may also be implemented as a dedicated logic circuit (e.g., an FPGA or ASIC) or a quantum simulator, or a combination of a dedicated logic circuit or a quantum simulator and one or more programmable digital and / or quantum computers.

[0092] For a system of one or more digital and / or quantum computers, being "configured" to perform a particular operation or action means that the system has installed thereon software, firmware, hardware, or a combination thereof that, when operated, causes the system to perform the operation or action. For one or more digital and / or quantum computer programs, being "configured" to perform a particular operation or action means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by the quantum computing device, cause the device to perform the operation or action.

[0093] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general-purpose or special-purpose digital and / or quantum processor, or both, or any other type of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0094] The essential elements of a digital and / or quantum computer are a central processing unit (CPU) for running or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented by or incorporated with specialized logic circuits or quantum simulators. Typically, a digital and / or quantum computer will also include or be operatively coupled to one or more mass storage devices (e.g., magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information) for storing digital and / or quantum data, to receive digital and / or quantum data from them, to transmit digital and / or quantum data to them, or both. However, a digital and / or quantum computer does not require such devices.

[0095] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media, and storage devices, including, for example: semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data for a long time with high fidelity and high efficiency, for example, a light-matter interface in which light is used for transmission and matter for storing and preserving quantum characteristics (such as superposition or quantum coherence) of quantum data.

[0096] The control of the various systems or portions thereof described in this specification may be implemented in a digital and / or quantum computer program product comprising instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification may each be implemented as an apparatus, method, or system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0097] Although this specification contains many specific implementation details, these should not be interpreted as limitations on the scope of what is claimed, but rather as descriptions of features specific to a particular embodiment. Certain features described in this specification in the context of separate implementations may also be implemented in combination in a single embodiment. Conversely, the various features described in the context of a single implementation may also be implemented individually or in any suitable sub-combination in multiple embodiments. Furthermore, although features may be described above as working in certain combinations and even initially claimed, in some cases, one or more features in a claimed combination may be deleted from that combination, and a claimed combination may point to a variant of a sub-combination or sub-combination.

[0098] Similarly, although operations are described in a particular order in the accompanying drawings, this should not be understood as requiring that the operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, in order to obtain the desired results. In some cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the above-described embodiments should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems may generally be integrated into a single software product or packaged into multiple software products.

[0099] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for measuring the energy of a chemical system, the method comprising: For a Hamiltonian describing a chemical system and comprising a sum of terms, wherein each term comprises i) a respective operator affecting a respective single-particle basis rotation and ii) one or more particle density operators, for each group comprising terms having the same operator affecting a respective single-particle basis rotation, repeatedly measuring the expectation values ​​of the terms comprised in the group comprises: performing corresponding single-particle basis rotations on a qubit system encoding a state of the chemical system via quantum computing; measuring the Jordan-Wigner transform of one or more particle density operators in the group under the computational basis to obtain corresponding measurement results of the group; Use the obtained measurements to calculate the total number of particles or spin components; determining whether the calculated total number of particles or spin components is equal to the corresponding target value; and In response to determining that the calculated total number of particles or spin components is equal to the corresponding target value, retaining the measurement results for use in determining the energy of the chemical system by classical calculation; and The obtained measurements are used by classical calculations to determine the energy of the chemical system.

2. The method according to claim 1, further comprising: In response to determining that the calculated total number of particles or spin component is not equal to the corresponding target value, the measurement result is discarded.

3. The method according to claim 1, wherein The Hamiltonian describing the chemical system includes a one-electron component and a two-electron component, and wherein the method further includes decomposing the Hamiltonian describing the chemical system into a sum of multiple terms by classical calculation, wherein the decomposition includes: diagonalizing each scalar coefficient in the two-electron component, comprising expressing each scalar coefficient in the two-electron component as a sum of second multinomials on a single-particle basis, each term in the sum of second multinomials comprising a product of a Hermitian coefficient matrix of a singleton operator formed by a first pair of spin-orbits, a matrix, and a Hermitian coefficient matrix of a singleton operator formed by a second pair of spin-orbits; For each term in the sum of multiple terms, determining a matrix that diagonalizes the singleton operators in the corresponding Hermitian coefficient matrix; and Use the matrix of diagonalized singleton operators to determine the corresponding operators that affect the corresponding basis rotations. The method according to claim 3 , further comprising discarding finite eigenvalues ​​that are smaller than a predetermined threshold.

5. The method of claim 1 , further comprising grouping diagonally decomposed Hamiltonian terms under the same single-particle basis, for each term in the decomposed Hamiltonian, comprising: Determine in which single-particle basis the term is diagonalized; as well as Assign the term to the group corresponding to the determined single-particle basis.

6. The method according to claim 1, wherein Measuring the expected values ​​of the items included in the group includes simultaneously measuring the expected values ​​of the items included in the group.

7. The method according to claim 1, wherein Performing the corresponding single-particle basis rotation includes applying a corresponding Givens rotation circuit to the qubit system.

8. The method according to claim 1, wherein Determining the energy of the chemical system using the obtained measurements by classical calculations involves: determining the average measurement corresponding to each group; and The determined average values ​​are added together.

9. The method according to claim 1, wherein Single particle bases include Gaussian or molecular orbital bases.

10. The method according to claim 1, wherein The Hamiltonian describing the chemical system includes a plurality of terms, each term including the product of one or more of the following: i) an annihilation operator of the corresponding spin-orbit, ii) a creation operator of the corresponding spin-orbit, and iii) a scalar coefficient given by a one-electron integral or a two-electron integral over basis functions in a single-particle basis.

11. The method according to claim 1, wherein The chemical system includes symmetrically stretched hydrogen chains, symmetrically stretched water molecules, or stretched nitrogen dimers.

12. The method according to claim 1, wherein Determining that the calculated total number of particles or spin components equals the corresponding target value includes determining that no error has occurred.

13. An apparatus comprising: quantum hardware; as well as One or more classic processors; The device is configured to perform operations, including: For a Hamiltonian describing a chemical system and comprising a sum of terms, wherein each term comprises i) a respective operator affecting a respective single-particle basis rotation and ii) one or more particle density operators, for each group comprising terms having the same operator affecting a respective single-particle basis rotation, repeatedly measuring the expectation values ​​of the terms comprised in the group comprises: performing corresponding single-particle basis rotations on a qubit system encoding a state of the chemical system via quantum computing; measuring the Jordan-Wigner transform of one or more particle density operators in the group under the computational basis to obtain corresponding measurement results of the group; Use the obtained measurements to calculate the total number of particles or spin components; determining whether the calculated total number of particles or spin components is equal to the corresponding target value; and In response to determining that the calculated total number of particles or spin components is equal to the corresponding target value, retaining the measurement results for use in determining the energy of the chemical system by classical calculation; and The obtained measurements are used by classical calculations to determine the energy of the chemical system.

14. The device according to claim 13, wherein The operations also include discarding the measurement result in response to determining that the calculated total number of particles or the spin component is not equal to the corresponding target value.

15. The device according to claim 13, wherein The Hamiltonian describing the chemical system includes a one-electron component and a two-electron component, and wherein the operation further includes decomposing the Hamiltonian describing the chemical system into a sum of multiple terms by classical calculation, wherein the decomposition includes: diagonalizing each scalar coefficient in the two-electron component, comprising expressing each scalar coefficient in the two-electron component as a sum of second multinomials on a single-particle basis, each term in the sum of second multinomials comprising a product of a Hermitian coefficient matrix of a singleton operator formed by a first pair of spin-orbits, a matrix, and a Hermitian coefficient matrix of a singleton operator formed by a second pair of spin-orbits; For each term in the sum of multiple terms, determining a matrix that diagonalizes the singleton operators in the corresponding Hermitian coefficient matrix; and Use the matrix of diagonalized singleton operators to determine the corresponding operators that affect the corresponding basis rotations.

16. The device according to claim 15, wherein The operations also include discarding finite feature values ​​that are less than a predetermined threshold.

17. The device according to claim 13, wherein The operation further includes grouping the terms of the decomposed Hamiltonian diagonally under the same single-particle basis, including, for each term in the decomposed Hamiltonian: Determine in which single-particle basis the term is diagonalized; and Assign the term to the group corresponding to the determined single-particle basis.

18. The device according to claim 13, wherein Measuring the expected values ​​of the items included in the group includes simultaneously measuring the expected values ​​of the items included in the group.

19. The device according to claim 13, wherein Performing the corresponding single-particle basis rotation includes applying a corresponding Givens rotation circuit to the qubit system.

20. The apparatus according to claim 13, wherein Determining the energy of the chemical system using the obtained measurements by classical calculations involves: determining the average measurement corresponding to each group; and The determined average values ​​are added together.

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