Efficient and Noise-Resistant Quantum Chemical Measurements

Abstract

Methods, systems, and apparatuses for measuring the energy of a quantum chemical system. In one aspect, a method includes: obtaining a Hamiltonian that describes a chemical system, where the Hamiltonian is represented in an orthonormal basis; decomposing the Hamiltonian into a sum of terms, where each term includes a corresponding operator that affects a corresponding single-particle basis rotation and one or more particle density operators; for each set of terms that includes terms having the same operator that affects a corresponding single-particle basis rotation, repeatedly measuring the expectation value of the terms included in the set, including: performing a corresponding single-particle basis rotation on a qubit system that encodes the state of the chemical system; and measuring the Jordan-Wigner transform of one or more particle density operators in the set to obtain a corresponding measurement result for the set; and using the obtained measurement results to determine the energy of the chemical system.

Description

Technical Field

[0001] This specification relates to quantum computing. Background Art

[0002] Quantum measurement is a core component of experimental quantum computing. Performing one round of measurement requires preparing a quantum system in a specific state and applying quantum operations and measurements with high-precision control to the quantum system. Therefore, performing one round of measurement is costly. Summary of the Invention

[0003] This specification describes a measurement strategy for quantum chemistry.

[0004] Generally, an 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 that describes the chemical system, wherein the Hamiltonian is represented in an orthonormal basis; by classical calculation, decomposing the Hamiltonian that describes the chemical system into a sum of terms, wherein each term includes i) a corresponding operator that affects the rotation of a corresponding single-particle basis and ii) one or more particle density operators; for each group of terms that include operators having the same effect on the rotation of the corresponding single-particle basis, repeatedly measuring the expected value of the terms included in the group, including: by quantum calculation, performing a corresponding single-particle basis rotation on a quantum bit system encoding the state of the chemical system; and measuring the Jordan-Wigner transformation of one or more particle density operators in the group in the computational basis to obtain a corresponding measurement result for the group; and by classical calculation, using the obtained measurement results to determine the energy of the chemical system.

[0005] Other embodiments of this aspect include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more 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, the software, firmware, hardware, or a combination thereof causing the system to perform the actions in 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 apparatus, cause the apparatus to perform the actions.

[0006] The foregoing and other embodiments may each optionally include, alone or in combination, one or more of the following features. In some embodiments, measuring an expected value of an item in a group further includes performing error mitigation through post selection. Performing error mitigation through post selection may include: calculating a total number of particles or a spin component using the obtained measurement results; determining whether the calculated total number of particles or spin component is equal to a corresponding target value; in response to determining that the calculated total number of particles or spin component is equal to the corresponding target value, providing the measurement results for determining the energy of a chemical system through classical calculation; and in response to determining that the calculated total number of particles or spin component is not equal to the corresponding target value, discarding the measurement results.

[0007] In some embodiments, the Hamiltonian describing the 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 terms through classical calculation includes: diagonalizing each scalar coefficient in the two - electron component, including: representing each scalar coefficient in the two - electron component as a second sum of terms on a single - particle basis, each term in the second sum of terms including the product of a Hermitian coefficient matrix of a monomer operator formed by a first pair of spin orbitals, a matrix, and a Hermitian coefficient matrix of a monomer operator formed by a second pair of spin orbitals; for each term in the sum of terms, determining the matrix of the monomer operator that diagonalizes the corresponding Hermitian coefficient matrix; and using the determined matrices of the diagonalized monomer operators to determine the corresponding operators that affect the corresponding basis rotation.

[0008] In some embodiments, the method further includes discarding finite eigenvalues that are less than a predetermined threshold.

[0009] In some embodiments, the method further includes grouping terms of the decomposed Hamiltonian that are diagonal in 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 a group corresponding to the determined single - particle basis.

[0010] In some embodiments, measuring an expected value of an item in a group includes simultaneously measuring the expected values of the items included in the group.

[0011] In some embodiments, performing the corresponding basis rotation includes applying a corresponding Givens rotation circuit to the qubit system.

[0012] In some embodiments, determining the energy of the chemical system through classical calculation using the obtained measurement results includes: determining an average measurement result corresponding to each group; and adding the determined average values.

[0013] In some embodiments, any orthonormal basis includes a Gaussian or molecular orbital basis.

[0014] In some embodiments, the Hamiltonian describing a chemical system includes a plurality of terms, each term including a product of one or more of the following: i) an annihilation operator of a corresponding spin orbital, ii) a creation operator of a corresponding spin orbital, and iii) a scalar coefficient given by a one - electron integral or a two - electron integral over basis functions in an orthonormal basis.

[0015] In some embodiments, the chemical system includes a symmetrically stretched hydrogen chain, a symmetrically stretched water molecule, or a stretched nitrogen dimer.

[0016] The subject matter described in this specification can be implemented in a particular manner so as to achieve one or more of the following advantages. The presently described disclosure represents a significant and widely applicable improvement over 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. Additionally, compared to measurements performed using naive strategies, the presently described measurement strategy can be implemented to perform measurements that are less sensitive to readout errors. Moreover, the presently described measurement strategy achieves a powerful form of error mitigation at minimal cost.

[0017] Details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 An example quantum computing system is depicted.

[0019] Figure 2 is a flow chart of an example process for measuring the energy of a chemical system.

[0020] Figures 3A - 3C Shows a plot of example measurement times versus the number of qubits / spin orbitals for different measurement strategies and different chemical systems.

[0021] Figure 4 is a flow chart of an example process for performing error mitigation using post - selection.

[0022] Figure 5 Shows a plot of the single - qubit depolarization probability versus the readout bit - flip probability for the ground - state measurement of a stretched chain of six - qubit hydrogen atoms.

[0023] Like reference numerals and names in the different figures indicate like elements. DETAILED DESCRIPTION

[0024] The variational quantum eigensolver (VQE) framework is an example of a promising path for effectively using small and noisy quantum devices to simulate quantum chemistry. The VQE method uses a quantum device as a co-processor that prepares a parameterized quantum wave function and measures the expected value of an observable. Combined with a classical optimization algorithm, it is possible to minimize the expected value of the Hamiltonian as a function of the parameters, thereby approximating the wave function, energy, and other properties of the ground state.

[0025] 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 expected value is calculated by Hamiltonian averaging, where the Hamiltonian is decomposed into a sum of operators that are easy to measure (e.g., Pauli strings), and the expected values of which are independently sampled by repeated measurements. When the measurements are optimally distributed among these easily sampled operators H l between, the upper bound on the total number of measurements M is M ≤ (∑ l |ω l | / ∈) 2 where H = ∑ l ω l H l represents the Hamiltonian whose expected value is estimated as ∑ l ω l <H l >, ω l represents a scalar, and ∈ represents the target accuracy. Techniques for evaluating the feasibility of VQE typically apply this upper bound, leading to the conclusion that chemical applications require a rather large number of measurements.

[0026] This specification describes an improved measurement strategy that is based on the decomposition of the two-electron integral tensor and does not rely on the properties of operators that are easy to measure (such as Pauli strings). The decomposition of the two-electron integral tensor results in a smaller set of terms to be measured separately and requires fewer repetitions to measure 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. Additionally, compared to existing measurement strategies, this measurement strategy is less sensitive to readout errors caused by long Jordan-Wigner strings. Furthermore, this measurement strategy provides a powerful form of error mitigation at minimal cost by allowing post-selection by simultaneously measuring the total particle number or spin operator at each measurement.

[0027] Example Hardware

[0028] Figure 1Depicts an example quantum computing system 100 suitable for implementing the measurement strategies described currently. 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 at one or more locations, where the systems, components, and techniques described below can be implemented.

[0029] System 100 may include quantum hardware 102 that communicates data with 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 within quantum hardware 102. For example, quantum hardware 102 may include one or more components for performing classical computing operations.

[0030] 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 the physical system to be probed or simulated. For example, it may be desired 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.

[0031] The system may generate output data, such as output data 108, representing the results of simulating the physical system of interest. Output data 108 may include data representing the properties of the determined physical system (e.g., the energy of the measured physical system), or data that can be used to determine the properties of the physical system (e.g., data representing raw measurement results). For example, as described above, in some embodiments, the physical system may be a chemical substance, such as an atom or molecule. In these cases, the data representing the simulation results can be used to determine the properties of the chemical substance, such as the rate of a chemical reaction, the electronic structure, and / or the optical / thermal properties.

[0032] System 100 is configured to perform classical computing using classical processor 104 and quantum hardware 102 in combination with quantum computing. Quantum hardware 102 includes components for performing quantum computing using quantum circuits. For example, quantum hardware 102 includes quantum system 120 and 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 computing. 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 computing being performed by quantum hardware 102. For example, the multi-level quantum subsystems may include qubits implemented via atomic, molecular, or solid-state quantum systems. In other examples, qubits may include, but are not limited to, superconducting qubits or semiconductor qubits.

[0033] The multi-level quantum subsystem can be frequency tunable. For example, each qubit can have an associated operating frequency, which can be adjusted, for example, by applying voltage pulses via one or more drive lines coupled to the qubit using one or more control devices 122. Example operating frequencies include qubit idle frequency, qubit interaction frequency, and qubit readout frequency. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to the corresponding idle frequency can put the qubit into a state in which it does not strongly interact with other qubits and can be used to perform single-qubit gates (e.g., as part of a quantum circuit). As another example, in the case where qubits interact via a coupler with a fixed coupling, the qubits can be configured to interact with each other by setting their respective operating frequencies to a certain gate-related frequency that is detuned from their common interaction frequency. In other cases, for example, when 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 achieve qubit interaction and then by setting the respective operating frequencies of the qubits to a certain gate-related frequency that is detuned from their common interaction frequency. Such interactions can be performed to execute multi-qubit gates (e.g., as part of a quantum circuit).

[0034] The control device 122 can also include a measurement device, such as a readout resonator. Measurement results (measurement data) obtained via the measurement device can be provided to the classical processor included in the quantum hardware 102 or the classical processor 104 for processing and analysis.

[0035] The classical processor 104 can include a Hamiltonian decomposition module 114, which is configured to process the 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, which is equivalent or approximate to the Hamiltonian expressed in the orthonormal basis.

[0036] The decomposed version of the Hamiltonian includes a sum of terms, where the sum of terms includes diagonal operators in the corresponding single-particle basis. For example, as described in more detail below with reference to Figure 2 Each term in the sum of terms can include i) a corresponding operator that affects the rotation of the corresponding basis and ii) one or more particle density operators, as described in more detail below with reference to Figure 2 An example process for decomposing a Hamiltonian that describes a chemical system into such a sum of terms is described in detail below.

[0037] The classical processor 104 may also include a basis rotation grouper 116 configured to group terms of the decomposed Hamiltonian that are diagonal in the same single-particle basis (composed of sums and products of particle density operators). The basis rotation grouper 116 is configured to provide data representing a group of terms 126 that are diagonal in the same single-particle basis to the quantum hardware 102. The quantum hardware 102 can then measure the expected value of the terms associated with each single-particle basis by applying a Givens rotation circuit that performs a change of basis prior to measuring the Jordan-Wigner transformed number operator in the computational basis. The following references Figure 2 describe in more detail an example process for grouping terms of the decomposed Hamiltonian and measuring the expected value of the terms in the group.

[0038] The classical processor 104 may also include a post-processing module 118 configured to process measurement results 128 received from the quantum hardware 102. For example, the post-processing module 118 may be configured to calculate the statistical mean of the measurement results. In some embodiments, the post-processing module 118 may also include a post-selection module 124 for performing error mitigation, as described in more detail below with reference to Figure 4 more detail.

[0039] Base Rotation Grouping Measurement Strategy

[0040] Figure 2 is a flowchart 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 system 100) appropriately programmed according to this specification may perform process 200.

[0041] The system obtains a Hamiltonian describing the quantum chemical system (step 202). The Hamiltonian may be obtained by a classical computing device. The Hamiltonian may be input by a user or may be extracted from another source (e.g., memory). The Hamiltonian describing the physical system may be represented in any orthonormal basis (e.g., Gaussian or general molecular orbital basis). The Hamiltonian may include a one-electron component and a two-electron component, where each term in the one-electron component and the two-electron component includes a corresponding scalar coefficient given by a known integral over basis functions depending on a particular discretization scheme. The following equation (1) gives an example Hamiltonian.

[0042]

[0043] In equation (1), a p and Annihilation and creation operators for the spin - orbit p, t p,q 、h pqrs represent scalar coefficients that can be given by known integrals over basis functions depending on a particular discretization scheme. In some embodiments, the Hamiltonian can describe a periodic physical system. In other embodiments, the Hamiltonian can describe an aperiodic physical system, such as a single molecule.

[0044] The Hamiltonian describing a quantum - chemical system can be mapped to a qubit Hilbert space using various methods for simulating indistinguishable fermions with resolvable 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.

[0045] The Jordan - Wigner transformation is attractive because it is simple and it allows for the explicit construction of many useful circuit primitives. However, a drawback of using the Jordan - Wigner transformation is that the Jordan - Wigner transformation maps an operator acting on a constant number of fermion modes to qubit operators, supporting at most N qubits. Under a simple readout error model, such as a symmetric bit - flip channel, a Pauli word supporting L qubits has L opportunities to produce an error with the opposite sign of the measured value. This drives the expectation - value measurement to zero by a multiplicative factor that is exponentially small in L. As described below, the system implementing the example process 200 can avoid this challenge without leaving the Jordan - Wigner framework, thus allowing the estimation of single - and two - particle fermion operator expectation values by separately measuring 1 - local qubit operators and 2 - local qubit operators.

[0046] The system performs classical calculations to decompose the Hamiltonian describing the chemical system into a sum of terms, where each term includes i) a corresponding operator affecting 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 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 equation (2) below

[0047]

[0048] where the operator U l represents an operator affecting the rotation of the corresponding single - particle basis, e.g., the single - particle basis rotation given by equation (3) below

[0049]

[0050] In some cases, it may be difficult to simulate the Hamiltonian describing a chemical system (e.g., the Hamiltonian in the form given in Equation (1)). For example, the Hamiltonian can describe an aperiodic system. A matrix representing the Hamiltonian (in a given basis) describing a periodic system typically includes an underlying structure that can be used to simplify the calculations associated with simulating the periodic system. For example, the repetitions in the underlying structure can be used to reduce the number of multiplication operations required to simulate the periodic system. However, a matrix representing the Hamiltonian (in a given basis) describing an aperiodic system may not include such an underlying structure. Thus, effectively simulating such a system can be challenging. However, by decomposing the Hamiltonian (represented in an orthonormal basis) into a sum of multiple terms as described above, the Hamiltonian is mapped to a combination of sub-Hamiltonians with a specific structure and represented in different corresponding bases, where simulation techniques can be effectively applied.

[0051] To decompose the Hamiltonian describing a chemical system into a sum of multiple terms, the system first diagonalizes the scalar coefficients h of the terms in the two-electron component of the Hamiltonian by representing each scalar coefficient as a sum of multiple terms on a one-particle basis pqrs . Each term in the sum of multiple terms includes the Hermitian coefficient matrix of a monomer operator formed by a first pair of spin orbitals matrix ω l and the product of the Hermitian coefficient matrix of a monomer operator formed by a second pair of spin orbitals . For example, the system can use Equation (4) below to diagonalize the scalar coefficient h of the two-electron component of the Hamiltonian pqrs .

[0052]

[0053] Then, the two-electron component of the Hamiltonian can be rewritten to obtain the Hamiltonian given in Equation (5) below.

[0054]

[0055] where, t pq and each (for a fixed l) are the Hermitian coefficient matrices of monomer operators.

[0056] Then, the system determines the matrices matrices to diagonalize the corresponding Hermitian coefficient matrices of the monomer operators in their respective one-particle Hilbert spaces. The system uses the determined matrices to determine the operators

[0057] In some embodiments, the system may discard finite eigenvalues that are 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 of the original Hamiltonian. The predetermined threshold may be preselected based on a specific quantum chemistry system (or equivalently, the Hamiltonian describing the quantum chemistry 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.

[0058] The system groups the terms of the decomposed Hamiltonian (step 206) that are diagonal in the same single-particle basis l. For example, for each term in the decomposed Hamiltonian given by the above equation (2), the system may determine in which single-particle basis the term is diagonalized and assign the term to the group corresponding to that single-particle basis.

[0059] Then, the system measures (step 206) the expectation value of the terms included in each group. Within a group, the measurement of the terms can be performed approximately simultaneously (e.g., within the limits of hardware precision).

[0060] To measure the expectation value of the terms included in a group, the system performs a basis rotation corresponding to the group on the qubit system encoding the state of the chemical system to be measured via quantum computing. For example, the system may apply a corresponding Givens rotation circuit to the qubit system. Then, the system measures the qubit system. This includes measuring the Jordan-Wigner transform of one or more particle density operators included in the terms in the computational basis to obtain the corresponding measurement results. That is, by performing a basis rotation, e.g., directly applying the corresponding U l circuit on the quantum state of the qubit system before measurement, the system can then simultaneously sample <n p > and <n p n q > expectation values to estimate the energy as where the subscript l on the expectation values indicates that they are sampled after applying the basis transformation U l .

[0061] <n p > and <n p n q > can be simultaneously sampled because under the Jordan-Wigner transform, this is a diagonal qubit operator. Thus, the system is able to sample all the terms in the Hamiltonian with only L + 1 different groups of terms. Fortunately, U l is extremely easy to implement, even on hardware with minimal connectivity. For example, any change in the single-particle basis can be achieved using (N2 be performed with (N - N) / 8 two-qubit gates and a gate depth of exactly N, even in the case of a qubit array with only linear connectivity.

[0062] The system can repeatedly (e.g., a predetermined number of times) measure the expected value of the terms included in the group to obtain multiple measurement results, and post-process the multiple measurement results to obtain a final measurement result. For example, the system can post-select measurement results from the multiple measurement results and / or determine the statistical average of the multiple measurement results. The following refers to Figure 4 Describe an example process for performing error mitigation using post-selection.

[0063] The system determines the energy of the chemical system (step 208) using the measurement results of each obtained group through classical computation, e.g., by summing the averages of the measurement results determined for each group. In some embodiments, the system can use the determined energy of the chemical system to perform further computations to determine the properties of the physical system. For example, the example process 200 can be used to determine the properties of a single molecule and simulate catalysts or drugs for performing materials science simulations or for determining the rate of a chemical reaction.

[0064] Figures 3A - 3C Shows the number of qubits / spin orbitals plotted against the example measurement time (the time required to obtain an estimate of the ground state energy) for different chemical systems and different measurement strategies / methods (i.e., the currently described measurement strategy (referred to as "basis rotation grouping"), three existing measurement strategies ("separate measurement", "Pauli word grouping", and "Pauli word grouping, RDM constraint"), and two upper bounds from the fermionic L1 norm and the qubit L1 norm).

[0065] Figures 3A - 3C The data plotted in is generated, for example, according to known specifications (such as those requiring each operator via to be measured as a fraction of the time f i ), assuming 10,000 circuit repetitions per second and an optimally distributed measurement, using the calculation of the variance of the expected value to determine the required measurement time. Since in practice the variance of each operator is not known in advance, an adaptive measurement scheme that assumes additional measurements are scheduled based on the observed sample variance can approximate the ideal partitioning of the measurement time, and for simplicity, Figures 3A - 3C only the numbers based on the ideal partitioning are presented. Assume the target accuracy corresponds to a 2σ error bar of 1.0 millihartree.

[0066] The computations performed to generate the data plotted in FIG. 3 are for symmetrically stretched hydrogen chains ( Figure 3A ) with different bond lengths and numbers of atoms, symmetrically stretched water molecules (Figure 3B ) and the stretched nitrogen dimer ( Figure 3C ) All calculations are performed under multiple basis sets. For systems with up to 20 qubits, all calculations use the configuration interaction singles and doubles (CISD) approximation for the ground state, and for cases with more than 20 qubits, they are performed using the Hartree - Fock state. The L1 - norm - based bounds are calculated in the fermionic Hilbert space and in terms of operators acting directly on the qubits.

[0067] Figure 3A Data generated for symmetrically stretched hydrogen chains with different bond lengths and numbers of atoms under multiple basis sets are shown. Under the STO - 3G, 6 - 31G, or cc - pVDZ basis sets, the hydrogen chains contain 2 to 10 atoms with symmetric inter - atomic spacings between 0.6 and . Calculations performed on systems that require the same number of qubits (spin - orbitals) are plotted together in columns and are slightly spread horizontally for visibility.

[0068] Figure 3B Data generated for symmetrically stretched water molecules under multiple basis sets are shown. The bonds in the water molecule are symmetrically stretched between 0.8 and . Water calculations are performed under the STO - 3G basis set with and without the 1s orbitals frozen on the nitrogen / oxygen molecule, and under the 6 - 31G basis set with the core frozen. Calculations performed on systems that require the same number of qubits (spin - orbitals) are plotted together in columns and are slightly spread horizontally for visibility.

[0069] Figure 3C Data generated for the stretched nitrogen dimer under multiple basis sets are shown. The spacing between the nitrogen atoms ranges from 0.9 to Nitrogen calculations are performed under the STO - 3G basis set with and without the 1s orbitals frozen on the nitrogen / oxygen molecule, and under the 6 - 31G basis set with the core frozen. Calculations performed on systems that require the same number of qubits (spin - orbitals) are plotted together in columns and are slightly spread horizontally for visibility.

[0070] Figures 3A - 3CEach of them demonstrates the effectiveness of the currently described measurement strategy compared to each of three existing measurement strategies and two L1 norm values. For example, for system sizes of 12 qubits in the case of a hydrogen chain, 24 qubits for a water molecule, and 16 qubits for a nitrogen dimer, the measurement time is reduced using the currently described basis rotation grouping method. Additionally, for an increasing number of qubits, i.e., for increasing system sizes, the relative improvement in measurement time between the currently described method and other existing methods also increases.

[0071] In addition to the reduction in measurement time, the measurement strategy described above with reference to example process 200 also has the additional benefits of reducing sensitivity to readout errors and enabling a powerful form of error mitigation through post-selection. These properties are a result of the fact that the Hamiltonian measured in example process 200 is measured only in terms of the particle density operator in different basis sets. As described above, applying the Jordan-Wigner transform to the terms of a quantum chemistry Hamiltonian results in operators that support up to N qubits. However, the particle density operator transforms more simply Thus, the measurement strategy described in example process 200 avoids the expansion of locality caused by long Jordan-Wigner strings, and the individual terms from the single-particle and two-particle components of the Hamiltonian can be measured by measuring 1-local qubit operators and 2-local qubit operators separately.

[0072] Furthermore, the currently described measurement strategy provides an opportunity for error mitigation. For example, when interested in a state with a definite eigenvalue of a symmetric operator (such as the total particle number or the z-component of the spin ), it is desirable to have a method to remove the components in some experimentally prepared state ρ that violate this constraint.

[0073] There are two strategies to achieve this goal. The first existing strategy involves: directly and non-destructively measuring the symmetric operator, and discarding those results where an undesired eigenvalue is observed, thereby projecting into the appropriate symmetric sector through post-selection. The difficulty in effectively performing these measurements limits the application of this strategy. Additionally, some implementations of this first existing strategy focus on using circuits of depth O(N) to measure and the parity, which may introduce further errors during their implementation. The second existing strategy builds on the first and uses additional measurements and classical post-processing to calculate the expectation value of the projected state without requiring additional circuit depth.

[0074] As referred to below Figure 4As described above, the currently described measurement strategy implements a new form of error mitigation based on post-selection and eliminates the challenges posed by the non-locality of operators transformed by the Jordan-Wigner transformation during measurement.

[0075] Error Mitigation by Post - selection

[0076] Figure 4 is a flowchart of an example process 400 for performing error mitigation using post-selection. As described above, example process 400 can be performed in conjunction with example process 200 for measuring the energy of a chemical system.

[0077] 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 system 100) appropriately programmed according to this specification can perform process 400.

[0078] The system obtains the measurement results generated during step 206 of example process 200 (step 402). That is, the system obtains the measurement results by applying the corresponding Givens rotation circuit to the qubit system to implement the corresponding basis rotation and measuring the Jordan-Wigner transformation of one or more number operators included in the terms in the computational basis, where the measurement results represent the results of measuring a group of terms having the same operator affecting the corresponding basis rotation.

[0079] The system uses the obtained measurement results to calculate the value of the total particle number or spin component (e.g., the z-component of the spin) (step 404). Since the obtained measurement results consist of bit strings, the system can calculate the total particle number by counting 1s or 0s, or calculate the value of the spin component by subtracting the sum of the particle density operators acting on the "up spin" spin orbitals from the sum of the particle density operators acting on the "down spin" spin orbitals (and dividing by 2).

[0080] The system determines whether the calculated value of the total particle number or spin component is equal to the corresponding target value. In response to determining that the calculated total particle number or spin number is equal to the corresponding target value, the system determines that no error has occurred and provides the measurement results for post-processing, as described above with reference to Figure 2 As described. In response to determining that the calculated total particle number or spin number is not equal to the corresponding target value, the system determines that an error has occurred and discards the measurement results.

[0081] Figure 5 shows a graph of the single-qubit depolarization probability versus the readout bit-flip probability for the ground state measurement of 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 diagrams A - D corresponding to different measurement strategies are shown. In each diagram, squares and numbers represent the absolute error in millihartrees.

[0082] Diagram A shows the error caused by the "Pauli grouping" measurement strategy, which involves simultaneously measuring compatible Pauli words in the usual molecular orbital basis. Diagram B shows the error when using the currently described "basis rotation" or "basis rotation grouping" scheme, which performs a change of the single - particle basis before measurement. Diagram C shows the error using the "Pauli grouping strategy" along with additional measurements and post - processing that effectively project the measured state onto the manifold with the correct parity of the total particle number and the current. Diagram D shows the error found when using the currently described basis rotation strategy and post - selecting on the observed correct particle number and the result. To approximate the true ansatz circuit, three identical random Givens rotation networks were simulated to act on the ground state before measurement.

[0083] Figure 5 It is shown that the amount of error caused by using the currently described measurement strategies (with and without post - selection techniques) can be less than the error caused by using the Pauli grouping measurement strategy (with and without post - selection techniques). Additionally, both the Pauli grouping strategy and the basis rotation strategy benefit from implementing post - selection error mitigation techniques. Moreover, although the currently claimed basis rotation strategy requires a circuit depth that is one - third of the circuit depth used by the Pauli grouping strategy, the remaining error after post - selection error mitigation is comparable in many cases and lower when the noise during measurement is the dominant error channel. Even without post - selection (where the currently described measurement strategy benefits from a strictly more powerful form of error mitigation), the locality of the currently described Jordan - Wigner - transformed operators also provides some benefits in suppressing the effects of readout errors.

[0084] The digital and / or quantum subject matter and the implementations of digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuits, in suitable quantum circuits, or more generally, in a quantum computing system, 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 of them. The term "quantum computing system" can include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0085] Embodiments of the digital and / or quantum subject matter described in this specification may 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, or to control the operation of, a data processing apparatus. The digital and / or quantum computer storage medium may 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 of them. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagated signal capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing apparatus.

[0086] The terms quantum information and quantum data refer to information or data carried, held, or stored by a quantum system, where the smallest non-trivial system is a qubit, i.e., the system that defines the unit of quantum information. It should be understood that the term "qubit" includes all quantum systems that can be suitably approximated as two-level systems in the corresponding context. Such quantum systems may include multi-level systems, e.g., having two or more energy levels. By way of example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational basis states are identified with the ground state and the first excited state; however, it should be understood that other settings where the computational states are identified with higher excited states are also possible.

[0087] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all kinds 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 dedicated logic circuitry, e.g., an FPGA (field-programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform universal quantum computing. In addition to the hardware, the apparatus may optionally include code that creates an execution environment for the digital and / or quantum computer program, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0088] A digital computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) can be written in any form of programming language (including compiled or interpreted languages or declarative or procedural languages), and it can 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) can 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 it can be written in a quantum programming language (e.g., QCL or Quipper).

[0089] A digital and / or quantum computer program may or may not correspond to a file in a file system. The program can be stored as part of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), stored in a single file dedicated to the program being discussed, or stored in multiple cooperating files (e.g., files that store one or more modules, subroutines, or portions of code). A digital and / or quantum computer program can 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 site or distributed across multiple sites 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). Generally, a digital data communication network cannot transmit quantum data, whereas a quantum data communication network can transmit both quantum data and digital data.

[0090] One or more digital and / or quantum computer programs can be executed by one or more programmable digital and / or quantum computers, optionally operating in conjunction with one or more digital and / or quantum processors, to perform functions by operating on input digital and quantum data and generating output, thereby performing the processes and logical flows described in this specification. The processes and logical flows can also be executed by special-purpose logic circuitry (e.g., FPGA or ASIC) or a quantum simulator, or a combination of special-purpose logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers, and the apparatus can also be implemented as special-purpose logic circuitry (e.g., FPGA or ASIC) or a quantum simulator, or a combination of special-purpose logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers.

[0091] 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 on it software, firmware, hardware, or a combination thereof, which, in operation, causes the system to perform those operations or actions. 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 those operations or actions. A quantum computer can receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform those operations or actions.

[0092] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs can be based on general-purpose or special-purpose digital and / or quantum processors or both, or any other type of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (such as photons), or a combination thereof.

[0093] The basic elements of a digital and / or quantum computer are a central processing unit for running or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory can be supplemented or incorporated with dedicated logic circuits or quantum simulators. Generally, a digital and / or quantum computer will also include or be operatively coupled to one or more mass storage devices (such as 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 it, or to transmit digital and / or quantum data to it, or both. However, a digital and / or quantum computer does not require such devices.

[0094] 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 a quantum memory is a device capable of storing quantum data with high fidelity and high efficiency for a long time, such as an optical-matter interface where light is used for transmission and a material with quantum characteristics (such as superposition or quantum coherence) for storing and preserving quantum data.

[0095] 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 that includes 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 be implemented, respectively, as apparatuses, methods, or systems 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.

[0096] Although this specification contains many specific implementation details, these should not be construed as limitations on the scope of what is claimed, but rather as descriptions of features specific to particular implementations. Certain features that are described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, the various features described in the context of a single implementation may also be implemented separately in multiple implementations or in any suitable sub-combination. Additionally, although features may be described above as acting in certain combinations and even initially claimed as such, in some cases, one or more features in the claimed combination may be excluded from the combination, and the claimed combination may be directed to a sub-combination or a variant of a sub-combination.

[0097] Similarly, although operations are depicted in the drawings in a particular order, this should not be understood as requiring that the operations be performed in the particular order shown or in sequential order, or that all of the illustrated operations be performed, to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Additionally, the separation of the various system modules and components in the above-described implementations should not be understood as required in all implementations, and it should be understood that the described program components and systems may generally be integrated in a single software product or packaged into multiple software products.

[0098] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the acts recited in the claims may be performed in a different order and still achieve the desired result. As one example, the processes described in the figures do not necessarily need the particular order or sequential order shown to achieve the desired result. 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: obtaining a Hamiltonian describing the chemical system, wherein the Hamiltonian is represented in an orthonormal basis; decomposing, by classical calculation, the Hamiltonian describing the chemical system into a sum of terms, wherein each term includes i) a corresponding operator affecting the rotation of a corresponding single-particle basis and ii) one or more particle density operators; for each group of terms including terms having the same operator affecting the rotation of a corresponding single-particle basis, repeatedly measuring the expected value of the terms included in the group, including: performing, by quantum calculation, a corresponding single-particle basis rotation on a qubit system encoding the state of the chemical system; and measuring, in the computational basis, the Jordan-Wigner transform of one or more particle density operators in the group to obtain a corresponding measurement result for the group; and determining, by classical calculation, the energy of the chemical system using the obtained measurement results.

2. The method according to claim 1, wherein measuring the expected value of the terms included in the group further includes performing error mitigation by post-selection.

3. The method according to claim 2, wherein performing error mitigation by post-selection includes: calculating the total number of particles or the spin component using the obtained measurement results; determining whether the calculated total number of particles or the spin component is equal to a corresponding target value; in response to determining that the calculated total number of particles or the spin component 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 the spin component is not equal to the corresponding target value, discarding the measurement results.

4. The method according to any one of claims 1 to 3, wherein the Hamiltonian describing the chemical system includes a single-electron component and a two-electron component, and wherein decomposing, by classical calculation, the Hamiltonian describing the chemical system into a sum of terms includes: diagonalizing each scalar coefficient in the two-electron component, including representing each scalar coefficient in the two-electron component as a second sum of terms on a single-particle basis, each term in the second sum of terms including the Hermitian coefficient matrix of a monomer operator formed by a first pair of spin orbitals, a matrix, and the product of the Hermitian coefficient matrix of a monomer operator formed by a second pair of spin orbitals; for each term in the sum of terms, determining the matrix of the monomer operator in the diagonalized corresponding Hermitian coefficient matrix; and using the determined matrix of the diagonalized monomer operator to determine the corresponding operator affecting the corresponding basis rotation.

5. The method according to claim 4, further comprising discarding finite eigenvalues less than a predetermined threshold.

6. The method according to any one of claims 1 to 3, further comprising grouping the terms of the decomposed Hamiltonian that are diagonal in the same single-particle basis, for each term in the decomposed Hamiltonian, including: determining in which single-particle basis the term is diagonalized; and assigning the term to the group corresponding to the determined single-particle basis.

7. The method according to any one of claims 1 to 3, wherein measuring the expected value of the terms included in the group includes simultaneously measuring the expected value of the terms included in the group.

8. The method according to any one of claims 1 to 3, wherein, performing the corresponding basis rotation includes applying a corresponding Givens rotation circuit to the qubit system.

9. The method according to any one of claims 1 to 3, wherein, determining the energy of the chemical system using the obtained measurement results through classical computing includes: determining the average measurement result corresponding to each group; and adding the determined average values.

10. The method according to any one of claims 1 to 3, wherein, any orthonormal basis includes a Gaussian or molecular orbital basis.

11. The method according to any one of claims 1 to 3, wherein, the Hamiltonian describing the chemical system includes a plurality of terms, each term including a product of one or more of the following: i) an annihilation operator of a corresponding spin orbital, ii) a creation operator of a corresponding spin orbital, and iii) a scalar coefficient given by a one - electron integral or a two - electron integral over basis functions in an orthonormal basis.

12. The method according to claim 1, wherein, the chemical system includes a symmetrically stretched hydrogen chain, a symmetrically stretched water molecule, or a stretched nitrogen dimer.

13. An apparatus for measuring the energy of a chemical system, comprising: quantum hardware; and one or more classical processors; wherein the apparatus is configured to perform operations including the method according to any of the preceding claims.

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