WO2022257316A1 - 量子体系基态能量估计方法及系统 - Google Patents

量子体系基态能量估计方法及系统 Download PDF

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WO2022257316A1
WO2022257316A1 PCT/CN2021/124392 CN2021124392W WO2022257316A1 WO 2022257316 A1 WO2022257316 A1 WO 2022257316A1 CN 2021124392 W CN2021124392 W CN 2021124392W WO 2022257316 A1 WO2022257316 A1 WO 2022257316A1
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pauli
quantum
string
target
qubit
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PCT/CN2021/124392
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French (fr)
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张士欣
万周全
张胜誉
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腾讯科技（深圳）有限公司
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Priority to JP2022567390A priority Critical patent/JP7471736B2/ja
Priority to US17/977,344 priority patent/US20230054868A1/en
Publication of WO2022257316A1 publication Critical patent/WO2022257316A1/zh

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/60—Quantum algorithms, e.g. based on quantum optimisation, quantum Fourier or Hadamard transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/20—Models of quantum computing, e.g. quantum circuits or universal quantum computers
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/70—Quantum error correction, detection or prevention, e.g. surface codes or magic state distillation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/08—Learning methods
    • G06N3/084—Backpropagation, e.g. using gradient descent

Definitions

  • the embodiments of the present application relate to the field of quantum technology, and in particular to a method and system for estimating the ground state energy of a quantum system.
  • VQE Very Quantum Eigensolver
  • a related technology proposes a scheme of using the Jastrow factor as a post-processing enhancement of the VQE.
  • the Jastrow factor to post-process the wave function output by the variable quantum circuit in VQE, in order to describe more quantum entanglement and correlation, so that the final estimated ground state energy is as close as possible to the real value.
  • Jastrow factor is more suitable for describing many-body associations, it is still not the most general form that classical postprocessing can have, so its expressive ability is weak, which will affect the accuracy of ground state energy estimation.
  • the embodiment of the present application provides a method and system for estimating the ground state energy of a quantum system. Described technical scheme is as follows:
  • a method for estimating the ground state energy of a quantum system is provided, the method is executed by a computer device, and the method includes:
  • the expected energy value in the quantum state is the summation result of the expected energy values of k Pauli strings obtained by decomposing the Hamiltonian, n is a positive integer, and k is a positive integer;
  • the expected energy value of the Hamiltonian satisfies a convergence condition
  • the expected energy value of the Hamiltonian satisfying the convergence condition is determined as the ground state energy of the target quantum system.
  • a device for estimating the ground state energy of a quantum system comprising:
  • the state acquisition module is used to obtain the output quantum states of the n qubits obtained after the input quantum states of the n qubits are transformed by the parameterized quantum circuit; wherein, the Hamiltonian of the target quantum system is in the
  • the expected energy value under the output quantum state of n qubits is the summation result of the expected energy values of k Pauli strings obtained by decomposing the Hamiltonian, n is a positive integer, and k is a positive integer;
  • a post-processing module configured to use a neural network to post-process the output quantum states of the n qubits, and calculate the expected energy value of the Hamiltonian according to the post-processing results of the neural network;
  • the optimizer module is used to adjust the parameters of the parameterized quantum circuit and the parameters of the neural network with the goal of convergence of the expected energy value of the Hamiltonian; the expected energy value of the Hamiltonian satisfies the convergence condition
  • the energy expectation value of the Hamiltonian satisfying the convergence condition is determined as the ground state energy of the target quantum system.
  • a computer device the computer device includes a processor and a memory, and a computer program is stored in the memory, and the computer program is loaded and executed by the processor to realize the above-mentioned method.
  • a computer-readable storage medium is provided, and a computer program is stored in the storage medium, and the computer program is loaded and executed by a processor to implement the above method.
  • a computer program product or computer program includes computer instructions, the computer instructions are stored in a computer-readable storage medium, and a processor reads from the The computer-readable storage medium reads and executes the computer instructions to implement the above method.
  • a system for estimating the ground state energy of a quantum system includes: a parameterized quantum circuit and a computer device, and the computer device includes a post-processing module and an optimizer module;
  • the parameterized quantum circuit is used to transform the input quantum states of n qubits to obtain the output quantum states of the n qubits; wherein, the Hamiltonian of the target quantum system is within the range of the n qubits
  • the expected energy value under the output quantum state is the summation result of the expected energy values of the k Pauli strings obtained by decomposing the Hamiltonian, n is a positive integer, and k is a positive integer;
  • the post-processing module is used to post-process the output quantum states of the n qubits by using a neural network, and calculate the expected energy value of the Hamiltonian according to the post-processing results of the neural network;
  • the optimizer module is used to adjust the parameters of the parameterized quantum circuit and the parameters of the neural network with the goal of converging the expected energy value of the Hamiltonian; wherein, the expected energy value of the Hamiltonian If the convergence condition is satisfied, the energy expectation value of the Hamiltonian satisfying the convergence condition is determined as the ground state energy of the target quantum system.
  • the neural network can play the role of a general function approximator, which has stronger expression ability and ground state energy approximation ability than the Jastrow factor, so that It helps to improve the accuracy of ground state energy estimation.
  • Fig. 1 is a schematic diagram of a VQNHE framework provided by an embodiment of the present application
  • Fig. 2 is the flowchart of the method for estimating the ground state energy of the quantum system provided by one embodiment of the present application;
  • Fig. 3 is a schematic diagram of a VQNHE framework provided by another embodiment of the present application.
  • Fig. 4 is a flowchart of a method for estimating the ground state energy of a quantum system provided by another embodiment of the present application.
  • Fig. 5 is a schematic diagram of a measurement circuit provided by an embodiment of the present application.
  • Fig. 6 is a schematic diagram of a measurement circuit provided by another embodiment of the present application.
  • Fig. 7 is a schematic diagram of the comparison of various schemes in molecular energy calculation shown in the present application.
  • Fig. 8 is a schematic diagram of the quantum circuit structure in molecular energy calculation shown in the present application.
  • Fig. 9 is a comparison diagram of the performance of VQE and VQNHE exemplarily shown in the present application on real hardware and a noise simulator;
  • Fig. 10 is a schematic diagram of the PQC circuit structure exemplarily shown in the present application.
  • Fig. 11 is a block diagram of a quantum system ground state energy estimation device provided by an embodiment of the present application.
  • Fig. 12 is a schematic structural diagram of a computer device provided by an embodiment of the present application.
  • Quantum computing Based on quantum logic computing, the basic unit of data storage is the quantum bit (qubit).
  • Qubit The basic unit of quantum computing. Traditional computers use 0 and 1 as the basic units of binary. The difference is that quantum computing can process 0 and 1 at the same time, and the system can be in the linear superposition state of 0 and 1:
  • ⁇ > ⁇
  • 2 represent the probability of being 0 and 1, respectively.
  • Quantum circuit A representation of a quantum general-purpose computer, which represents the hardware implementation of the corresponding quantum algorithm/program under the quantum gate model. If the quantum circuit contains adjustable parameters to control the quantum gate, it is called a parameterized quantum circuit (Parameterized Quantum Circuit, referred to as PQC) or a variable quantum circuit (Variational Quantum Circuit, referred to as VQC), both of which are the same concept .
  • PQC Parameterized Quantum Circuit
  • VQC Variational Quantum Circuit
  • Hamiltonian A Hermitian matrix that describes the total energy of a quantum system.
  • the Hamiltonian is a physical vocabulary and an operator that describes the total energy of a system, usually denoted by H.
  • Quantum Architecture Search Quantum Architecture Search, referred to as QAS: a general term for a series of work and programs that attempt to automate and programmatically search for the structure, mode, and arrangement of quantum circuits.
  • Traditional quantum structure search work usually uses greedy algorithm, reinforcement learning or genetic algorithm as its core technology.
  • Recently developed techniques include differentiable quantum structure searches and predictor-based structure search schemes.
  • Quantum-classical hybrid computing a calculation paradigm in which the inner layer uses quantum circuits (such as PQC) to calculate the corresponding physical quantities or loss functions, and the outer layer uses traditional classical optimizers to adjust the variational parameters of quantum circuits. Taking advantage of quantum computing is believed to be one of the important directions that has the potential to prove quantum superiority.
  • quantum circuits such as PQC
  • NISQ Noisy Intermediate-Scale Quantum
  • Quantum Error Mitigation Corresponding to Quantum Error Correction (Quantum Error Correction), it is a series of quantum error mitigation and noise suppression schemes with smaller resource costs under the hardware of the NISQ era. It requires significantly fewer resources than full quantum error correction, and may only be applicable to specific tasks rather than a general solution.
  • VQE Variational Quantum Eigensolver
  • Jastrow factor A factor commonly used in the proposed design of the variational Monte Carlo wave function, which is used to strengthen the wave function with no interaction in the mean field, in order to describe more quantum related information. Its basic form is where ⁇ is a variational parameter, Z is a quantum operator that gives ⁇ 1 eigenvalues on a measurement basis, k and l represent different qubit degrees of freedom, k represents the kth qubit, and l represents the lth quantum bit.
  • Non-unitary The so-called unitary matrix is a matrix that satisfies All matrices of , and all evolution processes directly allowed by quantum mechanics, can be described by unitary matrices.
  • U is a unitary matrix (Unitary Matrix), also known as a unitary matrix, a unitary matrix, etc., is the conjugate transpose of U.
  • matrices that do not meet this condition are non-unitary, which requires auxiliary means or even exponentially more resources to be realized experimentally, but non-unitary matrices often have stronger expressive power and faster ground state projection effects .
  • the above-mentioned “exponentially many resources” means that the demand for resources increases exponentially with the increase in the number of qubits.
  • the exponentially many resources can mean that the total number of quantum circuits that need to be measured is exponentially multiple, that is, the corresponding needs Exponentially much computation time.
  • Pauli string An item composed of the direct product of multiple Pauli operators at different lattice points.
  • a general Hamiltonian can usually be decomposed into a direct product of a set of Pauli strings.
  • the measurement of VQE is generally measured item by item according to the Pauli string decomposition.
  • Pauli operator also known as Pauli matrix, is a set of three 2 ⁇ 2 unitary Hermitian complex matrices (also known as unitary matrices), generally represented by the Greek letter ⁇ (sigma). Among them, the Pauli X operator is The Pauli Y operator is The Pauli Z operator is
  • UCC Unitary Coupled Cluster, unitary coupling cluster
  • hardware friendly hardware efficient proposed design: two different variational line structures of VQE.
  • the former borrows from the traditional variational numerical method of quantum chemistry coupled-cluster (coupled cluster), and the approximation effect is better, but it needs Trotter to decompose the corresponding exponential operator, so it requires higher quantum resources.
  • the latter adopts the strategy of directly densely arranging native quantum gate groups, which requires shallower circuits and lower requirements for quantum resources, but the corresponding expression and approximation capabilities are also worse than those proposed by UCC.
  • Bit string a string of numbers consisting of 0 and 1.
  • the classical results obtained by each measurement of the quantum circuit can be represented by 0 and 1 respectively according to the upper and lower spin configurations on the measurement basis, so that the total measurement result corresponds to a bit string.
  • the technical solution provided by this application helps to speed up and strengthen the development and design of variable quantum algorithms at the present stage.
  • the typical shortcomings of quantum hardware in the NISQ era are short coherence time and large quantum noise.
  • the traditional VQE scheme based on UCC often has high accuracy but requires high line depth, which is difficult to implement on a large scale on existing quantum hardware with coherence time.
  • the hardware-saving hypothesis can be used as a circuit structure through the close arrangement of native quantum gates.
  • the advantage is that the variational structure is easy to realize on quantum hardware, but the expressive ability and the approximation ability to the ground state are often unsatisfactory.
  • VQNHE Variational Quantum Neural network Hybrid Eigensolver
  • the technical solutions provided by this application can be applied to quantum hardware evaluation and actual production in the short to medium term. Its applications include, but are not limited to, the simulation and solution of a variety of ground states of Hamiltonians from condensed matter physics and quantum chemistry problems.
  • the technical solution provided by this application is also expected to further play a role in tasks supported by other variable quantum algorithms such as excited state search and quantum time-dependent evolution.
  • the technical solution provided by this application by further optimizing the neural network model, can achieve a certain effect of quantum error correction on the basis of no prior noise model, which further releases the huge potential of this solution in the NISQ era.
  • any VQE program (used to execute the measurement and estimation process under the entire system architecture) can be seamlessly transplanted to the VQNHE framework, which It can be provided and invoked as a quantum cloud service, and can be encapsulated into a very simple VQE-enhanced API (Application Programming Interface, application programming interface).
  • this scheme can be combined with the quantum structure search method to further adaptively construct the quantum circuit structure suitable for VQNHE.
  • FIG. 1 A VQNHE framework provided by an exemplary embodiment of the present application is shown in FIG. 1 , including a parameterized quantum circuit (PQC) 10 , a neural network 20 and an optimizer 30 .
  • the neural network 20 and the optimizer 30 may be functional modules deployed in computer equipment, and the optimizer 30 may also be called an optimizer module.
  • the computer device may be a classical computer that implements the method by executing a computer program through a processor, which has storage and computing capabilities.
  • the parameterized quantum circuit 10 is used to transform the input quantum states of n qubits to obtain the output quantum states of the n qubits, where n is a positive integer.
  • the expected energy value of the Hamiltonian of the target quantum system in the output quantum state of the n qubits is the summation result of the expected energy values of k Pauli strings obtained by decomposition of the Hamiltonian, where k is a positive integer.
  • the neural network 20 is used for post-processing the output quantum states of the n qubits. Based on the post-processing results of the neural network 20, the expected energy values of the k Pauli strings are obtained, and then the expected energy values of the Hamiltonian are calculated.
  • the optimizer 30 is used to adjust the parameters of the parameterized quantum circuit 10 and the parameters of the neural network 20 with the goal of converging the expected energy value of the Hamiltonian. When the energy expectation value of the Hamiltonian satisfies the convergence condition, the energy expectation value of the Hamiltonian meeting the convergence condition is determined as the ground state energy of the target quantum system.
  • the parameterized quantum circuit (PQC) 10 in the upper left corner of Figure 1 is consistent with that in the traditional VQE framework, and its output wave function
  • the role of the enhanced quantum-neural network hybrid wave function is obtained:
  • the following method can be adopted: for each Pauli string in the above k Pauli strings, measure the output quantum states of n qubits respectively
  • the bit string on the measurement base corresponding to the Pauli string, the metadata used to calculate the energy expectation value of the Pauli string is output by the neural network 20 according to the bit string, and then the bubble is calculated according to these metadata
  • the expected energy value of the Pauli string, and finally the energy expected value of the k Pauli strings is summed to obtain the expected energy value of the Hamiltonian.
  • the gradient-based optimizer 30 (such as Adam) developed by the classical machine learning community can be used to update the corresponding parameters, thereby completing a round of iterations of the quantum-classical hybrid computing paradigm until the obtained energy expectation converges , whose value can be used as an approximate estimate of the ground state of the Hamiltonian of the corresponding system.
  • Fig. 2 is a flowchart of a method for estimating the ground state energy of a quantum system provided by an embodiment of the present application.
  • the method can be applied to the VQNHE framework shown in FIG. 1 , for example, the execution subject of each step of the method can be a computer device.
  • the method may include the following steps (210-240):
  • Step 210 obtain the output quantum states of the n qubits obtained after the input quantum states of the n qubits are transformed by the parameterized quantum circuit; wherein, the Hamiltonian of the target quantum system is between the n qubits
  • the expected energy value in the output quantum state is the summation result of the expected energy values of k Pauli strings obtained by Hamiltonian decomposition, where n is a positive integer and k is a positive integer.
  • the input quantum states of n qubits are transformed through parameterized quantum circuits to obtain the output quantum states of the n qubits.
  • the input quantum state of the parameterized quantum circuit can generally use all 0 states, uniform superposition states or Hartree-Fock (Hartley-Fock) states, and the input quantum states are also called tentative states.
  • the Hamiltonian of the target quantum system can be decomposed into the direct product of k Pauli strings, k is usually an integer greater than 1, but in some special cases k can also be equal to 1, that is, the Hamiltonian of the target quantum system can be Think of it as a Pauli string.
  • the output quantum state of the target quantum system is approximated by the output of the parameterized quantum circuit, and the energy expectation value of the Hamiltonian of the target quantum system is estimated by measuring the output quantum state of the parameterized quantum circuit, and expressed as Minimizing the energy expectation is the optimization goal, and continuously optimizing the parameters of the parameterized quantum circuit to adjust its output quantum state, so that the energy expectation value of the Hamiltonian of the target quantum system in the output quantum state tends to be minimized, and finally the goal is obtained
  • Minimizing the energy expectation is the optimization goal, and continuously optimizing the parameters of the parameterized quantum circuit to adjust its output quantum state, so that the energy expectation value of the Hamiltonian of the target quantum system in the output quantum state tends to be minimized, and finally the goal is obtained
  • the ground state energy of a quantum system is obtained by measuring the output quantum state of the parameterized quantum circuit, and expressed as Minimizing the energy expectation is the optimization goal, and continuously optimizing the parameters of the parameterized quantum circuit to adjust its output quantum state, so that the energy expectation value of the Hamiltoni
  • a neural network is used to post-process the output quantum states of the n qubits, and an energy expectation value of the Hamiltonian is calculated according to the post-processing result of the neural network.
  • the neural network is used to post-process the wave function output by the parameterized quantum circuit.
  • the neural network can play the role of a general function approximator, which has a stronger expression than the Jastrow factor ability and ground state energy approximation ability, which helps to improve the accuracy of ground state energy estimation.
  • step 220 includes several sub-steps as follows:
  • Taylor expansion is performed on the post-processing operator corresponding to the neural network to obtain t Pauli strings, where t is a positive integer; k obtained by decomposing the t Pauli strings and the Hamiltonian of the target quantum system Pauli strings do direct product operation to generate multiple Pauli strings corresponding to the equivalent Hamiltonian of the target quantum system.
  • the equivalent Hamiltonian of the target quantum system is the direct product of multiple Pauli strings corresponding to it.
  • the maximum number of multiple Pauli strings corresponding to the equivalent Hamiltonian of the target quantum system is t ⁇ t ⁇ k.
  • c ijk... represents the coefficient corresponding to Z i Z j Z k ...
  • c ijk... is determined based on the parameters of the neural network
  • Z i is the Z Pauli operator on the i-th qubit
  • Z j is the Z Pauli operator on the jth qubit
  • Z k is the Z Pauli operator on the kth qubit, and so on.
  • the equivalent Hamiltonian of the target quantum system is equal to the direct product of t Pauli strings, the Hamiltonian of the target quantum system, and t Pauli strings, and the Hamiltonian of the target quantum system can be decomposed into k bubbles
  • the direct product of the Pauli strings therefore, it is necessary to measure the expected energy values corresponding to t ⁇ t ⁇ k Pauli strings at most. For each Pauli string in the t ⁇ t ⁇ k Pauli strings, multiple measurements are performed, and the energy calculation result is obtained based on the bit string obtained from each measurement, and then the energy calculation result obtained by the multiple measurements The energy calculation results are averaged to obtain the expected energy value of the Pauli string.
  • the energy expectation value of the equivalent Hamiltonian of the target quantum system on the output quantum state of the PQC is equal to the energy expectation value of the original Hamiltonian of the target quantum system on the post-processing wave function. Therefore, calculating the expected energy value of the original Hamiltonian of the target quantum system is equivalent to calculating the expected energy value of its equivalent Hamiltonian. And the energy expectation value of the equivalent Hamiltonian in The summation result of the expected energy values corresponding to t ⁇ t ⁇ k Pauli strings. For example, the expected energy value of the equivalent Hamiltonian is obtained by adding the expected energy values of t ⁇ t ⁇ k Pauli strings. It should be noted that the above addition may be direct addition or weighted summation, which is not limited in this application.
  • the specific structure of the neural network is not limited, and it may be a simple fully connected structure, or other more complex structures, which is not limited in the present application.
  • step 230 the parameters of the parameterized quantum circuit and the parameters of the neural network are adjusted with the goal of converging the energy expectation value of the Hamiltonian.
  • the derivatives of the expected energy value of the Hamiltonian with respect to the parameters of the parameterized quantum circuit and with respect to the parameters of the neural network are respectively calculated. Then, based on the derivative information, the parameters of the parameterized quantum circuit and the parameters of the neural network are adjusted respectively by using the gradient descent method, so that the energy expectation value of the Hamiltonian tends to be minimized.
  • the parameter optimization process of the parameterized quantum circuit and the parameter optimization process of the neural network can be performed simultaneously or sequentially, which is not limited in this application.
  • Step 240 in the case that the expected energy value of the Hamiltonian satisfies the convergence condition, determine the expected energy value of the Hamiltonian satisfying the convergence condition as the ground state energy of the target quantum system.
  • the minimum energy expectation value of the Hamiltonian is determined as the ground state energy of the target quantum system.
  • the embodiment of the present application uses a neural network to post-process the wave function output by the parameterized quantum circuit.
  • the neural network can function as a general function approximator, which has stronger expressive power and ground state energy approximation than the Jastrow factor ability, which helps to improve the accuracy of ground state energy estimation.
  • the VQNHE framework provided by another exemplary embodiment of the present application is shown in FIG. 3 , including a parameterized quantum circuit (PQC) 10 , a measurement circuit 40 , a neural network 20 and an optimizer 30 .
  • the neural network 20 and the optimizer 30 may be functional modules deployed in computer equipment, and the optimizer 30 may also be called an optimizer module.
  • the computer device may be a classical computer that implements the method by executing a computer program through a processor, which has storage and computing capabilities.
  • the measurement circuit 40 includes k groups of measurement circuits, and the k groups of measurement circuits are in one-to-one correspondence with the k Pauli strings obtained by decomposition of the Hamiltonian.
  • the parameterized quantum circuit 10 is used to transform the input quantum states of n qubits to obtain the output quantum states of the n qubits, where n is a positive integer.
  • the expected energy value of the Hamiltonian of the target quantum system in the output quantum state of the n qubits is the summation result of the expected energy values of k Pauli strings obtained by decomposition of the Hamiltonian, where k is a positive integer.
  • the measurement circuit corresponding to the target Pauli string is used to perform transformation processing corresponding to the target Pauli string on the output quantum states of n qubits, Get the transformed output quantum state.
  • the neural network 20 is used for post-processing the transformed output quantum state.
  • the energy expectation value of the target Pauli string is obtained.
  • For k Pauli strings perform the above operations respectively to obtain the expected energy values corresponding to the k Pauli strings, and then sum up to obtain the expected energy value of the Hamiltonian.
  • the optimizer 30 is used to adjust the parameters of the parameterized quantum circuit 10 and the parameters of the neural network 20 with the goal of converging the expected energy value of the Hamiltonian.
  • the energy expectation value of the Hamiltonian satisfies the convergence condition
  • the energy expectation value of the Hamiltonian meeting the convergence condition is determined as the ground state energy of the target quantum system.
  • Fig. 4 is a flowchart of a method for estimating the ground state energy of a quantum system provided by another embodiment of the present application.
  • the method can be applied to the VQNHE framework shown in Figure 3.
  • the execution subject of each step of the method can be a computer device, and the computer device can be a classical computer that executes a computer program through a processor to realize the method. It has storage and computing ability.
  • the method may include the following steps (410-480):
  • Step 410 obtain the output quantum states of the n qubits obtained after the input quantum states of the n qubits are transformed by the parameterized quantum circuit; wherein, the Hamiltonian of the target quantum system is between the n qubits
  • the expected energy value in the output quantum state is the summation result of the expected energy values of k Pauli strings obtained by Hamiltonian decomposition, where n is a positive integer and k is a positive integer.
  • the expected energy values corresponding to the k Pauli strings obtained by Hamiltonian decomposition are directly calculated, and then the Hamiltonian of the target quantum system is calculated according to the expected energy values corresponding to the k Pauli strings respectively. Quantity of energy expectations.
  • summation is performed on the expected energy values corresponding to the k Pauli strings respectively, and the obtained summation result is used as the expected energy value of the Hamiltonian of the target quantum system. It should be noted that the summation process here may be direct summation or weighted summation, which is not limited in the present application.
  • the energy expected values corresponding to the multiple Pauli strings are summed , to obtain the expected energy value of the equivalent Hamiltonian, and use the expected energy value of the equivalent Hamiltonian as the expected energy value of the Hamiltonian of the target quantum system.
  • This method is relatively complicated and inefficient because it may need to calculate the expected energy values corresponding to t ⁇ t ⁇ k Pauli strings at most.
  • it is only necessary to calculate the expected energy values corresponding to the k Pauli strings, which is simpler and more efficient.
  • Step 420 for the target Pauli string in the k Pauli strings, obtain the measurement circuit corresponding to the target Pauli string, and perform the transformation corresponding to the target Pauli string on the output quantum states of n qubits The transformed output quantum state obtained after processing.
  • the measurement circuit corresponding to the target Pauli string is used to perform the transformation corresponding to the target Pauli string on the output quantum states of n qubits processing to obtain the transformed output quantum state.
  • the expected energy values are obtained by measuring and estimating one by one.
  • the VQNHE framework shown in FIG. 3 includes k sets of measurement lines, and the k sets of measurement lines are in one-to-one correspondence with k Pauli strings.
  • the target Pauli string can be any one of the k Pauli strings, and when measuring and estimating the energy expectation value of the target Pauli string, use the measurement line pair corresponding to the target Pauli string.
  • the output quantum state of the parameterized quantum circuit is transformed corresponding to the target Pauli string to obtain the transformed output quantum state.
  • the purpose of this transformation step is to reduce resource consumption in the process of measurement and estimation.
  • the measurement circuit corresponding to the target Pauli string includes quantum gates corresponding to unsigned qubits other than signed qubits, so that the unsigned qubits are measured on the same measurement basis; wherein, A symbolic qubit is a qubit corresponding to a target Pauli operator in the target Pauli string among n qubits, and the measurement basis corresponding to the symbolic qubit is according to the bubble corresponding to the symbolic qubit in the target Pauli string The operator is determined.
  • the quantum gate corresponding to each unsigned qubit is a double-bit quantum gate, which acts on both the signed qubit and the unsigned qubit.
  • the target Pauli string is I 0 X 1 X 2 Y 3 I 4 , where the I operator can be ignored, so the target Pauli string can be recorded as X 1 X 2 Y 3.
  • the second qubit (corresponding to the Pauli operator X 1 ) can be used as a sign qubit, and the other qubits are unsigned qubits.
  • the measurement circuit 50 corresponding to the target Pauli string includes a double-bit control X gate acting on the second qubit (namely, the symbol qubit) and the third qubit (corresponding to the Pauli operator X 2 ) 51, and a double-bit control Y gate 52 acting on the second qubit (ie, the symbol qubit) and the fourth qubit (corresponding to the Pauli operator Y 3 ).
  • the measurement basis corresponding to the symbolic qubit is determined according to the Pauli operator corresponding to the symbolic qubit in the target Pauli string.
  • the second qubit is a symbolic qubit, which corresponds to the Pauli operator X 1 , which therefore corresponds to the measurement base X.
  • the above-mentioned same measurement base is the measurement base corresponding to the first Pauli operator, and the target Pauli operator is the second Pauli operator or the third Pauli operator; wherein, the first Pauli operator, The second Pauli operator and the third Pauli operator are different from each other, and for any one of the first Pauli operator, the second Pauli operator and the third Pauli operator, the bubble One of the Pauli X operator, the Pauli Y operator, and the Pauli Z operator.
  • the sign qubit is a qubit corresponding to the Pauli Y or Z operator; in the case where the above-mentioned same measurement base is the measurement base Y, the sign qubit The qubit is a certain qubit corresponding to the Pauli X or Z operator; in the case where the above-mentioned same measurement basis is the measurement basis Z, the symbol qubit is a certain qubit corresponding to the Pauli X or Y operator.
  • the quantum gate corresponding to the unsigned qubit is a double-bit control X gate;
  • the quantum gate corresponding to the non-sign qubit is a double-bit control Y gate; or, when the non-sign qubit is in the target Pauli string In the case of the Pauli Z operator, the quantum gate corresponding to the unsigned qubit is a double-bit control Z gate.
  • the measurement basis corresponding to the symbol qubit is the measurement basis corresponding to the Pauli X operator;
  • the measurement basis corresponding to the symbolic qubit is the measurement basis corresponding to the Pauli Y operator;
  • the measurement basis corresponding to the symbol qubit is the measurement basis corresponding to the Pauli Z operator.
  • Step 430 obtain the bit string of the transformed output quantum state on the specified measurement basis through measurement.
  • the measurement bases corresponding to the other unsigned qubits are the same.
  • the signed qubit corresponds to the measurement basis X
  • the other unsigned qubits correspond to the measurement basis Z.
  • Step 440 output metadata for calculating the expected energy value of the target Pauli string according to the bit string through the neural network.
  • the measured bit string is input to the neural network, the neural network performs forward calculation, and outputs the metadata used to calculate the energy expectation value of the target Pauli string.
  • Step 450 calculate the expected energy value of the target Pauli string according to the metadata.
  • the energy expectation value of the target Pauli string is calculated according to the following formula
  • f represents the neural network
  • s 0 represents the measurement result corresponding to the symbol qubit (its value is 0 or 1)
  • s represents the bit string
  • 0s 1:n-1 represents the corresponding value of the symbol qubit in the bit string s
  • the bit string obtained by setting the bits to 0 Indicates the bit string obtained by setting the bit corresponding to the symbol qubit in the bit string s to 1 and performing corresponding bit inversion on other bits according to the target Pauli string.
  • the so-called bit inversion is to change 0 to 1 and 1 to 0.
  • bit string s is s 0 s 1 s 2 s 3 s 4
  • symbol qubit is the second qubit
  • the bit string 0s 1:n-1 is s 0 0s 2 s 3 s 4
  • Step 460 calculating the expected energy value of the Hamiltonian according to the expected energy values of the k Pauli strings.
  • the expected energy value of the k Pauli strings is added to obtain the expected energy value of the Hamiltonian. It should be noted that the above addition may be direct addition or weighted summation, which is not limited in this application.
  • step 470 the parameters of the parameterized quantum circuit and the parameters of the neural network are adjusted with the goal of converging the energy expectation value of the Hamiltonian.
  • Step 480 in the case that the expected energy value of the Hamiltonian satisfies the convergence condition, determine the expected energy value of the Hamiltonian satisfying the convergence condition as the ground state energy of the target quantum system.
  • Steps 470-480 are the same as steps 230-240 in the embodiment shown in FIG. 2 .
  • Steps 470-480 are the same as steps 230-240 in the embodiment shown in FIG. 2 .
  • steps 470-480 are the same as steps 230-240 in the embodiment shown in FIG. 2 .
  • steps 470-480 are the same as steps 230-240 in the embodiment shown in FIG. 2 .
  • steps 470-480 are the same as steps 230-240 in the embodiment shown in FIG. 2 .
  • Steps 470-480 are the same as steps 230-240 in the embodiment shown in FIG. 2 .
  • the quantum gate corresponding to the unsigned qubit is equivalently replaced by the sign corresponding to the measurement result corresponding to the unsigned qubit.
  • the target Pauli string is I 0 I 1 Y 2 Z 3 X 4 , where the I operator can be ignored, so the target Pauli string can be recorded as Y 2 Z 3 X 4 , assuming that the If the measurement is performed on the measurement basis Z, the third qubit (corresponding to the Pauli operator Y 2 ) can be used as a sign qubit, and the other qubits are unsigned qubits.
  • the measurement circuit 60 corresponding to the target Pauli string should include a two-bit control Z acting on the third qubit (that is, the symbol qubit) and the fourth qubit (corresponding to the Pauli operator Z 3 ).
  • the above-mentioned two-bit control Z-gate acting on the third qubit (ie, the symbol qubit) and the fourth qubit (corresponding to the Pauli operator Z 3 ) can be omitted , and the symbol 1-2s 3 corresponding to the measurement result s 3 corresponding to the fourth qubit is used for equivalent replacement.
  • the goal we need to optimize is the normalized energy expectation in is any Pauli string.
  • the expected energy value of the Hamiltonian it can always be decomposed into a simple summation of the expected energy values of multiple Pauli strings. Therefore, our measurement and estimation scheme only needs to solve the expected estimation problem of a single Pauli string.
  • ⁇ s ⁇ s
  • ⁇ > represents the probability amplitude of the wave function output by the parameterized quantum circuit PQC on the measurement basis.
  • the implementation strategy corresponding to this formula is very simple: directly measure the bit string s on the PQC measurement base, and then calculate the mean value of
  • the Pauli string to be estimated Contains only the Pauli Z operator (and optionally the I operator ), that is, ⁇ s
  • s′> H s ⁇ ss′ (among them, s and s′ represent two bit strings, and ⁇ ss′ is the Kronecker function, only when s and s′ are the same is 1, and is 0 at other times, H s is the expectation of the Pauli string under the corresponding basis of s), then for the molecule in the above formula, we have: Its measurement strategy is completely similar to the estimation of the denominator, and the expectation of
  • the real difficulty of VQE post-processing which has been considered to consume exponential resources before, is when the Pauli string When the Pauli X or Y operator is included in .
  • our neural network post-processing is based on the measurement basis Z, all qubits need to be measured on the measurement basis Z to obtain the bit string s, and then input The neural network calculates the value of f(s).
  • a Pauli string containing the Pauli X or Y operator needs to be measured on the measurement basis X or Y to obtain the corresponding result of the corresponding qubit.
  • the post-processing neural network output f(s) is a real number (for the case of complex numbers, it will be explained below), it can be obtained:
  • the final probability amplitude ⁇ ⁇ ,s ⁇ ,s 1:n
  • ⁇ > is the probability amplitude of the PQC output wave function on the Pauli string eigenstate basis.
  • V a measurement circuit attached to the PQC (represented by U).
  • ⁇ > ⁇ s
  • the double-bit quantum corresponding to the non-signed qubit is equivalently replaced by the sign corresponding to the measurement result corresponding to the unsigned qubit, thereby helping to simplify the structure of the measurement circuit.
  • the measurement error is estimated as:
  • ⁇ n is the standard deviation corresponding to the expected n of the numerator distribution
  • ⁇ d is the standard deviation corresponding to the expected d of the denominator distribution.
  • the theoretical upper limit of the number of measurements required to achieve the corresponding accuracy in the case of VQNHE is 9r 8 /4 ⁇ 2 .
  • This value and the VQE ratio have only a polynomial dependence on the scope of the neural network function, and have nothing to do with the size of the system. Therefore, VQNHE can be implemented efficiently on quantum hardware. It is worth noting that the theoretical upper bound is relatively loose, and the number of additional measurements required in practical problems is much smaller than this value.
  • the theoretical derivation and experimental scheme under the VQNHE framework are mainly introduced and explained above with the output f(s) of the neural network as a real number.
  • the output f(s) of the neural network can take complex numbers, the VQNHE framework provided by this application can still be efficiently completed, and the corresponding derivation is as follows.
  • the measurement and estimation are the same as those introduced above, the only difference is that the factor is f * f to take the real part.
  • the measurement line is used to perform a transformation process corresponding to the Pauli string on the output quantum state of the PQC to obtain the transformed output quantum state.
  • This step of transformation can reduce the measurement and estimation process. Therefore, under the consumption of polynomial resources, the measurement and unbiased estimation of Pauli strings and even general Hamiltonian can be completed.
  • VQNHE VQNHE framework to optimally calculate the ground state energy values of the one-dimensional transverse field Ising model and the isotropic quantum Heisenberg model. Both models are calculated on 12 grid points, and the corresponding model Hamiltonian parameters are all 1 and periodic boundary conditions are used. The comparison between the results of VQNHE, VQE and strict results is shown in Table 1 below. Among them, both VQE and VQNHE are calculated using the same quantum circuit structure in the same model.
  • the VQNHE framework can also be applied to molecular energy calculations.
  • this energy with the energy obtained by VQE and the energy obtained by the HartreeFock (Hartley-Fock) mean field method.
  • Curve 72 corresponds to the energy obtained by VQE
  • curve 73 corresponds to the energy obtained by VQNHE.
  • the energy obtained by VQNHE basically coincides with the strict result. It can be seen from part (b) of Figure 7 that the optimized energy accuracy corresponding to VQNHE is more than an order of magnitude higher than that of VQE. Both VQNHE and VQE are calculated in the symmetry-reduced 4-qubit full active space on this problem.
  • the two algorithms adopt the same hardware-friendly proposed quantum circuit structure, and the quantum circuit structure can be shown in FIG. 8 .
  • the device and system embodiments correspond to the above-mentioned method embodiments and belong to the same inventive concept.
  • the method embodiments of the present application please refer to the method embodiments of the present application .
  • Fig. 11 is a block diagram of a device for estimating the ground state energy of a quantum system provided by an embodiment of the present application.
  • the device has the function of realizing the above-mentioned method example, and the function may be realized by hardware, or may be realized by executing corresponding software by the hardware.
  • the device may be a computer device, or may be set in the computer device.
  • the apparatus 1100 may include: a state acquisition module 1110 , a post-processing module 1120 and an optimizer module 1130 .
  • the state acquisition module 1110 is used to obtain the output quantum states of the n qubits obtained after the input quantum states of the n qubits are transformed by the parameterized quantum circuit; wherein, the Hamiltonian of the target quantum system is in the
  • the expected energy value of the output quantum state of the n qubits is the summation result of the expected energy values of the k Pauli strings obtained by decomposition of the Hamiltonian, where n is a positive integer and k is a positive integer.
  • the post-processing module 1120 is configured to post-process the output quantum states of the n qubits by using a neural network, and calculate the expected energy value of the Hamiltonian according to the post-processing results of the neural network.
  • the optimizer module 1130 is used to adjust the parameters of the parameterized quantum circuit and the parameters of the neural network with the goal of convergence of the expected energy value of the Hamiltonian; when the expected energy value of the Hamiltonian satisfies the convergence In the case of the condition, the energy expectation value of the Hamiltonian satisfying the convergence condition is determined as the ground state energy of the target quantum system.
  • the post-processing module 1120 includes: a decomposition unit, a measurement unit and a calculation unit;
  • the decomposition unit is configured to generate an equivalent Hamiltonian of the target quantum system according to the Pauli string obtained by decomposing the Hamiltonian and the Pauli string obtained by decomposing the post-processing operator corresponding to the neural network Multiple Pauli strings corresponding to the quantity;
  • the measurement unit is configured to, for each Pauli string in the plurality of Pauli strings, obtain the output quantum states of the n qubits by measurement on the measurement basis corresponding to the Pauli strings bit string of
  • the computing unit is configured to calculate the expected energy values corresponding to the multiple Pauli strings according to the bit strings respectively corresponding to the multiple Pauli strings, and obtain the energy expectations corresponding to the multiple Pauli strings respectively, The expected energy value of computes the expected energy value of the Hamiltonian.
  • the decomposition unit is used for:
  • the post-processing module includes: an acquisition unit, a measurement unit, a neural network unit, and a calculation unit;
  • the acquiring unit is configured to, for a target Pauli string in the k Pauli strings, acquire a measurement line corresponding to the target Pauli string, and execute the output quantum state of the n qubits A transformed output quantum state obtained after transformation processing corresponding to the target Pauli string;
  • the measurement unit is used to obtain the bit string of the transformed output quantum state on the specified measurement basis through measurement
  • the neural network unit is configured to output metadata for calculating the expected energy value of the target Pauli string according to the bit string through the neural network;
  • the calculation unit is configured to calculate the expected energy value of the target Pauli string according to the metadata, and calculate the expected energy value of the Hamiltonian according to the expected energy values of the k Pauli strings.
  • the measurement circuit corresponding to the target Pauli string includes quantum gates corresponding to unsigned qubits other than signed qubits, so that the unsigned qubits are measured on the same measurement basis; wherein , the sign qubit is the qubit corresponding to a target Pauli operator in the target Pauli string among the n qubits, and the measurement basis corresponding to the sign qubit is according to the sign qubit The corresponding Pauli operator in the target Pauli string is determined.
  • the same measurement base is the measurement base corresponding to the first Pauli operator, and the target Pauli operator is the second Pauli operator or the third Pauli operator; wherein, the first Pauli operator The Pauli operator, the second Pauli operator and the third Pauli operator are different from each other, and for the first Pauli operator, the second Pauli operator and the third Pauli operator Any Pauli operator among the Pauli operators is one of the Pauli X operator, the Pauli Y operator, and the Pauli Z operator.
  • the quantum gate corresponding to the unsigned qubit is a double-bit control X gate
  • the quantum gate corresponding to the unsigned qubit is a double-bit control Y gate
  • the quantum gate corresponding to the unsigned qubit is a double-bit control Z gate.
  • the measurement basis corresponding to the sign qubit is the measurement basis corresponding to the Pauli X operator ;
  • the measurement basis corresponding to the sign qubit is the measurement basis corresponding to the Pauli Y operator;
  • the measurement basis corresponding to the sign qubit is the measurement basis corresponding to the Pauli Z operator.
  • the calculation unit is used to calculate the expected energy value of the target Pauli string according to the following formula
  • f represents the neural network
  • s 0 represents the measurement result corresponding to the symbol qubit
  • s represents the bit string
  • 0s 1:n-1 represents the corresponding symbol qubit in the bit string
  • the bits of are set to 0 to obtain the bit string, Indicates the bit string obtained by setting the bit corresponding to the symbol qubit in the bit string to 1 and performing corresponding bit inversion on other bits according to the target Pauli string.
  • the unsigned qubit when the Pauli operator corresponding to the unsigned qubit in the target Pauli string is the same as the Pauli operator corresponding to the same measurement base, the unsigned qubit The quantum gate corresponding to the bit is equivalently replaced by the sign corresponding to the measurement result corresponding to the unsigned qubit.
  • the neural network is used to post-process the wave function output by the parameterized quantum circuit.
  • the neural network can function as a general function approximator, which has stronger expression ability and ground state energy approximation ability than the Jastrow factor. , which helps to improve the accuracy of ground state energy estimation.
  • An exemplary embodiment of the present application also provides a system for estimating the ground state energy of a quantum system, and the system includes: a parameterized quantum circuit and a computer device.
  • the computer device includes a post-processing module and an optimizer module.
  • the parameterized quantum circuit is used to transform the input quantum states of n qubits to obtain the output quantum states of the n qubits; wherein, the Hamiltonian of the target quantum system is within the range of the n qubits
  • the expected energy value in the output quantum state is the summation result of the expected energy values of the k Pauli strings obtained by decomposition of the Hamiltonian, where n is a positive integer and k is a positive integer.
  • the post-processing module is used to post-process the output quantum states of the n qubits by using a neural network, and calculate the expected energy value of the Hamiltonian according to the post-processing results of the neural network.
  • the optimizer module is used to adjust the parameters of the parameterized quantum circuit and the parameters of the neural network with the goal of converging the expected energy value of the Hamiltonian; wherein, the expected energy value of the Hamiltonian If the convergence condition is satisfied, the energy expectation value of the Hamiltonian satisfying the convergence condition is determined as the ground state energy of the target quantum system.
  • the post-processing module includes: a decomposition unit, a measurement unit and a calculation unit;
  • the decomposition unit is configured to generate an equivalent Hamiltonian of the target quantum system according to the Pauli string obtained by decomposing the Hamiltonian and the Pauli string obtained by decomposing the post-processing operator corresponding to the neural network Multiple Pauli strings corresponding to the quantity;
  • the measuring unit is configured to, for each Pauli string in the plurality of Pauli strings, measure the output quantum state of the n qubits on the measurement base corresponding to the Pauli string bit string;
  • the computing unit is configured to calculate the expected energy values corresponding to the multiple Pauli strings according to the bit strings respectively corresponding to the multiple Pauli strings, and obtain the energy expectations corresponding to the multiple Pauli strings respectively, The expected energy value of computes the expected energy value of the Hamiltonian.
  • the decomposition unit is used for:
  • the system further includes k sets of measurement lines
  • the post-processing module includes: an acquisition unit, a measurement unit, a neural network unit, and a calculation unit, the k sets of measurement lines and the k Pauli One-to-one correspondence between strings;
  • the measurement line corresponding to the target Pauli string is used to perform transformation processing corresponding to the target Pauli string on the output quantum states of the n qubits to obtain the transformed output quantum state;
  • the obtaining unit is used to obtain the transformed output quantum state
  • the measurement unit is used to measure and obtain the bit string of the transformed output quantum state on a specified measurement basis
  • the neural network unit is used to output metadata for calculating the expected energy value of the target Pauli string according to the bit string through the neural network;
  • the calculation unit is configured to calculate the expected energy value of the target Pauli string according to the metadata, and calculate the expected energy value of the Hamiltonian according to the expected energy values of the k Pauli strings.
  • the measurement circuit corresponding to the target Pauli string includes quantum gates corresponding to unsigned qubits other than signed qubits, so that the unsigned qubits are measured on the same measurement basis; wherein , the sign qubit is the qubit corresponding to a target Pauli operator in the target Pauli string among the n qubits, and the measurement basis corresponding to the sign qubit is according to the sign qubit The corresponding Pauli operator in the target Pauli string is determined.
  • the same measurement base is the measurement base corresponding to the first Pauli operator, and the target Pauli operator is the second Pauli operator or the third Pauli operator; wherein, the first Pauli operator The Pauli operator, the second Pauli operator and the third Pauli operator are different from each other, and for the first Pauli operator, the second Pauli operator and the third Pauli operator Any Pauli operator among the Pauli operators is one of the Pauli X operator, the Pauli Y operator, and the Pauli Z operator.
  • the quantum gate corresponding to the unsigned qubit is a double-bit control X gate
  • the quantum gate corresponding to the unsigned qubit is a double-bit control Y gate
  • the quantum gate corresponding to the unsigned qubit is a double-bit control Z gate.
  • the measurement basis corresponding to the sign qubit is the measurement basis corresponding to the Pauli X operator ;
  • the measurement basis corresponding to the sign qubit is the measurement basis corresponding to the Pauli Y operator;
  • the measurement basis corresponding to the sign qubit is the measurement basis corresponding to the Pauli Z operator.
  • the calculation unit is used to calculate the expected energy value of the target Pauli string according to the following formula
  • f represents the neural network
  • s 0 represents the measurement result corresponding to the symbol qubit
  • s represents the bit string
  • 0s 1:n-1 represents the corresponding symbol qubit in the bit string
  • the bits of are set to 0 to obtain the bit string, Indicates the bit string obtained by setting the bit corresponding to the symbol qubit in the bit string to 1 and performing corresponding bit inversion on other bits according to the target Pauli string.
  • the unsigned qubit when the Pauli operator corresponding to the unsigned qubit in the target Pauli string is the same as the Pauli operator corresponding to the same measurement base, the unsigned qubit The quantum gate corresponding to the bit is equivalently replaced by the sign corresponding to the measurement result corresponding to the unsigned qubit.
  • the division of the above-mentioned functional modules is used as an example for illustration.
  • the above-mentioned function allocation can be completed by different functional modules according to the needs.
  • the device and system provided by the above embodiments belong to the same idea as the method embodiments, and the specific implementation process thereof is detailed in the method embodiments, and will not be repeated here.
  • Fig. 12 is a schematic structural diagram of a computer device provided by an embodiment of the present application.
  • the computer device can be any electronic device with data storage and computing capabilities, and the computer device can be used to implement the method for estimating the ground state energy of a quantum system provided in the above embodiments. Specifically:
  • the computer device 1200 includes a central processing unit (such as a CPU (Central Processing Unit, central processing unit), GPU (Graphics Processing Unit, graphics processing unit) and FPGA (Field Programmable Gate Array, field programmable logic gate array) etc.) 1201, A system memory 1204 including a RAM (Random-Access Memory) 1202 and a ROM (Read-Only Memory) 1203, and a system bus 1205 connecting the system memory 1204 and the central processing unit 1201.
  • the computer device 1200 also includes a basic input/output system (Input Output System, I/O system) 1206 that helps to transmit information between various devices in the server, and is used to store an operating system 1213, an application program 1214 and other program modules 1215 mass storage device 1207.
  • I/O system Input Output System
  • the basic input/output system 1206 includes a display 1208 for displaying information and input devices 1209 such as a mouse and a keyboard for user input of information. Wherein, both the display 1208 and the input device 1209 are connected to the central processing unit 1201 through the input and output controller 1210 connected to the system bus 1205 .
  • the basic input/output system 1206 may also include an input-output controller 1210 for receiving and processing input from a keyboard, a mouse, or an electronic stylus and other devices. Similarly, input output controller 1210 also provides output to a display screen, printer, or other type of output device.
  • the mass storage device 1207 is connected to the central processing unit 1201 through a mass storage controller (not shown) connected to the system bus 1205 .
  • the mass storage device 1207 and its associated computer-readable media provide non-volatile storage for the computer device 1200 . That is to say, the mass storage device 1207 may include a computer-readable medium (not shown) such as a hard disk or a CD-ROM (Compact Disc Read-Only Memory, CD-ROM) drive.
  • a computer-readable medium such as a hard disk or a CD-ROM (Compact Disc Read-Only Memory, CD-ROM) drive.
  • Computer readable media may comprise computer storage media and communication media.
  • Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data.
  • Computer storage media include RAM, ROM, EPROM (Erasable Programmable Read-Only Memory, Erasable Programmable Read-Only Memory), EEPROM (Electrically Erasable Programmable Read-Only Memory, Electrically Erasable Programmable Read-Only Memory), flash memory or Other solid-state storage technologies, CD-ROM, DVD (Digital Video Disc, high-density digital video disc) or other optical storage, tape cartridges, tapes, disk storage or other magnetic storage devices.
  • the computer storage medium is not limited to the above-mentioned ones.
  • the above-mentioned system memory 1204 and mass storage device 1207 may be collectively referred to as memory.
  • the computer device 1200 can also run on a remote computer connected to the network through a network such as the Internet. That is, the computer device 1200 can be connected to the network 1212 through the network interface unit 1211 connected to the system bus 1205, or in other words, the network interface unit 1211 can also be used to connect to other types of networks or remote computer systems (not shown) .
  • the memory also includes a computer program, which is stored in the memory and configured to be executed by one or more processors, so as to realize the above-mentioned method for estimating the ground state energy of a quantum system.
  • a computer-readable storage medium in which a computer program is stored, and when the computer program is executed by a processor of a computer device, the above method for estimating the ground state energy of a quantum system is implemented.
  • the computer-readable storage medium may include: ROM (Read-Only Memory, read-only memory), RAM (Random-Access Memory, random access memory), SSD (Solid State Drives, solid state drive) or an optical disc, etc.
  • the random access memory may include ReRAM (Resistance Random Access Memory, resistive random access memory) and DRAM (Dynamic Random Access Memory, dynamic random access memory).
  • a computer program product or computer program comprising computer instructions stored in a computer readable storage medium.
  • the processor of the computer device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the computer device executes the above-mentioned method for estimating the ground state energy of a quantum system.
  • the "plurality” mentioned herein refers to two or more than two.
  • “And/or” describes the association relationship of associated objects, indicating that there may be three types of relationships, for example, A and/or B may indicate: A exists alone, A and B exist simultaneously, and B exists independently.
  • the character "/” generally indicates that the contextual objects are an "or” relationship.
  • the numbering of the steps described herein only exemplarily shows a possible sequence of execution among the steps. In some other embodiments, the above-mentioned steps may not be executed according to the order of the numbers, such as two different numbers The steps are executed at the same time, or two steps with different numbers are executed in the reverse order as shown in the illustration, which is not limited in this embodiment of the present application.

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Abstract

本申请公开了一种量子体系基态能量估计方法及系统，涉及量子技术领域。本申请通过采用神经网络对PQC输出的波函数做后处理，该神经网络能够起到一个通用函数近似器的作用，其相比于Jastrow因子具有更强的表达能力和基态能量近似能力，从而有助于提升基态能量估计的准确度。并且，本申请通过在PQC之后添加测量线路，利用该测量线路对PQC的输出量子态执行与泡利字符串相对应的变换处理，得到变换后的输出量子态，这一步变换能够减少测量估计过程中的资源消耗，从而能够在多项式资源的消耗下，完成对泡利字符串乃至一般哈密顿量的测量和无偏估计。

Description

量子体系基态能量估计方法及系统
本申请要求于2021年06月07日提交的申请号为202110634389.5、发明名称为“量子体系基态能量估计方法及系统”的中国专利申请的优先权，其全部内容通过引用结合在本申请中。
技术领域
本申请实施例涉及量子技术领域，特别涉及一种量子体系基态能量估计方法及系统。
背景技术
目前，提出了一种VQE(Variational Quantum Eigensolver，变分量子本征求解器)的方案来对量子系统进行基态能量估计。VQE通过变分量子线路实现对量子系统进行基态能量估计，是一种典型的量子经典混合计算范式。
为了对VQE的输出性能做进一步的增强，提升基态能量估计的准确度，相关技术提出了利用Jastrow因子作为VQE后处理增强的方案。通过利用Jastrow因子对VQE中变分量子线路输出的波函数做后处理，以期刻画更多的量子纠缠和关联关系，从而使得最终估算出的基态能量与真实值尽可能地接近。
然而，Jastrow因子虽然比较适合刻画多体关联，但仍不是经典后处理可以具有的最普遍形式，因此其表达能力较弱，这会影响到基态能量估计的准确度。
发明内容
本申请实施例提供了一种量子体系基态能量估计方法及系统。所述技术方案如下：
根据本申请实施例的一个方面，提供了一种量子体系基态能量估计方法，所述方法由计算机设备执行，所述方法包括：
获取经参数化量子线路对n个量子比特的输入量子态进行变换处理后得到的所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数；
采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值；
以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；
在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
根据本申请实施例的一个方面，提供了一种量子体系基态能量估计装置，所述装置包括：
状态获取模块，用于获取经参数化量子线路对n个量子比特的输入量子态进行变换处理后得到的所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数；
后处理模块，用于采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值；
优化器模块，用于以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
根据本申请实施例的一个方面，提供了一种计算机设备，所述计算机设备包括处理器和存储器，所述存储器中存储有计算机程序，所述计算机程序由所述处理器加载并执行以实现 上述方法。
根据本申请实施例的一个方面，提供了一种计算机可读存储介质，所述存储介质中存储有计算机程序，所述计算机程序由处理器加载并执行以实现上述方法。
根据本申请实施例的一个方面，提供了一种计算机程序产品或计算机程序，所述计算机程序产品或计算机程序包括计算机指令，所述计算机指令存储在计算机可读存储介质中，处理器从所述计算机可读存储介质读取并执行所述计算机指令，以实现上述方法。
根据本申请实施例的一个方面，提供了一种量子体系基态能量估计系统，所述系统包括：参数化量子线路和计算机设备，所述计算机设备包括后处理模块和优化器模块；
所述参数化量子线路用于对n个量子比特的输入量子态进行变换处理，得到所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数；
所述后处理模块用于采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值；
所述优化器模块用于以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；其中，在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
本申请实施例提供的技术方案可以包括如下有益效果：
通过采用神经网络对参数化量子线路输出的波函数做后处理，该神经网络能够起到一个通用函数近似器的作用，其相比于Jastrow因子具有更强的表达能力和基态能量近似能力，从而有助于提升基态能量估计的准确度。
附图说明
为了更清楚地说明本申请实施例中的技术方案，下面将对实施例描述中所需要使用的附图作简单地介绍，显而易见地，下面描述中的附图仅仅是本申请的一些实施例，对于本领域普通技术人员来讲，在不付出创造性劳动的前提下，还可以根据这些附图获得其他的附图。
图1是本申请一个实施例提供的VQNHE框架的示意图；
图2是本申请一个实施例提供的量子体系基态能量估计方法的流程图；
图3是本申请另一个实施例提供的VQNHE框架的示意图；
图4是本申请另一个实施例提供的量子体系基态能量估计方法的流程图；
图5是本申请一个实施例提供的测量线路的示意图；
图6是本申请另一个实施例提供的测量线路的示意图；
图7是本申请示例性示出的在分子能量计算上各种方案的比对示意图；
图8是本申请示例性示出的在分子能量计算上量子线路结构的示意图；
图9是本申请示例性示出的VQE和VQNHE在真实硬件和带噪声模拟器上表现的对比图；
图10是本申请示例性示出的PQC线路结构的示意图；
图11是本申请一个实施例提供的量子体系基态能量估计装置的框图；
图12是本申请一个实施例提供的计算机设备的结构示意图。
具体实施方式
为使本申请的目的、技术方案和优点更加清楚，下面将结合附图对本申请实施方式作进一步地详细描述。
在对本申请技术方案进行介绍之前，先对本申请中涉及的一些关键术语进行解释说明。
1.量子计算：基于量子逻辑的计算方式，存储数据的基本单元是量子比特(qubit)。
2.量子比特：量子计算的基本单元。传统计算机使用0和1作为二进制的基本单元。不 同的是量子计算可以同时处理0和1，系统可以处于0和1的线性叠加态：|ψ>＝α|0>+β|1>，这边α,β代表系统在0和1上的复数概率幅。它们的模平方|α| 2,|β| 2分别代表处于0和1的概率。
3.量子线路：量子通用计算机的一种表示，代表了相应量子算法/程序在量子门模型下的硬件实现。若量子线路中包含可调的控制量子门的参数，则被称为参数化的量子线路(Parameterized Quantum Circuit，简称PQC)或变分量子线路(Variational Quantum Circuit，简称VQC)，两者为同一概念。
4.哈密顿量：描述量子系统总能量的一个厄密共轭的矩阵。哈密顿量是一个物理词汇，是一个描述系统总能量的算符，通常以H表示。
5.本征态：对于一个哈密顿量矩阵H，满足方程：H|ψ>＝E|ψ>的解称之为H的本征态|ψ>，具有本征能量E。基态则对应了量子系统能量最低的本征态。
6.量子结构搜索(Quantum Architecture Search，简称QAS)：尝试对量子线路的结构、模式和排布进行自动化和程序化搜索的一系列工作和方案的总称。传统上量子结构搜索的工作通常会采用贪心算法、强化学习或基因算法作为其核心技术。最新发展起来的技术包括可微量子结构搜索和基于预测器的结构搜索方案。
7.量子经典混合计算：一种内层利用量子线路(如PQC)进行计算得出相应物理量或损失函数，外层用传统的经典优化器调节量子线路变分参数的计算范式，可以最大限度地发挥量子计算的优势，被相信是有潜力证明量子优势的重要方向之一。
8.NISQ(Noisy Intermediate-Scale Quantum)：近期中等规模有噪声的量子硬件，是量子计算发展现在所处的阶段和研究的重点方向。这一阶段量子计算暂时由于规模和噪声的限制，无法作为通用计算的引擎应用，但在部分问题上，已经可以实现超越最强经典计算机的结果，这通常被称作量子霸权或量子优势。
9.量子误差消除(Quantum Error Mitigation)：与量子纠错(Quantum Error Correction)相对应，是一系列NISQ时代硬件下的更小资源代价的量子错误缓解和噪声抑制方案。相比完整的量子纠错所需的资源明显减少，同时可能只适用于特定任务，而非通用方案。
10.变分量子本征求解器(Variational Quantum Eigensolver，简称VQE)：通过变分线路(即PQC/VQC)实现特定量子系统基态能量的估计，是一种典型的量子经典混合计算范式，在量子化学领域有广泛的应用。
11.Jastrow因子：常见于变分蒙特卡洛波函数拟设中的一种因子，用来对平均场无相互作用的波函数做加强，以期刻画更多的量子关联信息。其基本形式为
Figure PCTCN2021124392-appb-000001
其中φ是变分参数，Z是在测量基上给出±1本征值的量子算符，k和l代表不同的量子比特自由度，k代表第k个量子比特，l代表第l个量子比特。
12.非幺正：所谓幺正矩阵，即是满足
Figure PCTCN2021124392-appb-000002
的全部矩阵，所有量子力学直接允许的演化过程，都可以通过幺正矩阵描述。其中，U为幺正矩阵(Unitary Matrix)，也称为酉矩阵、么正矩阵等，
Figure PCTCN2021124392-appb-000003
是U的共轭转置。另外，不满足该条件的矩阵则是非幺正的，其需要通过辅助手段甚至指数多的资源才可在实验上实现，但非幺正矩阵往往具有更强的表达能力和更快的基态投影效果。上述“指数多的资源”是指资源的需求量随着量子比特数量的增加，呈指数级增加，该指数多的资源可以是指需要测量的量子线路的总数是指数多个，也即相应需要指数多的计算时间。
13.泡利字符串(Pauli string)：在不同格点多个泡利算符的直积组成的项，一般的哈密顿量通常可以拆解为一组泡利字符串的直积。VQE的测量也一般都按照泡利字符串分解来逐项测量。
14.泡利算符：也称为泡利矩阵，是一组三个2×2的幺正厄米复矩阵(又称酉矩阵)，一 般都以希腊字母σ(西格玛)来表示。其中，泡利X算符为
Figure PCTCN2021124392-appb-000004
泡利Y算符为
Figure PCTCN2021124392-appb-000005
泡利Z算符为
Figure PCTCN2021124392-appb-000006
15.UCC(Unitary Coupled Cluster，幺正耦合簇)拟设(ansatz)和硬件友好(hardware efficient)拟设：两种VQE不同的变分线路结构。前者借鉴了量子化学的传统变分数值方法coupled-cluster(耦合簇)，近似效果较好，但需要Trotter(特洛特)分解相应的指数算符，从而对量子资源要求较高。后者采取直接密排原生量子门组的策略，需要的线路较浅，量子资源要求较低，但对应的表达和近似能力也较UCC拟设更差。
16.比特字符串(bit string)：一串由0，1组成的数字。对量子线路每次测量得到的经典结果，可以根据在测量基上的自旋构型的上下分别由0，1表示，从而总的一次测量结果对应一个比特字符串。
本申请提供的技术方案，有助于加快和加强现阶段变分量子算法的开发和设计。NISQ时代量子硬件的典型缺点就是相干时间短且量子噪声大，相应的我们需要充分考虑量子硬件本身具体特征的情况下，尽量减小量子线路的深度。传统的基于UCC拟设的VQE方案往往准确度较高但是对线路深度要求很高，很难在现有相干时间的量子硬件上大规模实现。与之对比，硬件节约假设可以通过原生量子门密排作为线路结构，其好处是变分结构很容易在量子硬件上实现，但表达能力和对基态的近似能力往往不能令人满意。本申请提供的技术方案可以称为变分量子-神经网络混合本征求解器(Variational Quantum Neural network Hybrid Eigensolver，简称VQNHE)，恰恰很好地解决了这一对矛盾。通过神经网络支撑的通用非幺正后处理，我们可以在较浅的变分量子线路上，加强出超越物理/化学精度要求的基态能量近似。因此该方案特别适合于现阶段量子硬件上的应用，从而加速有效量子优势的验证和商业化应用。
另外，本申请提供的技术方案，可在中短期内应用到量子硬件评估和实际生产中。其应用包括但不限于对多样的来自凝聚态物理和量子化学问题中的哈密顿量的基态进行模拟和求解。本申请提供的技术方案也有望进一步地在激发态查找和量子含时演化等其他变分量子算法支持的任务上发挥作用。此外本申请提供的技术方案，通过进一步优化神经网络模型，可以在无先验噪声模型的基础上实现一定的量子纠错的效果，这进一步释放了本方案在NISQ时代的巨大潜力。由于该算法可作为VQE增强的通用方案，所消耗的量子资源和一般VQE相同，那么任何VQE程序(用于执行整个系统架构下的测量估计流程)都可以无缝地移植到VQNHE框架，该框架可作为量子云服务提供和调用，且可以封装成非常简单的VQE增强的API(Application Programming Interface，应用程序编程接口)。此外该方案可以和量子结构搜索的方法结合，进一步自适应地构造适合VQNHE的量子线路结构。
本申请一示例性实施例提供的VQNHE框架如图1所示，包括参数化量子线路(PQC)10、神经网络20和优化器30。其中，神经网络20和优化器30可以是部署在计算机设备中的功能模块，优化器30也可以称为优化器模块。在本申请实施例中，计算机设备可以是通过处理器执行计算机程序以实现该方法的经典计算机，其具备存储和运算能力。参数化量子线路10用于对n个量子比特的输入量子态进行变换处理，得到该n个量子比特的输出量子态，n为正整数。目标量子系统的哈密顿量在该n个量子比特的输出量子态下的能量期望值，是哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，k为正整数。神经网络20用于对该n个量子比特的输出量子态进行后处理。基于神经网络20的后处理结果，得到k个泡利字符串的能量期望值，然后计算出哈密顿量的能量期望值。优化器30用于以哈密顿量的能量期望值收敛为目标，对参数化量子线路10的参数和神经网络20的参数进行调整。在哈密 顿量的能量期望值满足收敛条件的情况下，将满足收敛条件的哈密顿量的能量期望值，确定为目标量子系统的基态能量。
图1左上角的参数化量子线路(PQC)10和传统VQE框架中的一致，其输出的波函数|ψ>经过神经网络后处理算符
Figure PCTCN2021124392-appb-000007
的作用，得到经过加强的量子-神经网络混合波函数：
Figure PCTCN2021124392-appb-000008
为了可以测量估计|ψ f>对应的哈密顿量的能量期望值，可以采用如下方式：对于上述k个泡利字符串中的每一个泡利字符串，分别测量得到n个量子比特的输出量子态在该泡利字符串对应的测量基上的比特字符串，通过神经网络20根据该比特字符串输出用于计算该泡利字符串的能量期望值的元数据，然后根据这些元数据计算得到该泡利字符串的能量期望值，最终将k个泡利字符串的能量期望值进行求和处理，得到哈密顿量的能量期望值。一旦得到了哈密顿量的能量期望值，我们就能够分别应用参数平移和反向传播来计算该能量期望值相对于参数化量子线路10的参数θ和神经网络20的参数φ的导数。通过该导数信息，就可以使用经典机器学习社区发展出的基于梯度的优化器30(如Adam)来更新相应的参数，从而完成一轮量子-经典混合计算范式的迭代，直到得到的能量期望值收敛，其值可作为相应系统哈密顿量基态的近似估计。
图2是本申请一个实施例提供的量子体系基态能量估计方法的流程图。该方法可应用于图1所示的VQNHE框架中，例如该方法各步骤的执行主体可以是计算机设备。该方法可以包括如下几个步骤(210～240)：
步骤210，获取经参数化量子线路对n个量子比特的输入量子态进行变换处理后得到的该n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在该n个量子比特的输出量子态下的能量期望值，是哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数。
在本申请实施例中，通过参数化量子线路对n个量子比特的输入量子态进行变换处理，得到该n个量子比特的输出量子态。
参数化量子线路的输入量子态一般可以使用全0态、均匀叠加态或者Hartree-Fock(哈特里-福克)态，该输入量子态也被称作试探态。目标量子系统的哈密顿量可分解为k个泡利字符串的直积，k通常为大于1的整数，但在一些特殊情况下k也可以等于1，也即目标量子系统的哈密顿量可看作是一个泡利字符串。因此，在VQE框架中，用参数化量子线路的输出来近似目标量子系统的输出量子态，通过测量估计目标量子系统的哈密顿量在参数化量子线路的输出量子态下的能量期望值，并以最小化该能量期望值为优化目标，不断优化参数化量子线路的参数，以调整其输出量子态，使得目标量子系统的哈密顿量在该输出量子态下的能量期望值趋于最小，最终得到该目标量子系统的基态能量。
步骤220，采用神经网络对该n个量子比特的输出量子态进行后处理，根据该神经网络的后处理结果计算得到哈密顿量的能量期望值。
在本申请提供的VQNHE框架中，采用神经网络对参数化量子线路输出的波函数做后处理，该神经网络能够起到一个通用函数近似器的作用，其相比于Jastrow因子具有更强的表达能力和基态能量近似能力，从而有助于提升基态能量估计的准确度。
在一些实施例中，步骤220包括如下几个子步骤：
1.根据目标量子系统的哈密顿量分解得到的泡利字符串，以及神经网络对应的后处理算符分解得到的泡利字符串，生成目标量子系统的等效哈密顿量所对应的多个泡利字符串；
可选地，对神经网络对应的后处理算符做泰勒展开，得到t个泡利字符串，t为正整数；对该t个泡利字符串和目标量子系统的哈密顿量分解得到的k个泡利字符串做直积运算，生成目标量子系统的等效哈密顿量所对应的多个泡利字符串。其中，目标量子系统的等效哈密顿量，即为其所对应的多个泡利字符串的直积。
可选地，目标量子系统的等效哈密顿量所对应的多个泡利字符串的最大数量为t×t×k。
2.对于上述等效哈密顿量所对应的多个泡利字符串中的每一个泡利字符串，通过测量得 到n个量子比特的输出量子态在该泡利字符串对应的测量基上的比特字符串；
3.根据上述多个泡利字符串分别对应的比特字符串，计算得到该多个泡利字符串分别对应的能量期望值；
4.根据该多个泡利字符串分别对应的能量期望值，计算哈密顿量的能量期望值。
以测量基Z为例，神经网络对应的后处理算符
Figure PCTCN2021124392-appb-000009
的泰勒展开式如下：
Figure PCTCN2021124392-appb-000010
其中c ijk...代表Z iZ jZ k...对应的系数，且c ijk...是基于神经网络的参数确定的，Z i是第i个量子比特上的Z泡利算符，Z j是第j个量子比特上的Z泡利算符，Z k是第k个量子比特上的Z泡利算符，以此类推。通过上述泰勒展开，可以得到指数多个泡利字符串，也即t与量子比特的数量n是指数相关的。目标量子系统的等效哈密顿量等于t个泡利字符串、目标量子系统的哈密顿量以及t个泡利字符串的直积，而目标量子系统的哈密顿量又可以分解为k个泡利字符串的直积，因此，最多需要测量t×t×k个泡利字符串分别对应的能量期望值。对于该t×t×k个泡利字符串中的每个泡利字符串，分别进行多次测量，并基于每一次测量得到的比特字符串得到能量计算结果，然后将该多次测量得到的能量计算结果取平均值，得到该泡利字符串的能量期望值。上述目标量子系统的等效哈密顿量在PQC的输出量子态上的能量期望值值，等于目标量子系统的原始哈密顿量在后处理波函数上的能量期望值。因此，计算目标量子系统的原始哈密顿量的能量期望值，相当于计算其等效哈密顿量的能量期望值。而该等效哈密顿量的能量期望值
Figure PCTCN2021124392-appb-000011
其中
Figure PCTCN2021124392-appb-000012
对应t×t×k个泡利字符串的能量期望值的求和结果。例如，将t×t×k个泡利字符串的能量期望值相加，即得到该等效哈密顿量的能量期望值。需要说明的是，上述相加可以是直接相加，也可以是加权求和，本申请对此不作限定。
在本申请实施例中，对神经网络的具体结构不作限定，其可以是简单的全连接结构，也可以是其他较为复杂的结构，本申请对此不作限定。
步骤230，以哈密顿量的能量期望值收敛为目标，对参数化量子线路的参数和神经网络的参数进行调整。
可选地，分别计算哈密顿量的能量期望值相对于参数化量子线路的参数的导数，以及相对于神经网络的参数的导数。然后基于该导数信息，采用梯度下降法分别对参数化量子线路的参数和神经网络的参数进行调整，以使得哈密顿量的能量期望值趋于最小。其中，参数化量子线路的参数优化过程和神经网络的参数优化过程，两者可以同步进行，也可以先后进行，本申请对此不作限定。
步骤240，在哈密顿量的能量期望值满足收敛条件的情况下，将满足收敛条件的哈密顿量的能量期望值，确定为目标量子系统的基态能量。
最后，将该哈密顿量的能量期望值的最小值，确定为目标量子系统的基态能量。
本申请实施例采用神经网络对参数化量子线路输出的波函数做后处理，该神经网络能够起到一个通用函数近似器的作用，其相比于Jastrow因子具有更强的表达能力和基态能量近似能力，从而有助于提升基态能量估计的准确度。
本申请另一示例性实施例提供的VQNHE框架如图3所示，包括参数化量子线路(PQC)10、测量线路40、神经网络20和优化器30。其中，神经网络20和优化器30可以是部署在计算机设备中的功能模块，优化器30也可以称为优化器模块。在本申请实施例中，计算机设备可以是通过处理器执行计算机程序以实现该方法的经典计算机，其具备存储和运算能力。测量线路40包括k组测量线路，该k组测量线路和哈密顿量分解得到的k个泡利字符串一一对应。参数化量子线路10用于对n个量子比特的输入量子态进行变换处理，得到该n个量子 比特的输出量子态，n为正整数。目标量子系统的哈密顿量在该n个量子比特的输出量子态下的能量期望值，是哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，k为正整数。对于该k个泡利字符串中的目标泡利字符串，该目标泡利字符串对应的测量线路用于对n个量子比特的输出量子态执行与目标泡利字符串相对应的变换处理，得到变换后的输出量子态。神经网络20用于对该变换后的输出量子态进行后处理。基于神经网络20的后处理结果，得到目标泡利字符串的能量期望值。对于k个泡利字符串，分别执行上述操作，得到该k个泡利字符串分别对应的能量期望值，然后求和得到哈密顿量的能量期望值。优化器30用于以哈密顿量的能量期望值收敛为目标，对参数化量子线路10的参数和神经网络20的参数进行调整。在哈密顿量的能量期望值满足收敛条件的情况下，将满足收敛条件的哈密顿量的能量期望值，确定为目标量子系统的基态能量。
图4是本申请另一个实施例提供的量子体系基态能量估计方法的流程图。该方法可应用于图3所示的VQNHE框架中，例如该方法各步骤的执行主体可以是计算机设备，计算机设备可以是通过处理器执行计算机程序以实现该方法的经典计算机，其具备存储和运算能力。该方法可以包括如下几个步骤(410～480)：
步骤410，获取经参数化量子线路对n个量子比特的输入量子态进行变换处理后得到的该n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在该n个量子比特的输出量子态下的能量期望值，是哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数。
在本实施例中，直接计算出哈密顿量分解得到的k个泡利字符串分别对应的能量期望值，然后根据该k个泡利字符串分别对应的能量期望值，计算得到目标量子系统的哈密顿量的能量期望值。可选地，对该k个泡利字符串分别对应的能量期望值进行求和处理，将得到的求和结果作为目标量子系统的哈密顿量的能量期望值。需要说明的是，此处的求和处理可以是直接相加，也可以是加权求和，本申请对此不作限定。
在上文实施例中，通过计算目标量子系统的等效哈密顿量分解得到的多个泡利字符串分别对应的能量期望值，将该多个泡利字符串分别对应的能量期望值进行求和处理，得到等效哈密顿量的能量期望值，并将该等效哈密顿量的能量期望值，作为目标量子系统的哈密顿量的能量期望值。这种方式由于最多可能需要计算t×t×k个泡利字符串分别对应的能量期望值，较为复杂低效。本实施例提供的方式，通过引入测量线路，实现仅需计算k个泡利字符串分别对应的能量期望值即可，更加简单高效。
步骤420，对于k个泡利字符串中的目标泡利字符串，获取经目标泡利字符串对应的测量线路，对n个量子比特的输出量子态执行与目标泡利字符串相对应的变换处理后得到的变换后的输出量子态。
可选地，对于k个泡利字符串中的目标泡利字符串，采用目标泡利字符串对应的测量线路，对n个量子比特的输出量子态执行与目标泡利字符串相对应的变换处理，得到变换后的输出量子态。
对于哈密顿量分解得到的k个泡利字符串，逐个测量估计得到其能量期望值。图3所示的VQNHE框架包括k组测量线路，该k组测量线路和k个泡利字符串一一对应。目标泡利字符串可以是该k个泡利字符串中的任意一个泡利字符串，在对目标泡利字符串的能量期望值进行测量估计时，使用该目标泡利字符串对应的测量线路对参数化量子线路的输出量子态执行与目标泡利字符串相对应的变换处理，得到变换后的输出量子态。这一步变换的目的是为了减少测量估计过程中的资源消耗，具体原理请见下文的推导分析。
在示例性实施例中，目标泡利字符串对应的测量线路包括除符号量子比特之外的非符号量子比特对应的量子门，以使得非符号量子比特在同一种测量基上进行测量；其中，符号量子比特是n个量子比特中与目标泡利字符串中的一个目标泡利算符对应的量子比特，该符号量子比特对应的测量基根据符号量子比特在目标泡利字符串中对应的泡利算符确定。每个非 符号量子比特对应的量子门是一个双比特量子门，其同时作用在符号量子比特和该非符号量子比特上。
以图5所示的测量线路为例，目标泡利字符串为I 0X 1X 2Y 3I 4，其中I算符可以忽略，因此该目标泡利字符串可记为X 1X 2Y 3，假设要在测量基Z上进行测量，则可以将第2个量子比特(对应泡利算符X 1)作为符号量子比特，其他量子比特即为非符号量子比特。此时，该目标泡利字符串对应的测量线路50包括作用在第2个量子比特(即符号量子比特)和第3个量子比特(对应泡利算符X 2)上的双比特控制X门51，以及作用在第2个量子比特(即符号量子比特)和第4个量子比特(对应泡利算符Y 3)上的双比特控制Y门52。另外，符号量子比特对应的测量基根据该符号量子比特在目标泡利字符串中对应的泡利算符确定，在本例中，第2个量子比特为符号量子比特，其对应泡利算符X 1，因此其对应测量基X。
可选地，上述同一种测量基为第一泡利算符对应的测量基，目标泡利算符为第二泡利算符或第三泡利算符；其中，第一泡利算符、第二泡利算符和第三泡利算符互不相同，且对于第一泡利算符、第二泡利算符和第三泡利算符中的任一泡利算符，是泡利X算符、泡利Y算符和泡利Z算符中的一个。也即，在上述同一种测量基为测量基X的情况下，符号量子比特为泡利Y或Z算符对应的某一个量子比特；在上述同一种测量基为测量基Y的情况下，符号量子比特为泡利X或Z算符对应的某一个量子比特；在上述同一种测量基为测量基Z的情况下，符号量子比特为泡利X或Y算符对应的某一个量子比特。
可选地，对于非符号量子比特，在非符号量子比特在目标泡利字符串中对应泡利X算符的情况下，该非符号量子比特对应的量子门为双比特控制X门；在非符号量子比特在目标泡利字符串中对应泡利Y算符的情况下，该非符号量子比特对应的量子门为双比特控制Y门；或者，在非符号量子比特在目标泡利字符串中对应泡利Z算符的情况下，该非符号量子比特对应的量子门为双比特控制Z门。
可选地，对于符号量子比特，在符号量子比特在目标泡利字符串中对应泡利X算符的情况下，该符号量子比特对应的测量基为泡利X算符对应的测量基；在符号量子比特在目标泡利字符串中对应泡利Y算符的情况下，该符号量子比特对应的测量基为泡利Y算符对应的测量基；在符号量子比特在目标泡利字符串中对应泡利Z算符的情况下，该符号量子比特对应的测量基为泡利Z算符对应的测量基。
步骤430，通过测量得到变换后的输出量子态在指定测量基上的比特字符串。
在这一组指定测量基中，除符号量子比特对应的测量基之外，其余非符号量子比特对应的测量基相同。例如，图5中，符号量子比特对应测量基X，其余非符号量子比特均对应测量基Z。
步骤440，通过神经网络根据该比特字符串，输出用于计算目标泡利字符串的能量期望值的元数据。
将测量得到的比特字符串输入至神经网络，由神经网络进行前向计算，输出用于计算目标泡利字符串的能量期望值的元数据。
步骤450，根据该元数据计算得到目标泡利字符串的能量期望值。
可选地，按照如下公式计算目标泡利字符串的能量期望值
Figure PCTCN2021124392-appb-000013
Figure PCTCN2021124392-appb-000014
其中，f表示神经网络，s 0表示符号量子比特对应的测量结果(其值为0或1)，s表示比特字符串，0s 1:n-1表示将比特字符串s中符号量子比特对应的比特位设为0得到的比特字符串，
Figure PCTCN2021124392-appb-000015
表示将比特字符串s中符号量子比特对应的比特位设为1且对其他比特位按照目标 泡利字符串做相应的比特反转后得到的比特字符串。所谓比特反转，即将0变为1，将1变为0。
以图5为例，比特字符串s为s 0s 1s 2s 3s 4，符号量子比特为第2个量子比特，因此将比特字符串s中符号量子比特对应的比特位设为0得到的比特字符串0s 1:n-1为s 00s 2s 3s 4，将比特字符串s中符号量子比特对应的比特位设为1且对其他比特位进行比特反转后得到的比特字符串
Figure PCTCN2021124392-appb-000016
为
Figure PCTCN2021124392-appb-000017
将s 、0s 1:n-1和
Figure PCTCN2021124392-appb-000018
分别输入神经网络，由神经网络输出f(s)、f(0s 1:n-1)和
Figure PCTCN2021124392-appb-000019
的值，而后代入上述公式即可计算出该泡利字符串X 1X 2Y 3的能量期望值
Figure PCTCN2021124392-appb-000020
即
Figure PCTCN2021124392-appb-000021
步骤460，根据k个泡利字符串的能量期望值计算哈密顿量的能量期望值。
例如，将k个泡利字符串的能量期望值相加，得到哈密顿量的能量期望值。需要说明的是，上述相加可以是直接相加，也可以是加权求和，本申请对此不作限定。
步骤470，以哈密顿量的能量期望值收敛为目标，对参数化量子线路的参数和神经网络的参数进行调整。
步骤480，在哈密顿量的能量期望值满足收敛条件的情况下，将满足收敛条件的哈密顿量的能量期望值，确定为目标量子系统的基态能量。
步骤470-480与图2实施例中的步骤230-240相同，具体可参见上文介绍说明，本实施例对此不再赘述。
在示例性实施例中，为了进一步简化测量线路的结构，在非符号量子比特在目标泡利字符串中对应的泡利算符，与上述同一种测量基对应的泡利算符相同的情况下，该非符号量子比特对应的量子门采用非符号量子比特对应的测量结果所对应的符号进行等效代替。
以图6为例，目标泡利字符串为I 0I 1Y 2Z 3X 4，其中I算符可以忽略，因此该目标泡利字符串可记为Y 2Z 3X 4，假设要在测量基Z上进行测量，则可以将第3个量子比特(对应泡利算符Y 2)作为符号量子比特，其他量子比特即为非符号量子比特。此时，该目标泡利字符串对应的测量线路60应当包括作用在第3个量子比特(即符号量子比特)和第4个量子比特(对应泡利算符Z 3)上的双比特控制Z门，以及作用在第3个量子比特(即符号量子比特)和第5个量子比特(对应泡利算符X 4)上的双比特控制X门61。但是，为了进一步简化测量线路60的结构，可以省去上述作用在第3个量子比特(即符号量子比特)和第4个量子比特(对应泡利算符Z 3)上的双比特控制Z门，而采用该第4个量子比特对应的测量结果s 3所对应的符号1-2s 3进行等效代替。
如果不进行等效替代，该泡利字符串Y 2Z 3X 4的能量期望值的计算公式为
Figure PCTCN2021124392-appb-000022
经过等效替代之后，该泡利字符串的能量期望值的计算公式为
Figure PCTCN2021124392-appb-000023
下面，对加入测量线路以减少测量估计过程中的资源消耗的原理进行推导分析。
由于后处理算符的非幺正特性，我们需要优化的目标是归一化的能量期望
Figure PCTCN2021124392-appb-000024
其中
Figure PCTCN2021124392-appb-000025
是任意的泡利字符串。对于哈密顿量的能量期望值，其总是可以分解成多个泡利字符串的能量期望值的简单求和，因此我们的测量估计方案，解决单个泡利字符串的期望估算问题即可。
对于上式中的分母
Figure PCTCN2021124392-appb-000026
其中ψ s＝<s|ψ>代表了参数化量子线路PQC输出的波函数在测量基上的概率幅。这一公式对应的实现策略非常简单：直接在PQC测量基测量得到比特字符串s，然后计算多次测量结果的|f(s)| 2的均值即可，f(s)代表神经网络f输入比特字符串s相应输出的值。
以PQC测量基为测量基Z为例，如果待估计的泡利字符串
Figure PCTCN2021124392-appb-000027
中只包含泡利Z算符(可选地还包括I算符
Figure PCTCN2021124392-appb-000028
)，也即<s|H|s′>＝H sδ ss′(其中，s和s′代表两个比特字符串，δ ss′是克罗内克函数，只有当s和s′一样时为1，其他时候为0，H s是泡利字符串在s对应基下的期望)，那么对于上式中的分子，我们有：
Figure PCTCN2021124392-appb-000029
其测量策略和分母的估算完全类似，直接在PQC测量基测量得到比特字符串s之后，计算|f(s)| 2H s的期望。
VQE后处理真正的难点，也是之前一直被认为需要消耗指数资源才可以完成的是当泡利字符串
Figure PCTCN2021124392-appb-000030
中包含泡利X或者Y算符的时候。从最直接的视角来看，由于需要计算后处理的增强效应，由于我们的神经网络后处理建立在测量基Z上，所有量子比特都需要在测量基Z上去测量得到比特字符串s，然后输入神经网络计算f(s)的值。但另一方面，一个包含了泡利X或Y算符的泡利字符串，需要在测量基X或Y上去测量得到对应量子比特的相应结果。也即这里存在一个冲突：我们需要同时获取某几个量子比特上的X和Z在同一次测量中的值，这两者由于不对易(即XZ≠ZX)而不能同时获取，这也是之前的实现方案需要消耗指数资源的原因。
为了实现非幺正后处理的指数加速，考察泡利字符串的具体数学结构。本申请定义泡利字符串中对应X或Y算符的某一个量子比特为符号量子比特，并且为了便于在公式中体现，将该符号量子比特记为第0位，其对应的测量结果记为s 0。我们定义
Figure PCTCN2021124392-appb-000031
对应在泡利字符串作用下的比特字符串变换：
Figure PCTCN2021124392-appb-000032
其中S(s)对应相位因子，可能取值取决于具体的泡利算符，可能为±1，±i中的一个。考虑到
Figure PCTCN2021124392-appb-000033
我们有
Figure PCTCN2021124392-appb-000034
而泡利字符串的形式为：
Figure PCTCN2021124392-appb-000035
注意到这里的求和保持了符号量子比特固定在0，这样的求和我们之后简记为s∈{0,1} n-1。
该泡利字符串的全部本征值均为±1，对应的各2 n-1本征态分别为：
Figure PCTCN2021124392-appb-000036
Figure PCTCN2021124392-appb-000037
考虑后处理神经网络输出f(s)是实数(对于复数情形将在下文说明)，可得：
Figure PCTCN2021124392-appb-000038
最后的概率幅ψ ±,s＝<±,s 1:n|ψ>为PQC输出波函数在泡利字符串本征态基上的概率幅。为了实现在这组基上的测量，我们需要引入测量线路(以V表示)附加在PQC(以U表示)之后。若
Figure PCTCN2021124392-appb-000039
则我们恰好有<±,s 1:n-1|ψ>＝<s|VU(θ)|0>。也即我们需要构造测量线路V，对应的
Figure PCTCN2021124392-appb-000040
这样线路的构造方案如下：
1.对于泡利字符串中包含的除符号量子比特之外的非符号量子比特，我们作用一个双比特控制X/Y/Z门，具体的选择对应相应比特上的算符种类，控制比特均为符号量子比特。
可选地，在非符号量子比特在目标泡利字符串中对应的泡利算符，与上述同一种测量基对应的泡利算符相同的情况下，该非符号量子比特对应的双比特量子门采用该非符号量子比特对应的测量结果所对应的符号进行等效代替，从而有助于简化测量线路的结构。
2.除符号量子比特之外的非符号量子比特均在同一种测量基上进行测量，而符号量子比特对应的测量基根据该符号量子比特在泡利字符串中对应的泡利算符确定。
由上述的理论推导和实验方案构建可以看出，相较VQE，我们只需要额外的m-1个双比特量子门的量子资源，其中在非符号量子比特对应的同一种测量基为测量基Z的情况下，m是对应泡利字符串中包含泡利X和Y算符的个数(其他情况与此类似)。对于常见的短程相互作用，这一个数通常是O(1)的量级。因此整个VQNHE框架是否不需要指数时间所唯一需要分析的，就是测量误差的影响。我们接下来将对测量估计期望带来的随机误差进行分析，从而肯定地得出当前方案只需要多项式资源的结论。
对于标准VQE框架，测量误差估计为：
Figure PCTCN2021124392-appb-000041
其中p是测量泡利字符串对应+1的概率。为了达到估计泡利字符串的精度为1-ε，需要的测量次数为N＝4p(1-p)/ε 2，对于测量最困难的期望为0，p＝0.5的情形，需要的测量次数为1/ε 2的量级。
对于VQNHE框架的测量误差估计，由分子分布期望n和分母分布期望d的比值组成， 我们有：
Figure PCTCN2021124392-appb-000042
其中，δn是分子分布期望n对应的标准差，δd是分母分布期望d对应的标准差。我们考虑神经网络后处理的输出值被限制在1/r到r的范围内，则有1/d＜r 2，δd＜r 2/2，
Figure PCTCN2021124392-appb-000043
综合起来得到：
Figure PCTCN2021124392-appb-000044
也即VQNHE情形达到对应精度需要测量次数的理论上界为9r 8/4ε 2，这一数值和VQE比只有关于神经网络函数范围的多项式依赖，且与系统体系大小无关。因此VQNHE可以在量子硬件上高效实现。值得注意的是该理论上界比较松，实际问题中所需的额外测量次数要远小于这一数值。
另外，上文主要以神经网络的输出f(s)为实数形式对VQNHE框架下的理论推导和实验方案进行了介绍说明。对于神经网络的输出f(s)可以取复数的情形，本申请提供的VQNHE框架依旧可以高效地完成，相应的推导如下。
我们令
Figure PCTCN2021124392-appb-000045
将分子的估计分成如下两个部分：
Figure PCTCN2021124392-appb-000046
对于实部相关的部分，测量和估计与上文介绍内容相同，唯一区别是因子为f *f取实部。
对于虚部相关的部分，我们可以相似地转移到另一组基上测量：
Figure PCTCN2021124392-appb-000047
PQC输出态概率幅展开的这组新基为：
Figure PCTCN2021124392-appb-000048
Figure PCTCN2021124392-appb-000049
相类似地，我们需要构造测量线路V′以期
Figure PCTCN2021124392-appb-000050
该测量线路V′的构造规则和实数情形也类似，唯一的区别是，如果泡利字符串在符号量子比特位是Y(X)算符，我们最后在符号量子比特X(-Y)的基上测量。
本申请实施例通过在PQC之后添加测量线路，利用该测量线路对PQC的输出量子态执行与泡利字符串相对应的变换处理，得到变换后的输出量子态，这一步变换能够减少测量估计过程中的资源消耗，从而能够在多项式资源的消耗下，完成对泡利字符串乃至一般哈密顿量的测量和无偏估计。
下面，对应用本申请提供的VQNHE框架到具体的模型研究的案例进行示例性说明。
我们分别考虑量子自旋模型和分子模型，这两类在凝聚态物理和量子化学领域的典型问题。此外我们还将展示该VQNHE框架在实际量子硬件上的执行效果。
案例一：VQNHE框架在横场伊辛模型和海森堡模型上的计算。
我们用VQNHE框架优化计算了一维横场伊辛模型和各向同性量子海森堡模型的基态能量值。两模型都在12个格点上计算，且相应的模型哈密顿量参数均为1并取周期性边界条件。VQNHE、VQE的结果与严格结果的对比如下表1所示。其中VQE和VQNHE在同一模型中均采用了相同的量子线路结构来计算。
表1
模型 VQE VQNHE 严格结果
横场伊辛模型 -14.914 -15.319 -15.3226
海森堡模型 -21.393 -21.546 -21.5496
案例二：VQNHE框架计算LiH(氢化锂)分子的解离曲线。
VQNHE框架也可应用在分子能量计算上。本例中我们用VQNHE框架来计算不同原子距离对应的LiH体系的基态能量。我们同时将该能量和VQE得到的能量，以及HartreeFock(哈特里-福克)平均场方法得到的能量进行了对比，结果如图7中(a)部分所示，曲线71对应HartreeFock(哈特里-福克)平均场方法得到的能量，曲线72对应VQE得到的能量，曲线73对应VQNHE得到的能量，该VQNHE得到的能量与严格结果基本重合。从图7中(b)部分可以看到，VQNHE对应的优化能量精度比VQE高出了一个数量级以上。VQNHE和VQE在该问题上都是在对称性约化的4量子比特的完全活性空间计算。两算法采用相同的硬件友好拟设的量子线路结构，该量子线路结构可以如图8所示。
案例三：VQNHE框架在真实硬件和带噪声模拟器上的表现。
为了考察VQNHE框架在具有测量误差和量子硬件噪声的非理想情形下的表现，我们在真实的IBM量子硬件上和量子噪声仿真模型上，运行了VQNHE算法。相应的VQE和VQNHE获得的结果如图9所示。我们的测试模型为5格点开边界条件的横场伊辛模型，相应的PQC线路结构如图10所示。
可以看出无论是在理想模型，还是在真实硬件，VQNHE框架得出的结果都远远好于利用了同样数量量子资源的VQE框架。与此同时，在同样是8192次测量的前提下，VQNHE方法并没有引入显著增加的测量误差。图9中的线条91是真实的基态能量和理想情况下VQNHE收敛的能量(二者基本重合)，线条92是理想情况下VQE的优化能量。值得指出的是，这里我们利用了基于测量比特字符串真实数据，对神经网络后处理部分进行了再优化。我们发现偏离理想情况下最优的神经网络，反而可以给出最低的能量估计。也就是说，神经 网络的后处理部分可以自适应地考虑进部分量子噪声的效应，具有一定的QEM(Quantum Error Mitigation，量子误差消除)的自然属性。
以下是本申请的装置和系统实施例，该装置和系统实施例与上述方法实施例相对应，属于同一发明构思，对于装置和系统实施例中未详细说明的细节，可参见本申请方法实施例。
图11是本申请一个实施例提供的量子体系基态能量估计装置的框图。该装置具有实现上述方法示例的功能，所述功能可以由硬件实现，也可以由硬件执行相应的软件实现。该装置可以是计算机设备，也可以设置在计算机设备中。如图11所示，该装置1100可以包括：状态获取模块1110、后处理模块1120和优化器模块1130。
状态获取模块1110，用于获取经参数化量子线路对n个量子比特的输入量子态进行变换处理后得到的所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数。
后处理模块1120，用于采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值。
优化器模块1130，用于以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
在示例性实施例中，所述后处理模块1120包括：分解单元、测量单元和计算单元；
所述分解单元，用于根据所述哈密顿量分解得到的泡利字符串和所述神经网络对应的后处理算符分解得到的泡利字符串，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串；
所述测量单元，用于对于所述多个泡利字符串中的每一个泡利字符串，通过测量得到所述n个量子比特的输出量子态在所述泡利字符串对应的测量基上的比特字符串；
所述计算单元，用于根据所述多个泡利字符串分别对应的比特字符串，计算得到所述多个泡利字符串分别对应的能量期望值，根据所述多个泡利字符串分别对应的能量期望值计算所述哈密顿量的能量期望值。
可选地，所述分解单元，用于：
对所述神经网络对应的后处理算符做泰勒展开，得到t个泡利字符串，t为正整数；
对所述t个泡利字符串和所述哈密顿量分解得到的k个泡利字符串做直积运算，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串。
在示例性实施例中，所述后处理模块包括：获取单元、测量单元、神经网络单元和计算单元；
所述获取单元，用于对于所述k个泡利字符串中的目标泡利字符串，获取经所述目标泡利字符串对应的测量线路，对所述n个量子比特的输出量子态执行与所述目标泡利字符串相对应的变换处理后得到的变换后的输出量子态；
所述测量单元，用于通过测量得到所述变换后的输出量子态在指定测量基上的比特字符串；
所述神经网络单元，用于通过所述神经网络根据所述比特字符串，输出用于计算所述目标泡利字符串的能量期望值的元数据；
所述计算单元，用于根据所述元数据计算得到所述目标泡利字符串的能量期望值，根据所述k个泡利字符串的能量期望值计算所述哈密顿量的能量期望值。
可选地，所述目标泡利字符串对应的测量线路包括除符号量子比特之外的非符号量子比特对应的量子门，以使得所述非符号量子比特在同一种测量基上进行测量；其中，所述符号 量子比特是所述n个量子比特中与所述目标泡利字符串中的一个目标泡利算符对应的量子比特，所述符号量子比特对应的测量基根据所述符号量子比特在所述目标泡利字符串中对应的泡利算符确定。
可选地，所述同一种测量基为第一泡利算符对应的测量基，所述目标泡利算符为第二泡利算符或第三泡利算符；其中，所述第一泡利算符、所述第二泡利算符和所述第三泡利算符互不相同，且对于所述第一泡利算符、所述第二泡利算符和所述第三泡利算符中的任一泡利算符，是泡利X算符、泡利Y算符和泡利Z算符中的一个。
可选地，在所述非符号量子比特在所述目标泡利字符串中对应泡利X算符的情况下，所述非符号量子比特对应的量子门为双比特控制X门；或者，
在所述非符号量子比特在所述目标泡利字符串中对应泡利Y算符的情况下，所述非符号量子比特对应的量子门为双比特控制Y门；或者，
在所述非符号量子比特在所述目标泡利字符串中对应泡利Z算符的情况下，所述非符号量子比特对应的量子门为双比特控制Z门。
可选地，在所述符号量子比特在所述目标泡利字符串中对应泡利X算符的情况下，所述符号量子比特对应的测量基为所述泡利X算符对应的测量基；或者，
在所述符号量子比特在所述目标泡利字符串中对应泡利Y算符的情况下，所述符号量子比特对应的测量基为所述泡利Y算符对应的测量基；或者，
在所述符号量子比特在所述目标泡利字符串中对应泡利Z算符的情况下，所述符号量子比特对应的测量基为所述泡利Z算符对应的测量基。
可选地，所述计算单元用于按照如下公式计算所述目标泡利字符串的能量期望值
Figure PCTCN2021124392-appb-000051
Figure PCTCN2021124392-appb-000052
其中，f表示所述神经网络，s 0表示所述符号量子比特对应的测量结果，s表示所述比特字符串，0s 1:n-1表示将所述比特字符串中所述符号量子比特对应的比特位设为0得到的比特字符串，
Figure PCTCN2021124392-appb-000053
表示将所述比特字符串中所述符号量子比特对应的比特位设为1且对其他比特位按照所述目标泡利字符串做相应的比特反转后得到的比特字符串。
可选地，在所述非符号量子比特在所述目标泡利字符串中对应的泡利算符，与所述同一种测量基对应的泡利算符相同的情况下，所述非符号量子比特对应的量子门采用所述非符号量子比特对应的测量结果所对应的符号进行等效代替。
本申请通过采用神经网络对参数化量子线路输出的波函数做后处理，该神经网络能够起到一个通用函数近似器的作用，其相比于Jastrow因子具有更强的表达能力和基态能量近似能力，从而有助于提升基态能量估计的准确度。
本申请一示例性实施例还提供了一种量子体系基态能量估计系统，所述系统包括：参数化量子线路和计算机设备。所述计算机设备包括后处理模块和优化器模块。
所述参数化量子线路用于对n个量子比特的输入量子态进行变换处理，得到所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数。
所述后处理模块用于采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值。
所述优化器模块用于以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；其中，在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
在示例性实施例中，所述后处理模块包括：分解单元、测量单元和计算单元；
所述分解单元，用于根据所述哈密顿量分解得到的泡利字符串和所述神经网络对应的后处理算符分解得到的泡利字符串，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串；
所述测量单元，用于对于所述多个泡利字符串中的每一个泡利字符串，测量得到所述n个量子比特的输出量子态在所述泡利字符串对应的测量基上的比特字符串；
所述计算单元，用于根据所述多个泡利字符串分别对应的比特字符串，计算得到所述多个泡利字符串分别对应的能量期望值，根据所述多个泡利字符串分别对应的能量期望值计算所述哈密顿量的能量期望值。
可选地，所述分解单元，用于：
对所述神经网络对应的后处理算符做泰勒展开，得到t个泡利字符串，t为正整数；
对所述t个泡利字符串和所述哈密顿量分解得到的k个泡利字符串做直积运算，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串。
在示例性实施例中，所述系统还包括k组测量线路，所述后处理模块包括：获取单元、测量单元、神经网络单元和计算单元，所述k组测量线路和所述k个泡利字符串一一对应；
所述目标泡利字符串对应的测量线路，用于对所述n个量子比特的输出量子态执行与所述目标泡利字符串相对应的变换处理，得到变换后的输出量子态；
所述获取单元用于获取所述变换后的输出量子态；
所述测量单元用于测量得到所述变换后的输出量子态在指定测量基上的比特字符串；
所述神经网络单元用于通过所述神经网络根据所述比特字符串，输出用于计算所述目标泡利字符串的能量期望值的元数据；
所述计算单元用于根据所述元数据计算得到所述目标泡利字符串的能量期望值，根据所述k个泡利字符串的能量期望值计算所述哈密顿量的能量期望值。
可选地，所述目标泡利字符串对应的测量线路包括除符号量子比特之外的非符号量子比特对应的量子门，以使得所述非符号量子比特在同一种测量基上进行测量；其中，所述符号量子比特是所述n个量子比特中与所述目标泡利字符串中的一个目标泡利算符对应的量子比特，所述符号量子比特对应的测量基根据所述符号量子比特在所述目标泡利字符串中对应的泡利算符确定。
可选地，所述同一种测量基为第一泡利算符对应的测量基，所述目标泡利算符为第二泡利算符或第三泡利算符；其中，所述第一泡利算符、所述第二泡利算符和所述第三泡利算符互不相同，且对于所述第一泡利算符、所述第二泡利算符和所述第三泡利算符中的任一泡利算符，是泡利X算符、泡利Y算符和泡利Z算符中的一个。
可选地，在所述非符号量子比特在所述目标泡利字符串中对应泡利X算符的情况下，所述非符号量子比特对应的量子门为双比特控制X门；或者，
在所述非符号量子比特在所述目标泡利字符串中对应泡利Y算符的情况下，所述非符号量子比特对应的量子门为双比特控制Y门；或者，
在所述非符号量子比特在所述目标泡利字符串中对应泡利Z算符的情况下，所述非符号量子比特对应的量子门为双比特控制Z门。
可选地，在所述符号量子比特在所述目标泡利字符串中对应泡利X算符的情况下，所述符号量子比特对应的测量基为所述泡利X算符对应的测量基；或者，
在所述符号量子比特在所述目标泡利字符串中对应泡利Y算符的情况下，所述符号量子 比特对应的测量基为所述泡利Y算符对应的测量基；或者，
在所述符号量子比特在所述目标泡利字符串中对应泡利Z算符的情况下，所述符号量子比特对应的测量基为所述泡利Z算符对应的测量基。
可选地，所述计算单元用于按照如下公式计算所述目标泡利字符串的能量期望值
Figure PCTCN2021124392-appb-000054
Figure PCTCN2021124392-appb-000055
其中，f表示所述神经网络，s 0表示所述符号量子比特对应的测量结果，s表示所述比特字符串，0s 1:n-1表示将所述比特字符串中所述符号量子比特对应的比特位设为0得到的比特字符串，
Figure PCTCN2021124392-appb-000056
表示将所述比特字符串中所述符号量子比特对应的比特位设为1且对其他比特位按照所述目标泡利字符串做相应的比特反转后得到的比特字符串。
可选地，在所述非符号量子比特在所述目标泡利字符串中对应的泡利算符，与所述同一种测量基对应的泡利算符相同的情况下，所述非符号量子比特对应的量子门采用所述非符号量子比特对应的测量结果所对应的符号进行等效代替。
需要说明的是，上述实施例提供的装置和系统，在实现其功能时，仅以上述各功能模块的划分进行举例说明，实际应用中，可以根据需要而将上述功能分配由不同的功能模块完成，即将设备的内部结构划分成不同的功能模块，以完成以上描述的全部或者部分功能。另外，上述实施例提供的装置和系统，与方法实施例属于同一构思，其具体实现过程详见方法实施例，这里不再赘述。
图12是本申请一个实施例提供的计算机设备的结构示意图。该计算机设备可以是任何具备数据存储和运算能力的电子设备，该计算机设备可用于实施上述实施例中提供的量子体系基态能量估计方法。具体来讲：
该计算机设备1200包括中央处理单元(如CPU(Central Processing Unit，中央处理器)、GPU(Graphics Processing Unit，图形处理器)和FPGA(Field Programmable Gate Array，现场可编程逻辑门阵列)等)1201、包括RAM(Random-Access Memory，随机存储器)1202和ROM(Read-Only Memory，只读存储器)1203的系统存储器1204，以及连接系统存储器1204和中央处理单元1201的系统总线1205。该计算机设备1200还包括帮助服务器内的各个器件之间传输信息的基本输入/输出系统(Input Output System，I/O系统)1206，和用于存储操作系统1213、应用程序1214和其他程序模块1215的大容量存储设备1207。
在一些实施例中，该基本输入/输出系统1206包括有用于显示信息的显示器1208和用于用户输入信息的诸如鼠标、键盘之类的输入设备1209。其中，该显示器1208和输入设备1209都通过连接到系统总线1205的输入输出控制器1210连接到中央处理单元1201。该基本输入/输出系统1206还可以包括输入输出控制器1210以用于接收和处理来自键盘、鼠标、或电子触控笔等多个其他设备的输入。类似地，输入输出控制器1210还提供输出到显示屏、打印机或其他类型的输出设备。
该大容量存储设备1207通过连接到系统总线1205的大容量存储控制器(未示出)连接到中央处理单元1201。该大容量存储设备1207及其相关联的计算机可读介质为计算机设备1200提供非易失性存储。也就是说，该大容量存储设备1207可以包括诸如硬盘或者CD-ROM(Compact Disc Read-Only Memory，只读光盘)驱动器之类的计算机可读介质(未示出)。
不失一般性，该计算机可读介质可以包括计算机存储介质和通信介质。计算机存储介质 包括以用于存储诸如计算机可读指令、数据结构、程序模块或其他数据等信息的任何方法或技术实现的易失性和非易失性、可移动和不可移动介质。计算机存储介质包括RAM、ROM、EPROM(Erasable Programmable Read-Only Memory，可擦写可编程只读存储器)、EEPROM(Electrically Erasable Programmable Read-Only Memory，电可擦写可编程只读存储器)、闪存或其他固态存储技术，CD-ROM、DVD(Digital Video Disc，高密度数字视频光盘)或其他光学存储、磁带盒、磁带、磁盘存储或其他磁性存储设备。当然，本领域技术人员可知该计算机存储介质不局限于上述几种。上述的系统存储器1204和大容量存储设备1207可以统称为存储器。
根据本申请实施例，该计算机设备1200还可以通过诸如因特网等网络连接到网络上的远程计算机运行。也即计算机设备1200可以通过连接在该系统总线1205上的网络接口单元1211连接到网络1212，或者说，也可以使用网络接口单元1211来连接到其他类型的网络或远程计算机系统(未示出)。
所述存储器还包括计算机程序，该计算机程序存储于存储器中，且经配置以由一个或者一个以上处理器执行，以实现上述量子体系基态能量估计方法。
在示例性实施例中，还提供了一种计算机可读存储介质，所述存储介质中存储有计算机程序，所述计算机程序在被计算机设备的处理器执行时实现上述量子体系基态能量估计方法。
可选地，该计算机可读存储介质可以包括：ROM(Read-Only Memory，只读存储器)、RAM(Random-Access Memory，随机存储器)、SSD(Solid State Drives，固态硬盘)或光盘等。其中，随机存取记忆体可以包括ReRAM(Resistance Random Access Memory，电阻式随机存取记忆体)和DRAM(Dynamic Random Access Memory，动态随机存取存储器)。
在示例性实施例中，还提供了一种计算机程序产品或计算机程序，所述计算机程序产品或计算机程序包括计算机指令，所述计算机指令存储在计算机可读存储介质中。计算机设备的处理器从所述计算机可读存储介质中读取所述计算机指令，所述处理器执行所述计算机指令，使得所述计算机设备执行上述量子体系基态能量估计方法。
应当理解的是，在本文中提及的“多个”是指两个或两个以上。“和/或”，描述关联对象的关联关系，表示可以存在三种关系，例如，A和/或B，可以表示：单独存在A，同时存在A和B，单独存在B这三种情况。字符“/”一般表示前后关联对象是一种“或”的关系。另外，本文中描述的步骤编号，仅示例性示出了步骤间的一种可能的执行先后顺序，在一些其它实施例中，上述步骤也可以不按照编号顺序来执行，如两个不同编号的步骤同时执行，或者两个不同编号的步骤按照与图示相反的顺序执行，本申请实施例对此不作限定。
以上所述仅为本申请的示例性实施例，并不用以限制本申请，凡在本申请的精神和原则之内，所作的任何修改、等同替换、改进等，均应包含在本申请的保护范围之内。

Claims (20)

  1. 一种量子体系基态能量估计方法，所述方法由计算机设备执行，所述方法包括：
    获取经参数化量子线路对n个量子比特的输入量子态进行变换处理后得到的所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数；
    采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值；
    以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；
    在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
  2. 根据权利要求1所述的方法，其中，所述采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值，包括：
    根据所述哈密顿量分解得到的泡利字符串和所述神经网络对应的后处理算符分解得到的泡利字符串，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串；
    对于所述多个泡利字符串中的每一个泡利字符串，通过测量得到所述n个量子比特的输出量子态在所述泡利字符串对应的测量基上的比特字符串；
    根据所述多个泡利字符串分别对应的比特字符串，计算得到所述多个泡利字符串分别对应的能量期望值；
    根据所述多个泡利字符串分别对应的能量期望值，计算所述哈密顿量的能量期望值。
  3. 根据权利要求2所述的方法，其中，所述根据所述哈密顿量分解得到的泡利字符串和所述神经网络对应的后处理算符分解得到的泡利字符串，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串，包括：
    对所述神经网络对应的后处理算符做泰勒展开，得到t个泡利字符串，t为正整数；
    对所述t个泡利字符串和所述哈密顿量分解得到的k个泡利字符串做直积运算，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串。
  4. 根据权利要求1所述的方法，其中，所述采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值，包括：
    对于所述k个泡利字符串中的目标泡利字符串，获取经所述目标泡利字符串对应的测量线路，对所述n个量子比特的输出量子态执行与所述目标泡利字符串相对应的变换处理后得到的变换后的输出量子态；
    通过测量得到所述变换后的输出量子态在指定测量基上的比特字符串；
    通过所述神经网络根据所述比特字符串，输出用于计算所述目标泡利字符串的能量期望值的元数据；
    根据所述元数据计算得到所述目标泡利字符串的能量期望值；
    根据所述k个泡利字符串的能量期望值计算所述哈密顿量的能量期望值。
  5. 根据权利要求4所述的方法，其中，所述目标泡利字符串对应的测量线路包括除符号 量子比特之外的非符号量子比特对应的量子门，以使得所述非符号量子比特在同一种测量基上进行测量；其中，所述符号量子比特是所述n个量子比特中与所述目标泡利字符串中的一个目标泡利算符对应的量子比特，所述符号量子比特对应的测量基根据所述符号量子比特在所述目标泡利字符串中对应的泡利算符确定。
  6. 根据权利要求5所述的方法，其中，所述同一种测量基为第一泡利算符对应的测量基，所述目标泡利算符为第二泡利算符或第三泡利算符；其中，所述第一泡利算符、所述第二泡利算符和所述第三泡利算符互不相同，且对于所述第一泡利算符、所述第二泡利算符和所述第三泡利算符中的任一泡利算符，是泡利X算符、泡利Y算符和泡利Z算符中的一个。
  7. 根据权利要求5所述的方法，其中，
    在所述非符号量子比特在所述目标泡利字符串中对应泡利X算符的情况下，所述非符号量子比特对应的量子门为双比特控制X门；或者，
    在所述非符号量子比特在所述目标泡利字符串中对应泡利Y算符的情况下，所述非符号量子比特对应的量子门为双比特控制Y门；或者，
    在所述非符号量子比特在所述目标泡利字符串中对应泡利Z算符的情况下，所述非符号量子比特对应的量子门为双比特控制Z门。
  8. 根据权利要求5所述的方法，其中，
    在所述符号量子比特在所述目标泡利字符串中对应泡利X算符的情况下，所述符号量子比特对应的测量基为所述泡利X算符对应的测量基；或者，
    在所述符号量子比特在所述目标泡利字符串中对应泡利Y算符的情况下，所述符号量子比特对应的测量基为所述泡利Y算符对应的测量基；或者，
    在所述符号量子比特在所述目标泡利字符串中对应泡利Z算符的情况下，所述符号量子比特对应的测量基为所述泡利Z算符对应的测量基。
  9. 根据权利要求5所述的方法，其中，所述根据所述元数据计算得到所述目标泡利字符串的能量期望值，包括：
    按照如下公式计算所述目标泡利字符串的能量期望值
    Figure PCTCN2021124392-appb-100001
    Figure PCTCN2021124392-appb-100002
    其中，f表示所述神经网络，s 0表示所述符号量子比特对应的测量结果，s表示所述比特字符串，0s 1:n-1表示将所述比特字符串中所述符号量子比特对应的比特位设为0得到的比特字符串，
    Figure PCTCN2021124392-appb-100003
    表示将所述比特字符串中所述符号量子比特对应的比特位设为1且对其他比特位按照所述目标泡利字符串做相应的比特反转后得到的比特字符串。
  10. 根据权利要求5所述的方法，其中，在所述非符号量子比特在所述目标泡利字符串中对应的泡利算符，与所述同一种测量基对应的泡利算符相同的情况下，所述非符号量子比特对应的量子门采用所述非符号量子比特对应的测量结果所对应的符号进行等效代替。
  11. 一种量子体系基态能量估计装置，所述装置包括：
    状态获取模块，用于获取经参数化量子线路对n个量子比特的输入量子态进行变换处理 后得到的所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数；
    后处理模块，用于采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值；
    优化器模块，用于以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
  12. 一种计算机设备，所述计算机设备包括处理器和存储器，所述存储器中存储有计算机程序，所述计算机程序由所述处理器加载并执行以实现如权利要求1至10任一项所述的方法。
  13. 一种计算机可读存储介质，所述存储介质中存储有计算机程序，所述计算机程序由处理器加载并执行以实现如权利要求1至10任一项所述的方法。
  14. 一种计算机程序产品或计算机程序，所述计算机程序产品或计算机程序包括计算机指令，所述计算机指令存储在计算机可读存储介质中，处理器从所述计算机可读存储介质读取并执行所述计算机指令，以实现如权利要求1至10任一项所述的方法。
  15. 一种量子体系基态能量估计系统，所述系统包括：参数化量子线路和计算机设备，所述计算机设备包括后处理模块和优化器模块；
    所述参数化量子线路用于对n个量子比特的输入量子态进行变换处理，得到所述n个量子比特的输出量子态；其中，目标量子系统的哈密顿量在所述n个量子比特的输出量子态下的能量期望值，是所述哈密顿量分解得到的k个泡利字符串的能量期望值的求和结果，n为正整数，k为正整数；
    所述后处理模块用于采用神经网络对所述n个量子比特的输出量子态进行后处理，根据所述神经网络的后处理结果计算得到所述哈密顿量的能量期望值；
    所述优化器模块用于以所述哈密顿量的能量期望值收敛为目标，对所述参数化量子线路的参数和所述神经网络的参数进行调整；其中，在所述哈密顿量的能量期望值满足收敛条件的情况下，将满足所述收敛条件的所述哈密顿量的能量期望值，确定为所述目标量子系统的基态能量。
  16. 根据权利要求15所述的系统，其中，所述后处理模块包括：分解单元、测量单元和计算单元；
    所述分解单元，用于根据所述哈密顿量分解得到的泡利字符串和所述神经网络对应的后处理算符分解得到的泡利字符串，生成所述目标量子系统的等效哈密顿量所对应的多个泡利字符串；
    所述测量单元，用于对于所述多个泡利字符串中的每一个泡利字符串，通过测量得到所述n个量子比特的输出量子态在所述泡利字符串对应的测量基上的比特字符串；
    所述计算单元，用于根据所述多个泡利字符串分别对应的比特字符串，计算得到所述多个泡利字符串分别对应的能量期望值，根据所述多个泡利字符串分别对应的能量期望值计算所述哈密顿量的能量期望值。
  17. 根据权利要求15所述的系统，其中，所述系统还包括k组测量线路，所述后处理模块包括测量单元、神经网络单元和计算单元，所述k组测量线路和所述k个泡利字符串一一对应；
    所述目标泡利字符串对应的测量线路，用于对所述n个量子比特的输出量子态执行与所述目标泡利字符串相对应的变换处理，得到变换后的输出量子态；
    所述测量单元用于通过测量得到所述变换后的输出量子态在指定测量基上的比特字符串；
    所述神经网络单元用于通过所述神经网络根据所述比特字符串，输出用于计算所述目标泡利字符串的能量期望值的元数据；
    所述计算单元用于根据所述元数据计算得到所述目标泡利字符串的能量期望值，根据所述k个泡利字符串的能量期望值计算所述哈密顿量的能量期望值。
  18. 根据权利要求17所述的系统，其中，所述目标泡利字符串对应的测量线路包括除符号量子比特之外的非符号量子比特对应的量子门，以使得所述非符号量子比特在同一种测量基上进行测量；其中，所述符号量子比特是所述n个量子比特中与所述目标泡利字符串中的一个目标泡利算符对应的量子比特，所述符号量子比特对应的测量基根据所述符号量子比特在所述目标泡利字符串中对应的泡利算符确定。
  19. 根据权利要求18所述的系统，其中，所述计算单元用于按照如下公式计算所述目标泡利字符串的能量期望值
    Figure PCTCN2021124392-appb-100004
    Figure PCTCN2021124392-appb-100005
    其中，f表示所述神经网络，s 0表示所述符号量子比特对应的测量结果，s表示所述比特字符串，0s 1:n-1表示将所述比特字符串中所述符号量子比特对应的比特位设为0得到的比特字符串，
    Figure PCTCN2021124392-appb-100006
    表示将所述比特字符串中所述符号量子比特对应的比特位设为1且对其他比特位按照所述目标泡利字符串做相应的比特反转后得到的比特字符串。
  20. 根据权利要求18所述的系统，其中，在所述非符号量子比特在所述目标泡利字符串中对应的泡利算符，与所述同一种测量基对应的泡利算符相同的情况下，所述非符号量子比特对应的量子门采用所述非符号量子比特对应的测量结果所对应的符号进行等效代替。
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