Method and device for determining ground state energy of target molecular system

Abstract

The invention provides a method and device for determining the ground state energy of a target molecular system.The method for determining the ground state energy of the target molecular system.The method comprises the steps that the first Hamiltonian amount of the target molecular system is determined according to the Born Olympic approximation method; based on a preset mass ratio parameter of electrons and atomic nucleuses in the target molecular system and the first Hamiltonian, determining a first correction term through preset perturbation expansion, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian; and determining ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and a preset variable component sub-line.

Description

Technical Field

[0001] The present invention relates to the field of quantum chemistry, and in particular to a method and device for determining the ground state energy of a target molecular system. Background Art

[0002] The Born-Oppenheimer approximation (BOA), also known as the Born-Oppenheimer hypothesis, is a crucial approximation in quantum chemistry and molecular physics. This approximation assumes that the motion of electrons and nuclei can be considered separately, since nuclei are much more massive than electrons and therefore move much more slowly. This significant difference in mass and velocity allows electrons to quickly achieve a stable quantum state against a relatively stationary background of nuclei. However, at high energies, temperatures, or under strongly correlated conditions, nuclear quantum effects can become significant, causing errors in the Born-Oppenheimer (BOA) approximation to become non-negligible. Furthermore, as the precision of observations of molecular systems continues to improve, even small approximate errors can lead to observable deviations.

[0003] Currently, on quantum devices in the noisy intermediate-scale quantum era (NISQ), traditional quantum algorithms often directly assume the Born-Oppenheimer approximation to simplify the analysis of the Hamiltonian and wave functions of molecular systems. Consequently, the accuracy of these methods is limited by the Born-Oppenheimer approximation. As the requirements for observational precision in molecular systems continue to increase, the advantages of quantum algorithms over classical ones are becoming increasingly difficult to demonstrate.

[0004] Therefore, a method and device for determining the ground state energy of a target molecular system are needed. Summary of the Invention

[0005] The purpose of the present invention is to provide a method and device for determining the ground state energy of a target molecular system. By combining classical optimization problems and quantum eigenvalue solving algorithms through quantum variational methods, accurate simulation can be achieved on NISQ devices, which is more operational and feasible than fault-tolerant quantum algorithms.

[0006] In a first aspect, the present invention provides a method for determining the ground state energy of a target molecular system, comprising:

[0007] determining a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determining a first correction term through a preset perturbation expansion based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0008] The ground state energy corresponding to the second Hamiltonian is determined according to the second Hamiltonian and a preset variational quantum circuit.

[0009] Specifically, the variational quantum circuit includes a first number of working qubits and a second number of auxiliary qubits; and determining the ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and the preset variational quantum circuit includes:

[0010] Based on the preset test parameters, the working qubit is prepared into a test quantum state, and after applying the Hadamard gate to the auxiliary qubit, a controllable gate determined according to the second Hamiltonian is applied to the working qubit. Gate, where s is the binary representation of the auxiliary bit quantum basis vector, H is the second Hamiltonian, and i is the imaginary unit; an inverse Fourier transform operation QFT is applied to the auxiliary quantum bit. -1 , based on the measurement result of the working quantum bit, determine the ground state energy corresponding to the second Hamiltonian.

[0011] Specifically, the first correction term is expressed as:

[0012]

[0013] Wherein, λ is the momentum term of the nuclei in the target molecular system, M is the mass matrix, is the correction matrix, is the reduced Planck constant, is the imaginary unit, is the quality parameter, is the gradient operator.

[0014] Specifically, the correction matrix Expressed as:

[0015]

[0016] in, represents the quantum state, represents the first-order correction to the quantum state, i and j represent the serial number of the quantum state, R represents the coordinate of the atomic nucleus position, and r represents the coordinate of the electron position. is the first Hamiltonian, is the zero-order eigenvalue of the first Hamiltonian, is the volume element of three-dimensional space, is the correction matrix of quantum state i and quantum state j.

[0017] Specifically, the target molecule system includes: a target molecule system within a preset first temperature range.

[0018] Preferably, the method is performed by a mesoscale quantum device.

[0019] In a second aspect, the present invention provides a device for determining the ground state energy of a target molecular system, comprising:

[0020] a correction unit configured to determine a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determine a first correction term through a preset perturbation expansion based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian; and apply the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0021] The computing unit is configured to determine a ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and a preset variational quantum circuit.

[0022] In a third aspect, the present invention provides a storage medium for storing a program, wherein the program performs the following operations when executed:

[0023] determining a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determining a first correction term through a preset perturbation expansion based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0024] The ground state energy corresponding to the second Hamiltonian is determined according to the second Hamiltonian and a preset variational quantum circuit.

[0025] In a fourth aspect, the present invention provides a computing device comprising a quantum processor and a classical processor, wherein:

[0026] The classical processor is configured to determine a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determine a first correction term based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian through a preset perturbation expansion; and apply the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0027] The quantum processor is used to determine the ground state energy corresponding to the second Hamiltonian based on the second Hamiltonian and a preset variational quantum circuit.

[0028] Compared to existing technologies, this invention offers the following advantages: It utilizes a quantum variational algorithm for molecular dynamics simulations, employing an algorithm that goes beyond the BOA approximation for the nucleus' mass term, achieving higher accuracy than conventional simulation algorithms. Furthermore, the framework employed in this invention can be extended to dynamic calculations of interactions between light and heavy nuclei, as well as analysis of the dynamical behavior of nuclei in strongly correlated systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 A flow chart of a method for determining the ground state energy of a target molecular system provided by an embodiment of the present invention; Figure 2 A circuit diagram of the variational quantum algorithm (VQA) provided by an embodiment of the present invention; Figure 3 A structural diagram of a device for determining the ground state energy of a target molecular system provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0030] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments.

[0031] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be described below with reference to the accompanying drawings. It should be noted that similar reference numerals and letters represent similar items in the following drawings. Therefore, once an item is defined in one drawing, it does not need to be further defined or explained in subsequent drawings.

[0032] In the description of the embodiments of the present invention, words such as "exemplary," "for example," or "for example" are used to indicate examples, illustrations, or explanations. Any embodiment or design described as "exemplary," "for example," or "for example" in the embodiments of the present invention should not be construed as preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary," "for example," or "for example" is intended to present the relevant concepts in a concrete manner.

[0033] In molecular dynamics, the simulation of atomic nuclei's dynamics is a crucial issue. This technique involves simulating the dynamic behavior of atomic nuclei in molecular systems and calculating the ground-state energy and energy spectrum structure of molecules. This technique reveals the molecular structure under different conditions and helps us understand and predict the chemical properties of molecules. Therefore, it has broad application prospects in fields such as material synthesis, drug development, and catalyst design. For example, the calculation of ground-state energy can help predict the binding ability between drug molecules and biomolecules, thereby guiding the direction and strategy of drug design and accelerating the development of new drugs.

[0034] In practice, molecular dynamics simulations often use the Born-Oppenheimer approximation. This approximation divides the molecular system into two components: electrons and nuclei. It treats electron motion as rapid and adiabatically fixes the electron state at each instant of the nucleus's position. This simplifies the dynamical behavior of the nucleus to its evolution under a given electron density distribution, making it easier to solve using classical numerical methods. However, in some cases, particularly for simulations of large molecular systems or high-precision calculations, the Born-Oppenheimer approximation is insufficiently accurate, necessitating the combination of other techniques and methods to improve the accuracy and applicability of simulations.

[0035] The Born-Oppenheimer approximation is a key approximation in quantum chemistry, used to account for the coupling between electrons and nuclei in molecular systems. Because the mass difference between nuclei and electrons is approximately three orders of magnitude, this approximation assumes that the motion of nuclei is very slow relative to that of electrons. This allows the Schrödinger equation for molecular systems to be decomposed into two components, the electron and nucleus, with the nucleus' position treated as a parameter to solve for the electron's orbital motion. Furthermore, the electron's energy spectrum depends on the nuclear position parameter and can be analyzed as an equivalent potential energy surface for the nucleus, allowing for adiabatic solutions to the nuclear dynamics and energy spectrum. This approximation is widely applicable to most molecular systems, particularly small and medium-sized molecules and those operating at low temperatures, where its calculations offer good accuracy. However, for large molecules or strongly correlated systems, the Born-Oppenheimer approximation's quasiclassical treatment of nuclei can lead to significant errors, necessitating the use of methods beyond the Born-Oppenheimer approximation for analytical and numerical calculations. Existing exact factorization methods can formally decompose the time-dependent wave functions of atomic nuclei and electrons in molecules. Within this framework, the correction terms of the Born-Oppenheimer approximation can be processed using the perturbation expansion method, thus theoretically achieving higher-precision numerical simulations.

[0036] Quantum computers have a natural advantage in simulating quantum systems and can be applied to the dynamics of small molecule systems. However, simulating the dynamical evolution of the entire molecular Hamiltonian requires high depth and complexity of quantum circuits, making it difficult to implement on current noisy intermediate-scale quantum (NISQ) quantum hardware. Therefore, variational methods are often used to assist quantum computers in parameter optimization using classical algorithms. Quantum computers provide suitable ansatz (ansatz) algorithms that can efficiently solve for the expected value of quantum operators, achieving acceleration relative to classical algorithms.

[0037] To overcome the shortcomings of existing technologies, a method and apparatus for determining the ground-state energy of a target molecular system are proposed. This method employs quantum algorithms to achieve molecular dynamics simulations that go beyond the Born-Oppenheimer approximation. By combining classical optimization problems with quantum eigenvalue solvers (VQE) through quantum variational methods, it enables accurate simulations on NISQ devices. Compared to fault-tolerant quantum algorithms, this method offers greater operability and feasibility, achieving solution accuracy exceeding that of conventional methods.

[0038] Figure 1 Flowchart of a method for determining the ground state energy of a target molecular system provided by an embodiment of the present invention. Figure 1 As shown, the method comprises at least the following steps:

[0039] S101: Determine a first Hamiltonian of a target molecular system according to the Born-Oppenheimer approximation; determine a first correction term through a preset perturbation expansion based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian, and apply the first correction term to the first Hamiltonian to obtain a second Hamiltonian.

[0040] In one embodiment, the first correction term can be expressed as:

[0041]

[0042] Wherein, λ is the momentum term of the nuclei in the target molecular system, M is the mass matrix, is the correction matrix, is the reduced Planck constant, is the imaginary unit, is the quality parameter, is the gradient operator.

[0043] In a specific embodiment, the correction matrix It can be expressed as:

[0044]

[0045] in represents the quantum state, represents the first-order correction to the quantum state, i and j represent the serial number of the quantum state, R represents the coordinate of the atomic nucleus position, and r represents the coordinate of the electron position. is the first Hamiltonian, is the zero-order eigenvalue of the first Hamiltonian, is the volume element of three-dimensional space, is the correction matrix of quantum state i and quantum state j.

[0046] For example, an embodiment of the present invention is based on a perturbation expansion of exact factorization to achieve a solution to the ground state energy eigenvalue with an accuracy beyond the Born-Oppenheimer approximation. In the exact decomposition, the total wave function of the system is expressed as a function product of the electron and nuclear degrees of freedom, and the expressions of the Hamiltonian and the wave function are constructed through some precise mathematical relationships. Unlike the BO approximation, which treats the motion of electrons and nuclei separately, the exact decomposition emphasizes the overall correlation of the system and can handle electron-nuclear interactions without strict separation. In the Born-Oppenheimer approximation, it is assumed that the mass of the nucleus is much greater than the mass of the electron, so the kinetic energy of the nucleus can be ignored. In the perturbation method, for example, a small amount of To reduce the mass of the nucleus, specifically it can be expressed as the mass ratio of the electron and the nucleus, that is, when When the mass of the nucleus approaches infinity, the strict and exact decomposition returns to the lowest order result given by the Born-Oppenheimer approximation; and when the nucleus has a finite mass, the energy and wave function are By expanding it to different powers, we can get the results of the perturbation expansion. After retaining the lowest order correction, substitute , which is equivalent to the nucleus taking the actual physical value. The correction calculated by this perturbation expansion method consists of two parts: the diagonal Born-Oppenheimer correction term (DBOC) and the correction term for the nuclear mass. The correction for the nuclear mass is caused by the non-zero mass of the electron in the molecule. Its lowest-order correction can be expressed as:

[0047]

[0048] Wherein, λ is the momentum term of the nuclei in the target molecular system, M is the mass matrix, is the correction matrix, is the reduced Planck constant, is the imaginary unit, is the quality parameter, is the gradient operator.

[0049] Correction Matrix It can be expressed as:

[0050]

[0051] in, represents the quantum state, represents the first-order correction to the quantum state, i and j represent the serial number of the quantum state, R represents the coordinate of the atomic nucleus position, and r represents the coordinate of the electron position. is the first Hamiltonian, is the zero-order eigenvalue of the first Hamiltonian, is the volume element of three-dimensional space, is the correction matrix of quantum state i and quantum state j. The corrected second Hamiltonian is obtained by calculating the above formula.

[0052] S102: Determine a ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and a preset variational quantum circuit.

[0053] In one embodiment, the variational quantum circuit may include a first number of working qubits and a second number of auxiliary qubits. Furthermore, according to the second Hamiltonian and the preset variational quantum circuit, the specific method of determining the ground state energy corresponding to the second Hamiltonian may include the following process: based on preset test parameters, preparing the working qubit into a test quantum state, and, after applying a Hadamard gate to the auxiliary qubit, applying a controllable gate determined according to the second Hamiltonian to the working qubit. Gate, where s is the binary value of the auxiliary bit quantum basis vector, H is the second Hamiltonian, and i is the imaginary unit; an inverse Fourier transform operation QFT is applied to the auxiliary quantum bit. -1 , performing classical optimization on the measurement results of the working quantum bit to determine the ground state energy corresponding to the second Hamiltonian.

[0054] Figure 2 The variational quantum algorithm (VQA) circuit diagram provided by the embodiment of the present invention. Figure 2 As shown in the figure, the quantum circuit consists of two parts of quantum bits: n working bits and t auxiliary bits. To construct the test quantum state. Here is a parameter-dependent Unitary operations are used to generate specific quantum states, which will be used for subsequent energy calculations. Auxiliary bits: The initial state is usually , used to store phase information. First, a Hadamard gate is applied to the t auxiliary bits ( ), remove them from state into a uniform superposition state Substitute the corrected second Hamiltonian calculated by S101 into the controllable Door. Controllable Door (Controlled- gate) is an extended form of the basic quantum gate. gate, where s is the binary representation of the quantum basis vector of the auxiliary bit, H is the second Hamiltonian, and i is the imaginary unit. Then the quantum phase estimation (QPE) operation is performed. This involves State application controlled Operation, where is a parameterized operation on the working bit. This process encodes phase information related to the working bit state on the auxiliary bit. Finally, the inverse quantum Fourier transform (QFT) is applied to the auxiliary bit. This operation converts the state of the auxiliary bit into a binary representation of the phase, from which the phase information can be extracted. Through quantum phase estimation, the phase information related to the test quantum state can be obtained. This phase information can be further converted into energy expectation values. After obtaining the energy expectation value, the classical algorithm is used to optimize the Ansatz Parameters The goal of optimization is to minimize the expected value of energy. and recalculate the energy expectation value), and finally the ground state energy corresponding to the second Hamiltonian can be calculated, that is, the lowest energy state of the system.

[0055] In different scenarios, the target molecular system can be a target molecular system under high temperature, high energy or other conditions of strong correlation between atomic nuclei and electrons. In a specific embodiment, the target molecular system can include: a target molecular system within a preset first temperature range.

[0056] In different embodiments, the specific quantum device used to perform the method may vary. In one specific embodiment, the method may be performed by a medium-scale quantum device (NISQ).

[0057] According to yet another embodiment, an apparatus for determining a ground state energy of a target molecular system is provided. Figure 3 A structural diagram of a device for determining the ground state energy of a target molecular system provided by an embodiment of the present invention, such as Figure 3 As shown, the device 300 includes:

[0058] The correction unit 301 is configured to determine a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determine a first correction term based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian through a preset perturbation expansion; and apply the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0059] The calculation unit 302 is configured to determine the ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and a preset variational quantum circuit.

[0060] In one embodiment, the present invention provides a storage medium for storing a program, wherein the program performs the following operations when executed:

[0061] determining a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determining a first correction term through a preset perturbation expansion based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0062] The ground state energy corresponding to the second Hamiltonian is determined according to the second Hamiltonian and a preset variational quantum circuit.

[0063] In one embodiment, the present invention provides a computing device comprising a quantum processor and a classical processor, wherein:

[0064] The classical processor is configured to determine a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determine a first correction term based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian through a preset perturbation expansion; and apply the first correction term to the first Hamiltonian to obtain a second Hamiltonian;

[0065] The quantum processor is used to determine the ground state energy corresponding to the second Hamiltonian based on the second Hamiltonian and a preset variational quantum circuit.

[0066] It is understood that the method steps in the embodiments of the present invention can be implemented by hardware or by a processor executing software instructions. The software instructions can be composed of corresponding software modules, which can be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, hard disks, removable hard disks, CD-ROMs, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor so that the processor can read information from the storage medium and write information to the storage medium. Of course, the storage medium can also be an integral part of the processor. The processor and storage medium can be located in an ASIC.

[0067] In the above embodiments, all or part of the embodiments can be implemented using software, hardware, firmware, or any combination thereof. When implemented using software, all or part of the embodiments can be implemented in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the processes or functions described in accordance with the embodiments of the present invention are fully or partially generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted via the computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, optical fiber, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that can be accessed by a computer, or a data storage device such as a server or data center that integrates one or more available media. The available media can be magnetic media (e.g., floppy disks, hard disks, tapes), optical media (e.g., DVDs), or semiconductor media (e.g., solid-state drives (SSDs)).

[0068] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for determining the ground state energy of a target molecular system, wherein: include: Determine the first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; Based on a preset mass ratio parameter of electrons to atomic nuclei in the target molecular system and the first Hamiltonian, determining a first correction term through a preset perturbation expansion, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian; The ground state energy corresponding to the second Hamiltonian is determined according to the second Hamiltonian and a preset variational quantum circuit.

2. The method according to claim 1, wherein The variational quantum circuit includes a first number of working qubits and a second number of auxiliary qubits; and determining the ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and the preset variational quantum circuit includes: Based on the preset test parameters, the working qubit is prepared into a test quantum state, and after applying the Hadamard gate to the auxiliary qubit, a controllable gate determined according to the second Hamiltonian is applied to the working qubit. Gate, where s is the binary representation of the auxiliary bit quantum basis vector, H is the second Hamiltonian, and i is the imaginary unit; an inverse Fourier transform operation QFT is applied to the auxiliary quantum bit. -1 , based on the measurement result of the working quantum bit, determine the ground state energy corresponding to the second Hamiltonian.

3. The method according to claim 1, wherein The first correction term is expressed as: Wherein, λ is the momentum term of the nuclei in the target molecular system, M is the mass matrix, is the correction matrix, is the reduced Planck constant, is the imaginary unit, is the quality parameter, is the gradient operator.

4. The method according to claim 3, wherein the correction matrix Expressed as: in, represents the quantum state, represents the first-order correction to the quantum state, i and j represent the serial number of the quantum state, R represents the coordinate of the atomic nucleus position, and r represents the coordinate of the electron position. is the first Hamiltonian, is the zero-order eigenvalue of the first Hamiltonian, is the volume element of three-dimensional space, is the correction matrix of quantum state i and quantum state j.

5. The method according to claim 1, wherein The target molecule system includes: a target molecule system within a preset first temperature range.

6. The method according to claim 1, wherein The method is performed by a mesoscale quantum device.

7. A device for determining the ground state energy of a target molecular system, wherein: include: a correction unit configured to determine a first Hamiltonian of a target molecular system according to the Born-Oppenheimer approximation; Based on a preset mass ratio parameter of electrons to atomic nuclei in the target molecular system and the first Hamiltonian, determining a first correction term through a preset perturbation expansion, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian; The computing unit is configured to determine a ground state energy corresponding to the second Hamiltonian according to the second Hamiltonian and a preset variational quantum circuit.

8. A storage medium for storing a program, wherein the program performs the following operations when executed: determining a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determining a first correction term through a preset perturbation expansion based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian, and applying the first correction term to the first Hamiltonian to obtain a second Hamiltonian; The ground state energy corresponding to the second Hamiltonian is determined according to the second Hamiltonian and a preset variational quantum circuit.

9. A computing device comprising a quantum processor and a classical processor, wherein: The classical processor is configured to determine a first Hamiltonian of the target molecular system according to the Born-Oppenheimer approximation; determine a first correction term based on a preset mass ratio parameter of electrons to nuclei in the target molecular system and the first Hamiltonian through a preset perturbation expansion; and apply the first correction term to the first Hamiltonian to obtain a second Hamiltonian; The quantum processor is used to determine the ground state energy corresponding to the second Hamiltonian based on the second Hamiltonian and a preset variational quantum circuit.

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