A method, equipment, medium, and product for calibrating preheated Hamiltonian quantities.

Abstract

This invention discloses a method, apparatus, medium, and product for calibrating a preheated Hamiltonian, relating to the field of quantum state tomography for many-body systems. This invention employs a combination of spherical tensor operator expansion and discrete Fourier transform to accurately extract the component proportions corresponding to specific magnetic quantum numbers in a many-body quantum state. When the Hamiltonian is in the form of a single-body or two-body operator, this invention can effectively distinguish the coefficients of various magnetic quantum number operators in different angular motion quantum spaces. Compared to previous multi-quantum coherence techniques, it can obtain more accurate information about the Hamiltonian, thus uniquely determining the form of the system's Hamiltonian. Furthermore, due to the non-mixing property of the spherical tensor during rotation, this invention greatly simplifies the tomographic process of many-body Hamiltonians.

Description

Technical Field

[0001] This invention relates to the field of quantum state tomography for many-body systems, and in particular to a method, apparatus, medium, and product for calibrating a preheated Hamiltonian. Background Technology

[0002] Quantum state tomography (QST) refers to the measurement method used to obtain the density operator (or density matrix) of a system, and it has wide applications in many fields such as quantum computing, quantum precision measurement, and quantum simulation. Traditional QST generally obtains the entire density matrix of the system by measuring different components of the density matrix through universal quantum control of the system. However, for the preheated state of a many-body system, since universal quantum control of the system is currently not possible, existing techniques cannot achieve complete state tomography of the preheated state.

[0003] Current methods for tomography of preheated states in solid-state NMR (nuclear magnetic resonance) systems primarily employ the multiple quantum coherence scheme. This involves conducting separate experiments in two orthogonal directions, such as the x and y directions, using a series of global rotation pulses to obtain different FID (Free Induction Decay) sequences. Fourier transforms are then used to obtain the relative intensity ratios of different orders of quantum coherence. Examples include the literature "Encoding multiple quantum coherences in non-commuting bases" and "Spin counting experiments in the dipolar-ordered state." While the multiple quantum relative intensity ratios can reflect the multiple quantum coherence intensity ratios of the Hamiltonian, these ratios cannot completely determine the form of the Hamiltonian. Summary of the Invention

[0004] To address the aforementioned problems in the existing technology, this invention provides a method, equipment, medium, and product for calibrating preheated Hamiltonian quantities.

[0005] To achieve the above objectives, the present invention provides the following solution:

[0006] A method for calibrating the Hamiltonian in a preheated state, the method comprising:

[0007] The preheated quantum density of states operator is obtained by using the spherical tensor operator expansion method; the preheated quantum density of states operator is the expression of the initial preheated state of the system to be solved in the spherical tensor basis;

[0008] Based on the preheated quantum state density operator, the Euler angles are rotated globally within a set rotation range according to a set interval using global pulses, and the final state density operator to be measured is obtained under the action of the global rotation; the Euler angles include a first angle, a second angle, and a third angle;

[0009] Based on the measured final state density, the experimental signal is obtained through inversion measurement in solid-state nuclear magnetic resonance to generate an experimental signal set;

[0010] Based on the experimental signal set, a discrete two-dimensional Fourier transform is performed on the first angle and the second angle to obtain a linear equation;

[0011] Solve or fit the linear equation to restore the component intensities of different spherical tensors in order to determine the Hamiltonian to be measured.

[0012] Optionally, the preheated quantum state density operator is represented as:

[0013]

[0014] in, Represents the preheated quantum state density operator. Represents the unit operator, β T It is the thermodynamic inverse temperature. This represents the Hamiltonian.

[0015] Optionally, the spherical tensor basis adopts an irreducible spherical tensor basis; the irreducible spherical tensor operator of the single unit contains three irreducible spherical tensor operators; the three irreducible spherical tensor operators are determined by the Pauli matrix corresponding to the spin lattice point.

[0016] Optionally, the set rotation interval is [0, 2π).

[0017] Optionally, based on the experimental signal set, a discrete two-dimensional Fourier transform is performed on the first angle and the second angle to obtain a linear equation, specifically including:

[0018] Perform discrete two-dimensional Fourier transforms on the first angle and the second angle to transform the first angle and the second angle into the frequency domain corresponding to the magnetic quantum number, so as to obtain a set of equations about the magnetic quantum number;

[0019] A set of equations concerning magnetic quantum numbers is used as the linear equations.

[0020] Optionally, the linear equation is expressed as:

[0021]

[0022] Among them, F m,m'B represents the intensity of the components with the first magnetic quantum number m and the second magnetic quantum number m' in the Fourier spectrum. l,m B represents the component intensity of the sphere tensor corresponding to the first magnetic quantum number m. l,-m' This represents the component intensity of the sphere tensor corresponding to the second magnetic quantum number m'. These are the coefficients of the Wigner-d matrix, and β represents the third angle.

[0023] Optionally, in the process of realizing the global rotation of Euler angles within a set rotation range according to a set interval by global pulses based on the preheated quantum density of states operator, the global rotation of solid nuclear spin is realized by radio frequency pulses of a solid nuclear magnetic resonance spectrometer.

[0024] A computer device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor executing the computer program to implement the steps of the preheated Hamiltonian calibration method described in any of the preceding claims.

[0025] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the preheated Hamiltonian calibration method described in any of the preceding claims.

[0026] A computer program product includes a computer program that, when executed by a processor, implements the steps of the preheated Hamiltonian calibration method described in any of the preceding claims.

[0027] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0028] This invention employs a combination of spherical tensor operator expansion and discrete Fourier transform to accurately extract the component proportions corresponding to specific magnetic quantum numbers in many-body quantum states. When the Hamiltonian is in the form of a single-body or two-body operator, this invention can effectively distinguish the coefficients of each magnetic quantum number operator in different angular motion quantum spaces. Compared to previous multi-quantum coherence techniques, it can obtain more accurate information about the Hamiltonian, thus uniquely determining the form of the system's Hamiltonian. Furthermore, due to the non-mixing property of the spherical tensor during rotation, this invention can greatly simplify the tomographic process of many-body Hamiltonians. Attached Figure Description

[0029] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0030] Figure 1This is a flowchart of the preheated Hamiltonian calibration method provided in Embodiment 1 of the present invention;

[0031] Figure 2 The flowchart illustrates the implementation of the preheated Hamiltonian calibration method provided in Embodiment 1 of the invention. Detailed Implementation

[0032] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] The purpose of this invention is to provide a method, equipment, medium, and product for calibrating the Hamiltonian in a preheated state, which aims to uniquely determine the form of the Hamiltonian in the system.

[0034] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0035] Example 1

[0036] This embodiment provides a method for calibrating the Hamiltonian in a preheated state, such as... Figure 1 and Figure 2 As shown, the method includes:

[0037] Step 100: Using spherical tensor operator expansion, the preheated quantum density of states operator is obtained. The preheated quantum density of states operator is the expression of the initial preheated state of the system to be solved in the spherical tensor basis, and it is expressed as:

[0038]

[0039] in, Represents the preheated quantum state density operator. Represents the unit operator, β T It is the thermodynamic inverse temperature. K represents, T represents The Hamiltonian is represented as:

[0040]

[0041] In the formula, J (i,j) The interaction strength between spin lattice point (i) and spin lattice point (j) is represented by the coefficients mentioned above. Let denote an irreducible spherical tensor operator (ISTO) of order l and magnetic quantum number m, composed of spin lattice points (i) and (j). The component intensities B of a series of different spherical tensors are measured. l,m That is, the initial preheated state and Hamiltonian of the system are obtained.

[0042] In practical applications, the spherical tensor basis in this embodiment can be an irreducible spherical tensor basis. A single irreducible spherical tensor operator comprises three irreducible spherical tensor operators. These three irreducible spherical tensor operators are determined by the Pauli matrix corresponding to the spin lattice points. For example:

[0043]

[0044] Where i is the imaginary unit, The decibel represents the Pauli matrix corresponding to the spin lattice point j (the subscripts x, y, z represent the three directions of the Pauli matrix), denoted as . For three irreducible spherical tensor operators.

[0045] Higher-order spherical tensor operators can be constructed using single-spherical tensor operators and CG coefficients. For example, a second-order spherical tensor operator is represented as:

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055] ISTOs have the property of being mutually orthogonal, that is:

[0056]

[0057] Where, δ ij,pq The Kronecker symbol is represented if and only if i = p, j = q. ij,pq =1, otherwise δ ij,pq =0.

[0058] Using ISTO, any quantum state can be expanded into the form of an irreducible spherical tensor, represented as:

[0059]

[0060] Step 101: Based on the preheated quantum density of states operator, a global rotation of Euler angles is achieved within a set rotation interval using global pulses, and the final density of states to be measured is obtained under the action of the global rotation. The Euler angles include a first angle α, a second angle γ, and a third angle B. The global rotation is represented as follows: The operator representing the final density of states to be measured is as follows:

[0061] In practical applications, the global rotation of solid-state nuclear spin in the irreducible spherical tensor and global rotation can be realized using radio frequency pulses from a solid-state nuclear magnetic resonance spectrometer, as follows:

[0062]

[0063] The response of the spherical tensor operator to global rotation is:

[0064]

[0065] in, The Wigner-D matrix can be decomposed into Wigner-small d matrices:

[0066]

[0067] part The following are examples:

[0068]

[0069]

[0070]

[0071]

[0072]

[0073]

[0074] Step 102: Based on the final density of the measured state, the experimental signal is obtained through inversion measurement in solid-state nuclear magnetic resonance to generate the experimental signal set. The experimental signal is represented as follows:

[0075]

[0076] in, This indicates that for two matrices Find the trace of the product. These are the coefficients of the Wigner-d matrix, which can be determined by consulting the Wigner-d matrix coefficient table. The value of is given by J = ∑ (i,j) [J (i,j) ] 2 , where J is the interaction strength (i,j) The sum of squares.

[0077] In practical applications, several groups of α, β, γ can be selected at equal intervals within the rotation interval [0, 2π) according to the accuracy requirements, and the above steps 101 and 102 can be repeated to obtain a series of experimental signals S(α, β, γ) as experimental signal groups.

[0078] Step 103: Perform discrete two-dimensional Fourier transforms on the first and second angles based on the experimental signal set to obtain linear equations.

[0079] That is, based on the set of experimental signals S(α,β,γ) obtained in step 102, a discrete two-dimensional Fourier transform is performed on the two variables α and γ to transform the angles α and γ into the frequency domain corresponding to the magnetic quantum numbers m and m'. The resulting set of equations about m and m' after the Fourier transform is denoted as follows:

[0080]

[0081] Among them, F m,m' It represents the intensity of the m,m' components in the Fourier spectrum.

[0082] Step 104: Solve or fit a linear equation to reconstruct the component intensities of different spherical tensors to determine the Hamiltonian to be measured. That is, by solving or fitting the linear equation obtained in step 103, arbitrary coefficients B can be reconstructed. l,m This allows us to obtain the Hamiltonian to be measured.

[0083] In practical applications, the signal readout process of a solid-state NMR system is as follows:

[0084] Solid-state nuclear magnetic resonance (NMR) measures NMR signals using transverse induction coils, and its readout operator is as follows:

[0085]

[0086] The system evolves under the influence of the Hamiltonian. The FID signal can be obtained by performing a spectral integral on the final state, and the result can be obtained by integrating the Fourier transform of the time-domain signal.

[0087]

[0088] That is, direct measurement of the final state extracts the total intensity of the single quantum coherence terms of the single-unit operator. This invention is based on the tomography of the preheated Hamiltonian achieved by inversion readout. The quantum state is in the form obtained based on preheating and high temperature approximation, as shown in formula (1).

[0089] The Hamiltonian can also be expanded using the spherical tensor operator, as shown in formula (2).

[0090] The following measurement signal can be obtained through inversion measurement:

[0091]

[0092] in, For the aforementioned global rotation, define J = ∑ (i,j) [J (i,j) ] 2 It contributes an overall coefficient to the signal.

[0093] By selecting three angles α, β, and γ at equal intervals, and performing Fourier transforms on α and γ, signals at different m and m' can be extracted. For example, formula (8) is about B. l,m B l,-m The linear equation system of ' can be used to represent different B's. l,m Solve it.

[0094] Based on the above description, it can be concluded that the density operator of any quantum state can be written as the sum of several irreducible spherical tensors of different types. Different types of spherical tensors have different response properties to global rotation. By using a series of global rotation pulses to obtain different FID sequences of the system, and then using Fourier transform, the coefficients of different spherical tensors of the system's Hamiltonian can be obtained. Furthermore, these coefficients can uniquely determine the form of the system's density operator. At the same time, due to the irreducible nature of spherical tensors, spherical tensors of different orders will not mix during rotation, which can greatly simplify the pulse sequence required for state tomography.

[0095] In summary, compared to existing technologies, this invention employs an irreducible spherical tensor operator expansion combined with discrete Fourier transform to accurately extract the component proportions corresponding to specific magnetic quantum numbers in many-body quantum states. Furthermore, by utilizing the irreducible spherical tensor scheme for tomography of the preheated state, the tomography of the system's Hamiltonian is completed. When the Hamiltonian is in the form of a single-body or two-body operator, this invention can effectively distinguish the coefficients of various magnetic quantum number operators in different angular motion quantum spaces, providing more accurate Hamiltonian information compared to previous multi-quantum coherence techniques. Moreover, due to the non-mixing property of the irreducible spherical tensor during rotation, this invention can greatly simplify the tomography process of the many-body Hamiltonian.

[0096] Example 2

[0097] A computer device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the preheated Hamiltonian calibration method in Embodiment 1.

[0098] Example 3

[0099] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the preheated Hamiltonian calibration method in Embodiment 1.

[0100] Example 4

[0101] A computer program product includes a computer program that, when executed by a processor, implements the steps of the preheated Hamiltonian calibration method in Embodiment 1.

[0102] Example 5

[0103] A computer device, which may be a database, includes a processor, memory, input / output (I / O) interfaces, and a communication interface. The processor, memory, and I / O interfaces are connected via a system bus, and the communication interface is also connected to the system bus via the I / O interfaces. The processor provides computational and control capabilities. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The database stores pending transactions. The I / O interfaces facilitate information exchange between the processor and external devices. The communication interface allows communication with external terminals via a network connection. When executed by the processor, the computer program implements the preheating Hamiltonian calibration method described in Embodiment 1.

[0104] It should be noted that the object information (including but not limited to object device information, object personal information, etc.) and data (including but not limited to data used for analysis, data stored, data displayed, etc.) involved in this invention are all information and data authorized by the object or fully authorized by all parties, and the collection, use and processing of related data must comply with the relevant laws, regulations and standards of the relevant countries and regions.

[0105] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments described above. Any references to memory, databases, or other media used in the embodiments provided by this invention can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can take many forms, such as Static Random Access Memory (SRAM) or Dynamic Random Access Memory (DRAM). The databases involved in the embodiments provided by this invention may include at least one type of relational database and non-relational database. Non-relational databases may include, but are not limited to, blockchain-based distributed databases. The processors involved in the embodiments provided by this invention may be general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic devices, quantum computing-based data processing logic devices, etc., and are not limited to these.

[0106] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0107] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Similar or identical parts between the various embodiments can be referred to mutually. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for calibrating the Hamiltonian in a preheated state, characterized in that, The method includes: The preheated quantum density of states operator is obtained by using the spherical tensor operator expansion method; the preheated quantum density of states operator is the expression of the initial preheated state of the system to be solved in the spherical tensor basis; Based on the preheated quantum state density operator, the Euler angles are rotated globally within a set rotation range according to a set interval using global pulses, and the final state density operator to be measured is obtained under the action of the global rotation; the Euler angles include a first angle, a second angle, and a third angle; Based on the measured final state density operator, the experimental signal is obtained through inversion measurement in solid-state nuclear magnetic resonance to generate an experimental signal set; Based on the experimental signal set, a discrete two-dimensional Fourier transform is performed on the first angle and the second angle to obtain a linear equation; Solve or fit the linear equation to restore the component intensities of different spherical tensors in order to determine the Hamiltonian to be measured.

2. The preheated Hamiltonian calibration method according to claim 1, characterized in that, The preheated quantum state density operator is represented as follows: in, Represents the preheated quantum state density operator. Represents the unit operator, β T It is the thermodynamic inverse temperature. This represents the Hamiltonian.

3. The preheated Hamiltonian calibration method according to claim 1, characterized in that, The spherical tensor basis adopts an irreducible spherical tensor basis; the irreducible spherical tensor operator of the single entity contains three irreducible spherical tensor operators; the three irreducible spherical tensor operators are determined by the Pauli matrix corresponding to the spin lattice point.

4. The method for calibrating the preheated Hamiltonian quantity according to claim 1, characterized in that, The set rotation interval is [0, 2π].

5. The method for calibrating the preheated Hamiltonian quantity according to claim 1, characterized in that, Based on the experimental signal set, a discrete two-dimensional Fourier transform is performed on the first angle and the second angle to obtain a linear equation, specifically including: Perform discrete two-dimensional Fourier transforms on the first angle and the second angle to transform the first angle and the second angle into the frequency domain corresponding to the magnetic quantum number, so as to obtain a set of equations about the magnetic quantum number; A set of equations concerning magnetic quantum numbers is used as the linear equations.

6. The preheated Hamiltonian calibration method according to claim 5, characterized in that, The linear equation is expressed as: Among them, F m,m' B represents the intensity of the components with the first magnetic quantum number m and the second magnetic quantum number m' in the Fourier spectrum. l,m B represents the component intensity of the sphere tensor corresponding to the first magnetic quantum number m. l,-m' This represents the component intensity of the sphere tensor corresponding to the second magnetic quantum number m'. These are the coefficients of the Wigner-d matrix, and β represents the third angle.

7. The method for calibrating the preheated Hamiltonian quantity according to claim 1, characterized in that, In the process of realizing the global rotation of Euler angles within a set rotation range according to a set interval by global pulses based on the preheated quantum density of states operator, the global rotation of solid nuclear spin is realized by radio frequency pulses of solid nuclear magnetic resonance spectrometer.

8. A computer device, comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that the processor executes the computer program to implement the steps of the preheated Hamiltonian calibration method according to any one of claims 1-7.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the steps of the preheated Hamiltonian calibration method according to any one of claims 1-7.

10. A computer program product, comprising a computer program, characterized in that, When executed by a processor, the computer program implements the steps of the preheated Hamiltonian calibration method according to any one of claims 1-7.

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