Method and system for obtaining Gibbs state of quantum system
By using non-variant classic-quantum hybrid algorithm in quantum systems to set and solve the convex optimization problem, the problem of difficulty in optimizing when solving Gibbs states is solved, and the effect of efficiently calculating the global optimal solution is achieved.
Patent Information
- Application Number
- CN202311625674.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2023-11-30
- Publication Date
- 2025-05-30
AI Technical Summary
In the prior art, when solving the Gibbs state of the quantum system, the classical optimization part is difficult to optimize and is prone to fall into local minimum points, especially when there are many quantum circuit parameters.
Using a non-variant classic-quantum hybrid algorithm, the Hamiltonian of the target quantum system is obtained, and the initial state is evolved based on the Hamiltonian, the matrix elements of the Gramma matrix and Hamiltonian are measured, the convex optimization problem is set, and the convex optimization problem is solved by a classical computer to obtain the Gibbs state.
It realizes finding the global optimal solution in polynomial time, avoids the problem of falling into local minimum value points, and simplifies the time complexity of classical computer optimization.
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Figure CN120069114A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of quantum computing, and particularly relates to a method and system for obtaining the Gibbs state of a quantum system. Background Art
[0002] The Gibbs state is the thermodynamic equilibrium state and is the most important research object in statistical physics, with important applications in algorithm design, environmental science, and material design. In the prior art, the quantum algorithm that can solve the Gibbs state is the variational quantum algorithm. This algorithm is a classical-quantum hybrid algorithm. Its characteristics are that the required circuit depth is very shallow and error correction is not required. The idea of the variational algorithm is to parameterize the quantum circuit, optimize and update the parameters through a classical computer, and achieve the minimization of the free energy. It includes: 1. Using additional auxiliary qubits to store probability distribution information and obtaining samples of the probability distribution through measurement, so as to estimate the entropy on a classical computer; 2. Using a more complex circuit to directly estimate the entropy, combined with techniques such as Hamiltonian simulation, Fourier expansion of the logarithmic function of the quantum state, and quantum amplitude estimation.
[0003] In the prior art, the variational quantum algorithm is adopted, and its classical optimization is generally non-convex optimization, belonging to the NP-hard problem. When there are many quantum circuit parameters, the optimization difficulty is large, and it is easy to fall into local minimum points. And the quantum circuit parameters are often positively correlated with the expression ability. Summary of the Invention
[0004] In view of the above problems, the present invention proposes a method for obtaining the Gibbs state of a quantum system, including: obtaining the Hamiltonian of the target quantum system; evolving the initial state of the target quantum system based on the Hamiltonian to obtain a set of final states of the target quantum system; measuring the matrix elements of the Gram matrix corresponding to the final states and the matrix elements of the Hamiltonian to set a convex optimization problem; obtaining the optimal solution of the convex optimization problem; and obtaining the Gibbs state of the target quantum system based on the optimal solution.
[0005] In the method for obtaining the Gibbs state of a quantum system according to the present invention, the target quantum system is an n-qubit quantum system, having a density matrix Hamiltonian and the von Neumann entropy S = -tr(ρlog 2 ρ); the Gibbs state ρ Gibbs is the state corresponding to the lowest free energy F of the target quantum system, F = tr(ρH) - TS; by evolving the initial state of the target quantum system with an evolution time t j to obtain the final state {|ψ j >}, (j = 1, 2,... M), t j = jΔ t , Δ tis the preset unit evolution time, and M is the number of the final states.
[0006] The method for obtaining the Gibbs state of a quantum system according to the present invention, wherein based on {|ψi>}, the matrix elements G of the Gram matrix G are measured using a quantum computer ij and the matrix elements H of the Hamiltonian H ij , and the matrix elements G of G are obtained ij = <ψ i |ψ j ., H ij = <ψ i |H|ψ j >; an unitary matrix A is obtained, and A satisfies the diagonalization of G, G = AGA + ; a set of orthogonal bases |ψ' j > = ∑c j A ji |ψ i >; representing ρ = ∑β j > by |ψ' ij |ψ' i ><ψ' j |, then the linear combination form of H ij represents H' ij = ∑ k, l c k c l A ki A lj H kl , and tr(ρH) = ∑β ij H' ij ; G' is the diagonal matrix of G, c j , c k , c l are normalization coefficients, A ki , A lj are the matrix elements of A, and H kl are the matrix elements of H; setting the convex optimization problem as
[0007] The method for obtaining the Gibbs state of a quantum system according to the present invention solves the convex optimization problem through a classical computer to obtain the optimal solution β * , then the Gibbs state
[0008] The present invention also provides a system for obtaining the Gibbs state of a quantum system, including: an initialization module for obtaining the Hamiltonian of a target quantum system; a first solving module for evolving the initial state of the target quantum system with respect to the Hamiltonian to obtain a set of final states of the target quantum system; measuring the matrix elements of the Gram matrix corresponding to the final states and the matrix elements of the Hamiltonian to set a convex optimization problem; a second solving module for obtaining an optimal solution to the convex optimization problem; and obtaining the Gibbs state of the target quantum system based on the optimal solution.
[0009] The system for obtaining the Gibbs state of a quantum system according to the present invention is characterized in that the target quantum system is an n-bit quantum system with a density matrix Hamiltonian and the von Neumann entropy S = -tr(ρ log 2 ρ); the Gibbs state ρ Gibbs is the state corresponding to the lowest free energy F of the target quantum system, F = tr(ρH) - TS; the first solving module includes an evolution module for evolving the initial state of the target quantum system with an evolution time t j to obtain the final states {|ψj>}, (j = 1, 2,... M), where t j = jΔ t , Δ t is a preset unit evolution time and M is the number of the final states.
[0010] The system for obtaining the Gibbs state of a quantum system according to the present invention, wherein the first solving module further includes a setting module for setting the convex optimization problem; the setting module includes: a quantum measurement module for measuring the matrix elements G of the Gram matrix G and the matrix elements H of the Hamiltonian H ij based on {|ψi>} using a quantum computer to obtain the matrix elements G ij of G as G ij = <ψ i |ψ j >, H ij = <ψ i |H|ψ j >; a convex optimization problem setting module for setting the convex optimization problem; including obtaining a unitary matrix A that satisfies the diagonalization of G, G = AGA + ; generating a set of orthogonal bases |ψ' j > = ∑c j A ji |ψ i >; representing ρ = ∑β j |ψ' ij ><ψ' i | in terms of |ψ' j |, then H ijExpress H' in the form of a linear combination ij = ∑ k,l c k c l A ki A lj H kl , and tr(ρH) = ∑β ij H' ij ; G' is the diagonal matrix of G, c j , c k , c l are normalization coefficients, A ki , A lj are the matrix elements of A, H kl are the matrix elements of H; Set the convex optimization problem as
[0011] The method for obtaining the Gibbs state of a quantum system according to the present invention, wherein the second solving module includes: solving the convex optimization problem through a classical computer to obtain the optimal solution β * , then the Gibbs state
[0012] The present invention also proposes a computer-readable storage medium storing computer-executable instructions, characterized in that when the computer-executable instructions are executed, the method for obtaining the Gibbs state of a quantum system as described above is implemented.
[0013] The present invention also proposes a data processing device, including the computer-readable storage medium as described above. When the processor of the data processing device retrieves and executes the computer-executable instructions in the computer-readable storage medium, the Gibbs state of the target quantum system is obtained. BRIEF DESCRIPTION OF THE DRAWINGS
[0014] Figure 1 is a flowchart of the method for obtaining the Gibbs state of a quantum system according to the present invention.
[0015] Figure 2 is a schematic diagram of the data processing device according to the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS
[0016] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings. It should be understood that the specific implementation methods described herein are only used to explain the present invention and are not used to limit the present invention.
[0017] When the inventor was conducting research on Gibbs state preparation, it was found that all existing algorithms that can run on near-term quantum devices are variational quantum algorithms. When there are many quantum circuit parameters, it is difficult to optimize and it is easy to fall into local minimum points. Therefore, the inventor considered not using a variational classical-quantum hybrid algorithm. The inventor noticed that the free energy function is a convex function of the quantum state. Therefore, as long as a suitable set of ansatz quantum states is selected as the basis vectors and the expansion coefficients of the quantum state under this set of basis vectors are used as optimization variables, the problem of optimizing the free energy in a classical computer is transformed into a constrained convex optimization problem at this time. This greatly simplifies the time complexity of classical computer optimization. Compared with the variational quantum algorithm, assuming that the number of circuit parameters in the variational quantum algorithm is the same as the number of basis vectors of the ansatz quantum state in the present invention, the time complexity of classical computer optimization can be reduced from exponential level to polynomial level.
[0018] The purpose of the present invention is to solve the problem that the classical optimization part of the previous variational quantum algorithm for preparing the Gibbs state is difficult to optimize, and a new classical-quantum hybrid algorithm is proposed, and its classical optimization part is a convex optimization problem.
[0019] The present invention adopts a non-variational classical-quantum hybrid algorithm, does not adopt a parameterized circuit, and does not require parameter update. In addition, by optimizing the classical part into a convex optimization problem, the global optimal solution can be found in polynomial time and will not fall into local minimum points.
[0020] As Figure 1 shown, the method for obtaining the Gibbs state of a quantum system in the present invention specifically includes:
[0021] Consider an n-bit quantum system with a density matrix of The Gibbs state is the state corresponding to the lowest free energy F of the quantum system, where
[0022] F = tr(ρH) - TS (1)
[0023] H is the Hamiltonian of the quantum system, T is the temperature, and S = -tr(ρlog 2 ρ) is the von Neumann entropy of the quantum system. Considering a set of ansatz quantum states {|ψi>}, (j = 1, 2,..., M) as the basis vectors, the initial state can be evolved for t j = jΔ t to obtain, where Δ t is the unit evolution time selected artificially. The quantum circuit of the evolution process can be constructed using trotter decomposition.
[0024] After preparing {|ψi>}, the Hadamard Test technique can be used to measure on a quantum computer
[0025] G ij = <ψ i |ψ j >,H ij = <ψ i |H|ψ j > (2)
[0026] Since G is a Hermitian matrix, G can be diagonalized as:
[0027] G = AGA + (3)
[0028] G' is a diagonal matrix and A is a unitary matrix. After obtaining A, a set of orthogonal basis |ψ'> can be generated as: j > :
[0029] |ψ' j > = ∑c j A ji |ψ i > (4)
[0030] c j is the normalization coefficient. The density matrix ρ is expressed in the form composed of the orthogonal basis as:
[0031] ρ = ∑β ij |ψ' i ><ψ' j | (5)
[0032] Then the matrix representation H' of the Hamiltonian in this set of orthogonal basis ij can be expressed as a linear combination form of H ij as
[0033] H' ij = ∑ k,l c k c l A ki A lj H kl (6)
[0034] Thus, the first term of the free energy can be expressed as
[0035] tr(ρH) = ∑β ij H' ij (7)
[0036] The entropy can be expressed as
[0037] S = -tr(βlog 2 (β)) (8)
[0038] Finally, solve the following convex optimization problem on a classical computer
[0039]
[0040] Let the solution of the optimization problem be β * , then the Gibbs state is
[0041]
[0042] The overall algorithm flow is as follows:
[0043] Input: Initial value of variable β 0 , initial state of the quantum circuit
[0044]
[0045] 10. Diagonalize G according to formula (3) to obtain the expression of the orthogonal basis |ψ' i >
[0046] 11. Calculate H' according to formula (6) ij
[0047] 12. Solve the optimization problem formula (9) to obtain ρ Gibbs
[0048] Figure 2 is a schematic diagram of the data processing device of the present invention. As Figure 2 shown, the embodiment of the present invention also provides a computer-readable storage medium and a data processing device. The computer-readable storage medium of the present invention stores computer-executable instructions. When the computer-executable instructions are executed by the processor of the data processing device, the Gibbs state of the target quantum system is obtained. Those of ordinary skill in the art can understand that all or part of the steps in the above method can be completed by instructing relevant hardware (such as a processor, FPGA, ASIC, etc.) through a program. The program can be stored in a readable storage medium, such as a read-only memory, a disk, or an optical disc, etc. All or part of the steps of the above embodiments can also be implemented using one or more integrated circuits. Correspondingly, each module in the above embodiments can be implemented in the form of hardware, for example, by an integrated circuit to implement its corresponding function, or can be implemented in the form of a software function module, for example, by a processor executing a program / instruction stored in a memory to implement its corresponding function. The embodiments of the present invention are not limited to any specific form of the combination of hardware and software.
[0049] Compared with the prior art solution, in the method for obtaining the Gibbs state of a quantum system proposed by the present invention, by running and solving a convex optimization problem on a classical computer, the global optimal solution can be calculated more efficiently and will not fall into a local minimum point. In addition, a parameterized circuit is not used and parameter update is not required.
[0050] The above embodiments are only used to illustrate the present invention and are not intended to limit the present invention. Those of ordinary skill in the relevant technical field can also make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, all equivalent technical solutions also belong to the scope of the present invention. The patent protection scope of the present invention shall be defined by the claims.
Claims
1. A method for obtaining the Gibbs state of a quantum system, characterized in that, it includes: obtaining the Hamiltonian of the target quantum system; evolving the initial state of the target quantum system based on the Hamiltonian to obtain a set of final states of the target quantum system; measuring the matrix elements of the Gram matrix corresponding to the final states and the matrix elements of the Hamiltonian to set a convex optimization problem; obtaining the optimal solution of the convex optimization problem; obtaining the Gibbs state of the target quantum system based on the optimal solution.
2. The method for obtaining the Gibbs state of a quantum system according to claim 1, characterized in that, The target quantum system is an n - qubit quantum system, with a density matrix Hamiltonian and the von Neumann entropy S = -tr(ρ log 2 ρ); the Gibbs state ρ Gibbs is the state corresponding to the lowest free energy F of the target quantum system, where F = tr(ρH)-TS; By the initial state of the target quantum system evolve with the evolution time t j to obtain the final state {|ψ j >, (j = 1, 2, … M), t j = jΔ t , Δ t is the preset unit evolution time, and M is the number of the final states.
3. The method for obtaining the Gibbs state of a quantum system according to claim 2, characterized in that, Based on {|ψ i >}, use a quantum computer to measure the matrix element G ij of the Gram matrix G and the matrix element H ij of the Hamiltonian H, and obtain the matrix element G ij = <ψ i |ψ j >, H ij = <ψ i |H|ψ j >; Obtain the unitary matrix A, where A satisfies the diagonalization of G, G = AGA + ; Generate a set of orthogonal bases |ψ'> = ∑c j A j A ji |ψ i >; Represent ρ = ∑β j |ψ'><ψ' ij |ψ' i ><ψ' j |, then the linear combination form of H ij expresses H' ij = ∑ k,l c k c l A ki A lj H kl , and tr(ρH) = ∑β ij H' ij ; G' is the diagonal matrix of G, c j , c k , c l are normalization coefficients, A ki , A lj are the matrix elements of A, H kl are the matrix elements of H; Set the convex optimization problem as s.t. tr(β) = 1, β ≥ 0.
4. The method for obtaining the Gibbs state of a quantum system according to claim 3, characterized in that, Solving the convex optimization problem by a classical computer to obtain the optimal solution β * , then the Gibbs state 5. A system for obtaining the Gibbs state of a quantum system, characterized in that, it includes: an initialization module for obtaining the Hamiltonian of the target quantum system; a first solving module for evolving the initial state of the target quantum system based on the Hamiltonian to obtain a set of final states of the target quantum system; measuring the matrix elements of the Gram matrix corresponding to the final states and the matrix elements of the Hamiltonian to set a convex optimization problem; a second solving module for obtaining the optimal solution of the convex optimization problem; obtaining the Gibbs state of the target quantum system based on the optimal solution.
6. The system for obtaining the Gibbs state of a quantum system according to claim 5, characterized in that, The target quantum system is an n - qubit quantum system with a density matrix Hamiltonian and the von Neumann entropy S = -tr(ρ log 2 ρ); the Gibbs state ρ Gibbs is the state corresponding to the lowest free energy F of the target quantum system, where F = tr(ρH)-TS; The first calculation module includes an evolution module for evolving the initial state of the target quantum system with an evolution time t j to obtain the final state {|ψ j >}, (j = 1, 2, … M), where t j = jΔ t , Δ t is a preset unit evolution time and M is the number of the final states.
7. The system for obtaining the Gibbs state of a quantum system according to claim 6, characterized in that, the first solving module further includes a setting module for setting the convex optimization problem; the setting module includes: A quantum measurement module, for based on {|ψ i >}, using a quantum computer to measure the matrix elements G ij of the Gram matrix G and the matrix elements H ij of the Hamiltonian H, and obtaining the matrix element G ij = <ψ i |ψ j >, H ij = <ψ i |H|ψ j >; A convex optimization problem setting module for setting the convex optimization problem; including obtaining a unitary matrix A, where A satisfies the diagonalization of G, G = AGA + ; generating an orthonormal basis |ψ' j > = ∑c j A ji |ψ i >; representing ρ = ∑β j |ψ' ij ><ψ' i |, then the linear combination form of H j represents H' ij = ∑ ij c k,l c k c l A ki A lj H kl , and tr(ρH) = ∑β ij H' ij ; G' is the diagonal matrix of G, c j , c k , c l are normalization coefficients, A ki , A lj are matrix elements of A, H kl are matrix elements of H; setting the convex optimization problem as s.t. tr(β) = 1, β ≥ 0.
8. The method for obtaining the Gibbs state of a quantum system according to claim 7, characterized in that, The second solving module includes: solving the convex optimization problem by a classical computer to obtain the optimal solution β * , then the Gibbs state 9. A computer-readable storage medium storing computer-executable instructions, characterized in that, when the computer-executable instructions are executed, the method for obtaining the Gibbs state of a quantum system according to any one of claims 1 to 4 is implemented.
10. A data processing device includes the computer-readable storage medium according to claim 9. When the processor of the data processing device retrieves and executes the computer-executable instructions in the computer-readable storage medium, the Gibbs state of a quantum system is obtained.