Quantum process chromatography method based on improved error incidence matrix calculation

Abstract

The invention relates to a quantum process chromatography method based on improved error incidence matrix calculation, which comprises the following steps of: linearly transforming an initial state density matrix into a group of position matrixes, recombining a corresponding final state density matrix according to the same proportion, and solving an analytical solution of an error incidence matrix by utilizing the property of the position matrixes, the error incidence matrix is expressed as a function of the recombination matrix, and finally the recombination matrix of the final state matrix is substituted into the recombination matrix to obtain the error incidence matrix. The problem that an existing quantum process chromatography implementation scheme is difficult to expand to a large-scale high-dimensional measurement subsystem is solved, the number of initial states needing to be prepared when quantum process chromatography is executed is remarkably reduced by improving a calculation method of an error incidence matrix, and the data processing difficulty is lowered; therefore, the quantum process chromatography can be conveniently expanded to a large-scale high-dimensional measurement subsystem, and the calculation efficiency of the quantum process chromatography is improved.

Description

Technical Field

[0001] The invention relates to the technical fields of quantum computing, quantum communication and quantum measurement, and in particular to a quantum process tomography method based on improved error correlation matrix calculation. Background Art

[0002] Quantum process tomography plays an important role in the field of quantum information. It provides a basic solution for inferring the evolution of quantum systems. When performing quantum process tomography on a quantum process, different initial states need to be prepared and let different initial states experience the same quantum process. Then, the density matrix of the initial state and the corresponding final state is obtained using quantum state tomography. The specific situation of the quantum process is analyzed based on the density matrix of each initial state and final state, and the result is expressed as an error correlation matrix.

[0003] In existing quantum process tomography implementation schemes, as the size and dimension of the quantum system increase, the number of initial states required to be prepared and the difficulty of solving the error correlation matrix increase significantly, greatly limiting the expansion of quantum process tomography technology to large-scale high-dimensional quantum systems. Summary of the invention

[0004] In view of the problem that the existing quantum process tomography implementation scheme is difficult to expand to large-scale high-dimensional quantum systems, a quantum process tomography method based on improved error correlation matrix calculation is proposed. By improving the calculation method of the error correlation matrix, the number of initial states required to prepare when performing quantum process tomography is significantly reduced and the difficulty of data processing is reduced, thereby making it easier for quantum process tomography to expand to large-scale high-dimensional quantum systems and improving the computational efficiency of quantum process tomography.

[0005] The technical solution of the present invention is:

[0006] A quantum process tomography method based on improved error correlation matrix calculation includes the following steps:

[0007] First, the initial state density matrix is ​​linearly transformed into a set of position matrices, and the corresponding final state density matrix is ​​reorganized in the same proportion. Then, the properties of the position matrix are used to obtain the analytical solution of the error correlation matrix, and the error correlation matrix is ​​expressed as a function of the reorganization matrix. Finally, the reorganization matrix of the final state matrix is ​​substituted into the reorganization matrix to obtain the error correlation matrix, completing the quantum process tomography.

[0008] Further, the specific steps include:

[0009] Step 1, obtain the initial state and final state density matrix: For the quantum process experienced by m n-dimensional quantum states, first prepare (mn) 2 The initial state is obtained by using quantum state tomography to find the density matrix of the initial state and the density matrix of the final state after the quantum process.

[0010] Step 2: Linearly transform the initial state density matrix into a position matrix: Linearly combine the initial state density matrix and transform it into (mn) 2 Each position matrix is ​​represented as a linear combination of all initial density matrices, and a set of transformation coefficients is obtained;

[0011] Step 3, recombining the final state density matrix: using the transformation coefficients obtained in step 2 to recombine all the final state density matrices, we get (mn) 2 A reorganization matrix;

[0012] Step 4, solve the error correlation matrix expression: select a set of evolution basis vectors, find the analytical solution of the error correlation matrix under the basis vectors, and express this analytical solution as a function of all reorganization matrices; then substitute the reorganization matrix obtained in step 3 into the function expression of the error correlation matrix, calculate the error correlation matrix, and complete the quantum process tomography.

[0013] Furthermore, it also includes step 5: calculating the fidelity of the process and its corresponding ideal process. The experimental result of the final state density matrix in step 3 can be replaced by the theoretical result corresponding to the ideal process, and then steps 3 and 4 are repeated to obtain the error correlation matrix of the ideal process. Finally, the two error correlation matrices are multiplied and solved to obtain the fidelity.

[0014] The beneficial effects of the present invention are:

[0015] First, in the traditional technical solution, the quantum process tomography of the quantum process experienced by m n-dimensional quantum states requires the number of initial states to be prepared (n) 2m , with the increase of m or n, the number of initial states required to be prepared increases significantly; however, the present invention reduces the number of initial states required to be prepared when implementing quantum process tomography to (mn) 2 , the number of initial states required to be prepared increases linearly with the increase of m or n. Therefore, when the present invention is used to perform quantum process tomography on a large-scale high-dimensional quantum system, the number of initial states required to be prepared is greatly reduced, and accordingly, the number of times the quantum state tomography of the initial state and the final state is performed is also greatly reduced.

[0016] Second, in the traditional technical solution, to solve the error correlation matrix of the quantum process experienced by m n-dimensional quantum states, the steps of directly using the density matrix of the initial state and the final state to solve the error correlation matrix are cumbersome and the calculation complexity is high. To solve the error correlation matrix of the quantum process experienced by m n-dimensional quantum states, it is necessary to solve (n) 2m Number(n) 4m System of linear equations and 2(n) 4m Number(n) 8mIn the scheme designed by the present invention, only (mn) needs to be solved 2 pcs(mn) 2 By solving a set of linear equations with different variables, we can obtain the error correlation matrix of the quantum process experienced by m n-dimensional quantum states.

[0017] The above advantages enable the present invention to break through the limitation that traditional technical solutions are only suitable for small-scale, low-dimensional quantum systems, thereby expanding quantum process tomography technology to large-scale, high-dimensional quantum systems. DETAILED DESCRIPTION

[0018] The present invention is described in detail below in conjunction with specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and a specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0019] A quantum process tomography method based on improved error correlation matrix calculation does not directly use the density matrices of the initial and final states of the quantum system to solve the error correlation matrix, but first linearly transforms the initial state density matrix into a set of position matrices, and reorganizes the corresponding final state density matrix in the same proportion, then uses the properties of the position matrix to obtain the analytical solution of the error correlation matrix, expresses the error correlation matrix as a function of the reorganization matrix, and finally substitutes the reorganization matrix of the final state matrix into the reorganization matrix to obtain the error correlation matrix.

[0020] To perform quantum process tomography on a real quantum process, for the quantum process experienced by m n-dimensional quantum states, we only need to prepare (mn) 2 initial state, rather than the (n) required in traditional technical solutions 2m indivual.

[0021] To calculate the theoretical value of the error correlation matrix of an ideal process corresponding to a real quantum process, we only need to solve (mn) 2 pcs(mn) 2 By solving a set of linear equations with different variables, we can obtain the error correlation matrix corresponding to the evolution of the quantum system composed of m n-dimensional quantum states.

[0022] The fidelity between a real quantum process and its corresponding ideal process is measured and calculated, the error correlation matrix between the real process and the ideal process is obtained, and then the product of the two error correlation matrices is obtained and the trace is obtained.

[0023] A quantum process tomography method based on improved error correlation matrix calculation specifically includes the following steps:

[0024] Step 1, obtain the initial state and final state density matrix: For the quantum process experienced by m n-dimensional quantum states, first prepare (mn) 2The initial state is obtained by using quantum state tomography to find the density matrix of the initial state and the density matrix of the final state after the quantum process.

[0025] Step 2: Linearly transform the initial state density matrix into a position matrix: Linearly combine the initial state density matrix and transform it into (mn) 2 Each position matrix is ​​represented as a linear combination of all initial density matrices, and a set of transformation coefficients is obtained;

[0026] Step 3, recombining the final state density matrix: using the transformation coefficients obtained in step 2 to recombine all the final state density matrices, we get (mn) 2 A reorganization matrix;

[0027] Step 4, solve the error correlation matrix expression: select a set of evolution basis vectors, find the analytical solution of the error correlation matrix under the basis vectors, and express this analytical solution as a function of all reorganization matrices; then substitute the reorganization matrix obtained in step 3 into the function expression of the error correlation matrix, calculate the error correlation matrix, and complete the quantum process tomography.

[0028] Step 5. If you need to further calculate the fidelity of the process and its corresponding ideal process, you can replace the experimental results of the final state density matrix in step 3 (real process) with the theoretical results corresponding to the ideal process, and then repeat steps 3 and 4 to obtain the error correlation matrix of the ideal process. Finally, multiply the two error correlation matrices and solve them to obtain the fidelity.

[0029] The above-mentioned embodiment only expresses one implementation mode of the present invention, and its description is relatively specific and detailed, but it cannot be understood as limiting the scope of the invention patent. It should be pointed out that for ordinary technicians in this field, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be based on the attached claims.

Claims

1. A quantum process tomography method based on improved error correlation matrix calculation, characterized in that: The steps include: First, the initial state density matrix is ​​linearly transformed into a set of position matrices, and the corresponding final state density matrix is ​​reorganized in the same proportion. Then, the properties of the position matrix are used to obtain the analytical solution of the error correlation matrix, and the error correlation matrix is ​​expressed as a function of the reorganization matrix. Finally, the reorganization matrix of the final state matrix is ​​substituted into the reorganization matrix to obtain the error correlation matrix, completing the quantum process tomography.

2. The quantum process tomography method based on improved error correlation matrix calculation according to claim 1, characterized in that: The specific steps are as follows: Step 1, obtain the initial state and final state density matrix: For the quantum process experienced by m n-dimensional quantum states, first prepare (mn) 2 The initial state is obtained by using quantum state tomography to find the density matrix of the initial state and the density matrix of the final state after the quantum process. Step 2: Linearly transform the initial state density matrix into a position matrix: Linearly combine the initial state density matrix and transform it into (mn) 2 Each position matrix is ​​represented as a linear combination of all initial density matrices, and a set of transformation coefficients is obtained; Step 3, recombining the final state density matrix: Use the transformation coefficients obtained in step 2 to recombine all the final state density matrices to obtain (mn) 2 A reorganization matrix; Step 4, solve the error correlation matrix expression: select a set of evolution basis vectors, find the analytical solution of the error correlation matrix under the basis vectors, and express this analytical solution as a function of all reorganization matrices; then substitute the reorganization matrix obtained in step 3 into the function expression of the error correlation matrix, calculate the error correlation matrix, and complete the quantum process tomography.

3. The quantum process tomography method based on improved error correlation matrix calculation according to claim 2, characterized in that: It also includes step 5: calculating the fidelity of the process and its corresponding ideal process. The experimental result of the final state density matrix in step 3 can be replaced by the theoretical result corresponding to the ideal process, and then steps 3 and 4 are repeated to obtain the error correlation matrix of the ideal process. Finally, the two error correlation matrices are multiplied and solved to obtain the fidelity.