The Research On Quantum State Tomography Algorithm Based On Compressed Sensing

Quantum state tomography is the basis for the design and implementation of quantum computers,which aims to reconstruct an unknown quantum state through different measurements of multiple backups.The standard quantum state tomography technology requires an information complete measurement of the quantum state to be estimated.Therefore,the number of experimental settings and data processing time will increase exponentially with the number of qubits of the quantum system.Using compressed sensing,people can reconstruct the density matrix of the quantum state to be estimated efficiently with less measurements.In this way,quantum state tomography technology can be applied to high-dimensional quantum systems.This paper mainly studies the quantum state tomography problem based on compressed sensing under different noise conditions,and proposes effective density matrix reconstruction algorithms.This article mainly includes the following three aspects of research content:(1)First,the matrix truncation kernel norm is introduced to model the density matrix reconstruction problem.In the current research of quantum state tomography based on compressed sensing,people generally use the kernel norm of the matrix to approximate the rank norm of the matrix.Thus,the NP-hard non-convex optimization problem of minimizing the matrix rank norm is transformed into a convex optimization problem to solve.But the density matrix reconstructed by minimizing the kernel norm of the matrix may not be a good estimate of the original matrix.The kernel norm of a matrix is defined as the sum of all singular values,and the truncated kernel norm of the matrix considers the sum of the smallest singular values of the matrix.We try to estimate the density matrix by minimizing the truncated kernel norm of the matrix.(2)Secondly,for the different noises that appear in the quantum state tomography process,we respectively propose effective density matrix reconstruction algorithms.The noise includes the noise introduced from the preparation of the density matrix to the measurement and the noise introduced in the quantum measurement process.For different noise situations,we first transform the reconstruction problem into a convex optimization problem,and then use the alternating direction method of multipliers algorithm to solve it.Aiming at the quantum state constraint in the constraint condition,we decompose it into smaller sub-problems,and then use the singular value threshold algorithm and the iterative shrinking threshold algorithm to find an approximate solution.Therefore,the original density matrix can be reconstructed with higher accuracy efficiently.(3)Finally,we integrate the proposed quantum state tomography algorithm based on compressed sensing into the Qiskit platform of IBM.In this way,we can use the Qiskit plat- form to carry out quantum state tomography simulation experiments.Through experiments,we show the effectiveness of the algorithm.