An Efficient and Fast Quantum State Estimator With Sparse Disturbance

• Published in: IEEE transactions on cybernetics Vol. 49; no. 7; pp. 2546 - 2555
• Main Authors: Zhang, Jiaojiao, Cong, Shuang, Ling, Qing, Li, Kezhi
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.07.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Accuracy; Algorithms; Alternating direction method of multipliers (ADMM); Convergence; Convex functions; Density; Matrix methods; Optimization; Pollution measurement; Principal components analysis; Quantum computing; quantum state estimation (QSE); Quantum theory; Qubits (quantum computing); robust principal component analysis (RPCA); Robustness (mathematics); Sparse matrices; State estimation; Task analysis
• ISSN: 2168-2267; 2168-2275; 2168-2275
• DOI: 10.1109/TCYB.2018.2828498

A pure or nearly pure quantum state can be described as a low-rank density matrix, which is a positive semidefinite and unit-trace Hermitian. We consider the problem of recovering such a low-rank density matrix contaminated by sparse components, from a small set of linear measurements. This quantum state estimation task can be formulated as a robust principal component analysis (RPCA) problem subject to positive semidefinite and unit-trace Hermitian constraints. We propose an efficient and fast inexact alternating direction method of multipliers (I-ADMM), in which the subproblems are solved inexactly and hence have closed-form solutions. We prove global convergence of the proposed I-ADMM, and the theoretical result provides a guideline for parameter setting. Numerical experiments show that the proposed I-ADMM can recover state density matrices of 5 qubits on a laptop in 0.69 s, with <inline-formula> <tex-math notation="LaTeX">6 \times 10^{\mathbf {-4}} </tex-math></inline-formula> accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.