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 6 × 10-4 accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.

中文翻译：

---

一种高效快速的稀疏扰动量子态估计器


纯或接近纯的量子态可以描述为低阶密度矩阵，它是半正定的单位迹厄米特矩阵。我们考虑从一小组线性测量中恢复这样一个被稀疏分量污染的低秩密度矩阵的问题。该量子状态估计任务可以表述为受正半定和单位迹埃尔米特约束影响的鲁棒主成分分析（RPCA）问题。我们提出了一种高效且快速的不精确交替方向乘子法（I-ADMM），其中子问题得到不精确的求解，因此具有封闭式解。我们证明了所提出的 I-ADMM 的全局收敛性，理论结果为参数设置提供了指导。数值实验表明，所提出的 I-ADMM 可以在 0.69 秒内在笔记本电脑上恢复 5 个量子位的状态密度矩阵，使用 30% 压缩传感测量，精度为 6 × 10-4（保真度为 99.38%），优于现有算法。