We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.

翻译：我们提出了一种量子态层析成像的代数算法，该算法通过测量特定可观测量来估计底层密度矩阵的结构化元素。在低秩假设下，其余元素可仅通过标准数值线性代数运算获得。所提出的代数矩阵补全框架适用于一大类通用的低秩混合量子态，与现有先进方法相比，该框架在提供确定性恢复保证的同时具有更高的计算效率。