Tomography by Design: An Algebraic Approach to Low-Rank Quantum States
This work addresses the challenge of efficiently reconstructing low-rank quantum states, which is crucial for quantum computing and information processing, though it appears incremental as it builds on existing algebraic and matrix completion techniques.
The authors tackled the problem of quantum state tomography by introducing an algebraic algorithm that uses measurements of specific observables to estimate structured entries of the density matrix, achieving deterministic recovery guarantees and computational efficiency compared to state-of-the-art methods.
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.