Tomography by Design: An Algebraic Approach to Low-Rank Quantum States

2026-02-16

AI Summary

This work addresses the high computational cost and lack of deterministic recovery guarantees in conventional methods for low-rank mixed quantum state tomography. The authors propose an efficient reconstruction framework based on algebraic matrix completion, which acquires structured entries of the density matrix through measurements of specific observables and leverages low-rank priors to achieve global reconstruction via standard numerical linear algebra operations. Compared to existing approaches, the proposed method significantly improves computational efficiency while maintaining high reconstruction accuracy. Notably, it provides the first rigorous deterministic recovery guarantees for a broad class of low-rank mixed quantum states, thereby establishing a theoretically sound foundation for scalable quantum state tomography under realistic experimental constraints.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• quantum state tomography
• low-rank
• density matrix
• matrix completion
• algebraic approach

Innovation

• quantum state tomography
• low-rank
• algebraic matrix completion
• density matrix
• deterministic recovery