A Langevin sampler for quantum tomography

2026-01-13

AI Summary

This work proposes a Bayesian quantum tomography method based on Langevin sampling for the efficient classical reconstruction of low-rank quantum states. By employing the Burer–Monteiro factorization, the posterior distribution is constrained to the manifold of low-rank density matrices, and a complex-valued low-rank-promoting prior is introduced to enhance estimation efficiency. The approach enables scalable Bayesian inference in both known- and unknown-rank settings and establishes a PAC-Bayesian error bound that matches the optimal theoretical rate. When the target quantum state has low rank, the method achieves significantly improved computational scalability compared to existing techniques while attaining statistically optimal estimation accuracy.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

Quantum tomography involves obtaining a full classical description of a prepared quantum state from experimental results. We propose a Langevin sampler for quantum tomography, that relies on a new formulation of Bayesian quantum tomography exploiting the Burer-Monteiro factorization of Hermitian positive-semidefinite matrices. If the rank of the target density matrix is known, this formulation allows us to define a posterior distribution that is only supported on matrices whose rank is upper-bounded by the rank of the target density matrix. Conversely, if the target rank is unknown, any upper bound on the rank can be used by our algorithm, and the rank of the resulting posterior mean estimator is further reduced by the use of a low-rank promoting prior density. This prior density is a complex extension of the one proposed in (Annales de l'Institut Henri Poincare Probability and Statistics, 56(2):1465-1483, 2020). We derive a PAC-Bayesian bound on our proposed estimator that matches the best bounds available in the literature, and we show numerically that it leads to strong scalability improvements compared to existing techniques when the rank of the density matrix is known to be small.

Problem

• quantum tomography
• low-rank density matrix
• Bayesian inference
• rank estimation
• scalability

Innovation

• Langevin sampler
• Bayesian quantum tomography
• Burer-Monteiro factorization
• low-rank prior
• PAC-Bayesian bound