Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Langevin sampler for quantum tomography

Tameem Adel, Abhishek Agarwal, Stéphane Chrétien, Estelle Massart, Danila Mokeev, Ivan Rungger, Andrew Thompson

TL;DR

The paper introduces a Langevin-sampling approach for quantum state tomography that leverages a low-rank Burer-Monteiro factorization to reduce the parameter space from $d^2$ to $dr$, enabling scalable Bayesian tomography for small-rank density matrices. A novel complex spectral scaled Student’s $t$ prior promotes low rank, and a PAC-Bayesian bound shows the estimator achieves competitive rates with existing methods. Empirical results show that, when the target rank is small, the BM-Langevin sampler matches or surpasses the prob-estimator in accuracy while offering faster convergence and better scalability, especially with rank-aware priors and efficient real-valued gradient computations. The work provides a practical framework for high-dimensional quantum tomography with provable generalization guarantees and favorable computational properties, facilitating tomography for larger quantum systems. Overall, the approach advances scalable, uncertainty-aware quantum tomography by combining low-rank factorization, gradient-based sampling, and rigorous generalization bounds.

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.

A Langevin sampler for quantum tomography

TL;DR

The paper introduces a Langevin-sampling approach for quantum state tomography that leverages a low-rank Burer-Monteiro factorization to reduce the parameter space from to , enabling scalable Bayesian tomography for small-rank density matrices. A novel complex spectral scaled Student’s prior promotes low rank, and a PAC-Bayesian bound shows the estimator achieves competitive rates with existing methods. Empirical results show that, when the target rank is small, the BM-Langevin sampler matches or surpasses the prob-estimator in accuracy while offering faster convergence and better scalability, especially with rank-aware priors and efficient real-valued gradient computations. The work provides a practical framework for high-dimensional quantum tomography with provable generalization guarantees and favorable computational properties, facilitating tomography for larger quantum systems. Overall, the approach advances scalable, uncertainty-aware quantum tomography by combining low-rank factorization, gradient-based sampling, and rigorous generalization bounds.

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.
Paper Structure (19 sections, 7 theorems, 77 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 7 theorems, 77 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Abstract:
  2. Introduction
  3. Related works
  4. Contributions
  5. Bayesian quantum tomography
  6. A Burer-Monteiro formulation for Bayesian quantum tomography
  7. Implementation and numerical results
  8. Algorithmic implementation
  9. Data generation:
  10. Algorithms considered:
  11. Error measure:
  12. Comparison of our proposed BM-estimators with the prob-estimator
  13. Conclusions
  14. Appendix
  15. Proof of Lemma 1
  16. ...and 4 more sections

Key Result

Lemma 1

If $Y$ is a random $d \times r$ complex matrix having as density the function $\nu_{\theta}$, then the column vectors $y_i \in \mathbb{C}^d$ of $Y$, for $i = 1, \dots, r$, follow the $d$-variate complex scaled Student's $t$-distribution $(\sqrt{2/3} \theta)t_{3,d}$. As a consequence, it holds that $

Figures (3)

  • Figure 1: Comparison of the final accuracies of the different estimators considered.
  • Figure 2: Convergence of the Markov chain generated by each sampling algorithm, for a rank-2 target density matrix of different dimensions.
  • Figure 3: Relative error achieved by \ref{['algo:sampler']} when increasing $m$, the number of repetition of each experiment (blue). The orange line is the result of a linear regression of all points on the curve (omitting the first two).

Theorems & Definitions (13)

  • Lemma 1
  • Theorem 1: PAC-Bayesian bound
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 3 more