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

Shakir Showkat Sofi, Charlotte Vermeylen, Lieven De Lathauwer

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.

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

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.
Paper Structure (13 sections, 8 equations, 2 figures, 1 algorithm)

This paper contains 13 sections, 8 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Outline
  4. Low-rank QST formulation
  5. Efficient Low-Rank QST via Structured Measurements
  6. Algorithm
  7. Global Subspace from Local Constraints
  8. From Subspace Identification to Full State Reconstruction
  9. Summary of Recovery Conditions
  10. Classical Shadows for Structured Density Matrix Entries
  11. Experiments
  12. Results
  13. Conclusion

Figures (2)

  • Figure 1: Overlapping block diagonal pattern
  • Figure 2: The proposed method yields higher accuracy than both references while also enabling significantly faster reconstruction. Moreover, as $d$ grows, more measurements become available per parameter (with $D$ and $R$ fixed), leading to improved accuracy.