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Optimal Allocation of Pauli Measurements for Low-rank Quantum State Tomography

Zhen Qin, Casey Jameson, Zhexuan Gong, Michael B. Wakin, Zhihui Zhu

Abstract

The process of reconstructing quantum states from experimental measurements, accomplished through quantum state tomography (QST), plays a crucial role in verifying and benchmarking quantum devices. A key challenge of QST is to find out how the accuracy of the reconstruction depends on the number of state copies used in the measurements. When multiple measurement settings are used, the total number of state copies is determined by multiplying the number of measurement settings with the number of repeated measurements for each setting. Due to statistical noise intrinsic to quantum measurements, a large number of repeated measurements is often used in practice. However, recent studies have shown that even with single-sample measurements--where only one measurement sample is obtained for each measurement setting--high accuracy QST can still be achieved with a sufficiently large number of different measurement settings. In this paper, we establish a theoretical understanding of the trade-off between the number of measurement settings and the number of repeated measurements per setting in QST. Our focus is primarily on low-rank density matrix recovery using Pauli measurements. We delve into the global landscape underlying the low-rank QST problem and demonstrate that the joint consideration of measurement settings and repeated measurements ensures a bounded recovery error for all second-order critical points, to which optimization algorithms tend to converge. This finding suggests the advantage of minimizing the number of repeated measurements per setting when the total number of state copies is held fixed. Additionally, we prove that the Wirtinger gradient descent algorithm can converge to the region of second-order critical points with a linear convergence rate. We have also performed numerical experiments to support our theoretical findings.

Optimal Allocation of Pauli Measurements for Low-rank Quantum State Tomography

Abstract

The process of reconstructing quantum states from experimental measurements, accomplished through quantum state tomography (QST), plays a crucial role in verifying and benchmarking quantum devices. A key challenge of QST is to find out how the accuracy of the reconstruction depends on the number of state copies used in the measurements. When multiple measurement settings are used, the total number of state copies is determined by multiplying the number of measurement settings with the number of repeated measurements for each setting. Due to statistical noise intrinsic to quantum measurements, a large number of repeated measurements is often used in practice. However, recent studies have shown that even with single-sample measurements--where only one measurement sample is obtained for each measurement setting--high accuracy QST can still be achieved with a sufficiently large number of different measurement settings. In this paper, we establish a theoretical understanding of the trade-off between the number of measurement settings and the number of repeated measurements per setting in QST. Our focus is primarily on low-rank density matrix recovery using Pauli measurements. We delve into the global landscape underlying the low-rank QST problem and demonstrate that the joint consideration of measurement settings and repeated measurements ensures a bounded recovery error for all second-order critical points, to which optimization algorithms tend to converge. This finding suggests the advantage of minimizing the number of repeated measurements per setting when the total number of state copies is held fixed. Additionally, we prove that the Wirtinger gradient descent algorithm can converge to the region of second-order critical points with a linear convergence rate. We have also performed numerical experiments to support our theoretical findings.

Paper Structure

This paper contains 27 sections, 16 theorems, 83 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related works
  4. Notation
  5. Preliminary
  6. Quantum state and measurements
  7. Pauli observables
  8. Pauli basis measurement
  9. Statistical error in the empirical Pauli observables
  10. Recovery Guarantee for Pauli Observable Measurements
  11. Measurement setting and the matrix factorization approach
  12. Restricted isometry property with Pauli observable measurements
  13. Global landscape analysis
  14. Quality versus quantity of Pauli observable measurements
  15. Wirtinger Gradient Descent with Linear Convergence
  16. ...and 12 more sections

Key Result

Theorem 1

(informal version of upper bound recovery error of QST for Pauli measurements and Corollary LocalConvergence_Conclusion_1) Given an $n$-qubit density matrix $\boldsymbol{\rho}$ of rank $r$, randomly select $K$$n$-body Pauli observables from the full set $\{\boldsymbol{W}_i\}_{i=1}^{4^n}$ and perform

Figures (6)

  • Figure 1: Numerical computation of $\operatorname{\mathbb{E}} [(\sum_{j=1}^D \alpha_{k,j} \cdot \widehat{p}_{k,j} - \langle\boldsymbol{W}_k, \boldsymbol{\rho} \rangle )^2 ]$ for different $n$ and $M$.
  • Figure 2: Plot of $h(\delta_{2r})$ in \ref{['h-delta-function']}.
  • Figure 3: Performance comparison of Wirtinger GD, (a) for different $n$ with $r = 2$, $\mu = 0.3$, $K = 2000$ and $M = 100$, (b) for different $r$ with $n = 5$, $\mu = 0.3$, $K = 2000$ and $M = 100$.
  • Figure 4: Performance comparison of Wirtinger GD for different $N$ and $M$ with (a) $n = 4$, $r = 1$, (b) $n = 4$, $r = 2$.
  • Figure 5: Performance comparison of different algorithms with $\mu = 0.1$, $K = 4000$ and $M = 50$.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 1
  • Definition 1: Measurement setting
  • Theorem 2
  • Definition 2
  • Definition 3
  • Theorem 3
  • Lemma 2
  • Theorem 4
  • Lemma 3
  • ...and 17 more