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A Random Matrix Theory of Pauli Tomography

Nathan Keenan, John Goold, Alex Nico-Katz

TL;DR

This work develops a random matrix theory approach to quantum state tomography in informationally complete bases, revealing a deep connection between QST error matrices and the Gaussian Unitary Ensemble (GUE). It yields analytic expressions for the mean and variance of the trace-distance error and proves that, under broad conditions, functions of the empirical spectral measures of the tomography error can be computed by substituting GUE samples. The authors derive a tight non-adaptive Pauli tomography sample-complexity bound of $n = \Theta(10^N/ε^2)$ copies and show that rephysicalization does not improve this bound. Numerical results validate the analytic predictions for naïve and QWC protocols, supporting the robustness and broad applicability of the RMT framework for QST. The work lays a foundation for extending RMT treatments to broader QST protocols and spectral-function analyses.

Abstract

Quantum state tomography (QST), the process of reconstructing some unknown quantum state $\hatρ$ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error $Δ\hatρ= \hatρ-\hatρ^\prime$ in a QST reconstruction $\hatρ^\prime$ is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors $Δ\hatρ$ and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of $Δ\hatρ$ can be evaluated by substituting samples of an appropriate GUE for realizations of $Δ\hatρ$. This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance $\|Δ\hatρ\|_\mathrm{Tr}$ (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future.

A Random Matrix Theory of Pauli Tomography

TL;DR

This work develops a random matrix theory approach to quantum state tomography in informationally complete bases, revealing a deep connection between QST error matrices and the Gaussian Unitary Ensemble (GUE). It yields analytic expressions for the mean and variance of the trace-distance error and proves that, under broad conditions, functions of the empirical spectral measures of the tomography error can be computed by substituting GUE samples. The authors derive a tight non-adaptive Pauli tomography sample-complexity bound of copies and show that rephysicalization does not improve this bound. Numerical results validate the analytic predictions for naïve and QWC protocols, supporting the robustness and broad applicability of the RMT framework for QST. The work lays a foundation for extending RMT treatments to broader QST protocols and spectral-function analyses.

Abstract

Quantum state tomography (QST), the process of reconstructing some unknown quantum state from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error in a QST reconstruction is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of can be evaluated by substituting samples of an appropriate GUE for realizations of . This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future.

Paper Structure

This paper contains 16 sections, 11 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. The GUE in the Pauli Basis
  4. Reconstruction error as a likely sampling of the GUE
  5. Naïve Independent Measurements Tomographic Protocol
  6. QWC Tomographic Protocol
  7. Results
  8. Analytic Results for Quantum State Tomography Errors
  9. Independent and Correlated Coefficients
  10. Non-Adaptive Pauli Tomography Sample Complexity
  11. Comparison to the Maximum Likelihood Estimator
  12. Other Directions
  13. Conclusions
  14. Acknowledgements
  15. Author Contributions
  16. ...and 1 more sections

Key Result

Theorem 1

Given the Gaussian measure $\mathrm{d}\mu_\Sigma$ on $\mathbb{R}^{D^2}$ of eq:gaussian-measure with diagonal covariance matrix $\Sigma = \mathrm{diag}(v_1,v_2,\cdots,v_{D^2})$; then $\mathcal{A}_E$ defined with respect to an IC-basis $\{E_j\}$ with dual frame $\{F_j\}$ induces the Gaussian measure on $\mathcal{H}_D$, where $Q(H)$ is the positive-definite quadratic form given by:

Figures (3)

  • Figure 1: (a) Eigenvalue distributions of naive tomographic excess $\Delta\hat{\rho}$ for different shot counts $S$, overlayed with the GUE predicted Wigner semi-circles of radius $2/\sqrt{S}$. (b) Schematic for tomographic procedure. An a priori state is measured in an IC-basis, which leads to coefficients $\vec{a}$. The basis expansion map $\mathcal{A}(\vec{a})$ then gives us an estimate $\hat{\rho}'$ of $\hat{\rho}$. If we then want to ensure that our estimator for $\hat{\rho}$ is physical, we project back to $\mathrm{phys}_D$ via $\Pi$ which depends on the rephysicalization scheme.
  • Figure 2: Numerical analyses of the naïve procedure applied to the GHZ and identity state. System size $N=10$ and $S=10^4$ shots taken throughout. (a) shows the eigenvalue distribution of a single realisation of $\Delta\hat{\rho}$ with the Wigner semicircle of radius $R=2/\sqrt{S}$ superimposed. (b) and (c) show numerically determined (rescaled) expectation values $\mathbb{E}[\|\Delta\hat{\rho}\|_\mathrm{Tr}] / 2^N$ with the derived analytic prediction (\ref{['eq:naive-res-mean']}) superimposed for the GHZ and identity state respectively; (d) and (e) show the corresponding variances $\mathrm{var}[\|\Delta\hat{\rho}\|_\mathrm{Tr}] 2^N$, again with the analytic prediction (\ref{['eq:naive-res-var']}) superimposed. Panels (b)-(e) evaluated on 100 realizations of $\Delta \hat{\rho}$, with error bars shown where visible.
  • Figure 3: Numerical analyses of the QWC procedure applied to the GHZ and identity state. System size $N=10$ and $S=10^4$ shots taken throughout. (a) shows the eigenvalue distribution of a single realisation of $\Delta\hat{\rho}$ with the Wigner semicircle of radius $R=2/\sqrt{S}$ superimposed; with the GHZ state showing some deformation away from GUE level statistics. (b) and (c) show numerically determined (rescaled) expectation values $\mathbb{E}[\|\Delta\hat{\rho}\|_\mathrm{Tr}] / (10/3)^{N/2}$ with the derived analytic prediction (\ref{['eq:soph-res-mean']}) superimposed for the GHZ and identity state respectively; (d) and (e) show the corresponding variances $\mathrm{var}[\|\Delta\hat{\rho}\|_\mathrm{Tr}] 2^N$, again with the analytic prediction (\ref{['eq:soph-res-var']}) superimposed. We remark on the excellent agreement of the GHZ state numerics with GUE-derived analytic predictions in particular, despite the disagreement between GHZ-sampled and Wigner semicircle level statistics shown in panel (a); this is a concrete realization of our \ref{['tm:vanishing']}. Panels (b)-(e) evaluated on 100 realizations of $\Delta \hat{\rho}$, with error bars shown where visible.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 1 more