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Optimal trace-distance bounds for free-fermionic states: Testing and improved tomography

Lennart Bittel, Antonio Anna Mele, Jens Eisert, Lorenzo Leone

TL;DR

This work derives optimal perturbation bounds that connect trace-distance errors to errors in the correlation matrix for free-fermionic (Gaussian) states, enabling robust learning and benchmarking of quantum systems. The authors establish tight bounds for both pure and mixed states, extend the analysis to quantify non-Gaussianity, and apply these insights to property testing and tomography. They prove hardness results for general free-fermionic testing while giving efficient, low-rank regimes and robust tomography with polynomials in the system size, achieving optimal $O(\varepsilon^{-2})$ scaling. The results are experimentally appealing for fermionic platforms (e.g., cold atoms, digital simulators) and provide practical tools for certifying Gaussianity and reconstructing states with provable guarantees even in the presence of noise.

Abstract

Free-fermionic states, also known as fermionic Gaussian states, represent an important class of quantum states ubiquitous in physics. They are uniquely and efficiently described by their correlation matrix. However, in practical experiments, the correlation matrix can only be estimated with finite accuracy. This raises the question: how does the error in estimating the correlation matrix affect the trace-distance error of the state? We show that if the correlation matrix is known with an error $\varepsilon$, the trace-distance error also scales as $\varepsilon$ (and vice versa). Specifically, we provide distance bounds between (both pure and mixed) free-fermionic states in relation to their correlation matrix distance. Our analysis also extends to cases where one state may not be free-fermionic. Importantly, we leverage our preceding results to derive significant advancements in property testing and tomography of free-fermionic states. Property testing involves determining whether an unknown state is close to or far from being a free-fermionic state. We first demonstrate that any algorithm capable of testing arbitrary (possibly mixed) free-fermionic states would inevitably be inefficient. Then, we present an efficient algorithm for testing low-rank free-fermionic states. For free-fermionic state tomography, we provide improved bounds on sample complexity in the pure-state scenario, substantially improving over previous literature, and we generalize the efficient algorithm to mixed states, discussing its noise-robustness.

Optimal trace-distance bounds for free-fermionic states: Testing and improved tomography

TL;DR

This work derives optimal perturbation bounds that connect trace-distance errors to errors in the correlation matrix for free-fermionic (Gaussian) states, enabling robust learning and benchmarking of quantum systems. The authors establish tight bounds for both pure and mixed states, extend the analysis to quantify non-Gaussianity, and apply these insights to property testing and tomography. They prove hardness results for general free-fermionic testing while giving efficient, low-rank regimes and robust tomography with polynomials in the system size, achieving optimal scaling. The results are experimentally appealing for fermionic platforms (e.g., cold atoms, digital simulators) and provide practical tools for certifying Gaussianity and reconstructing states with provable guarantees even in the presence of noise.

Abstract

Free-fermionic states, also known as fermionic Gaussian states, represent an important class of quantum states ubiquitous in physics. They are uniquely and efficiently described by their correlation matrix. However, in practical experiments, the correlation matrix can only be estimated with finite accuracy. This raises the question: how does the error in estimating the correlation matrix affect the trace-distance error of the state? We show that if the correlation matrix is known with an error , the trace-distance error also scales as (and vice versa). Specifically, we provide distance bounds between (both pure and mixed) free-fermionic states in relation to their correlation matrix distance. Our analysis also extends to cases where one state may not be free-fermionic. Importantly, we leverage our preceding results to derive significant advancements in property testing and tomography of free-fermionic states. Property testing involves determining whether an unknown state is close to or far from being a free-fermionic state. We first demonstrate that any algorithm capable of testing arbitrary (possibly mixed) free-fermionic states would inevitably be inefficient. Then, we present an efficient algorithm for testing low-rank free-fermionic states. For free-fermionic state tomography, we provide improved bounds on sample complexity in the pure-state scenario, substantially improving over previous literature, and we generalize the efficient algorithm to mixed states, discussing its noise-robustness.
Paper Structure (24 sections, 49 theorems, 247 equations, 3 figures, 4 algorithms)

This paper contains 24 sections, 49 theorems, 247 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perturbation bounds
  4. Property testing
  5. Efficient and robust tomography
  6. Related works
  7. Discussion and open questions
  8. Preliminaries
  9. Notation and basic definitions
  10. Free-fermionic states
  11. Particle-number preserving fermionic states
  12. Norm inequalities for free and non-free fermionic states
  13. General lemmas
  14. Trace-distance bounds for pure free-fermionic states
  15. Trace-distance bounds for mixed free-fermionic states
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\psi$ and $\phi$ be two pure free-fermionic states, then it holds that while for $\rho$ and $\sigma$ being two possibly mixed free-fermionic states, it holds that

Figures (3)

  • Figure 1: (a) Testing free-fermionic quantum states and improved tomography. (b) Any procedure aiming to solve the property testing problem for possibly mixed free-fermionic states must have a sample complexity scaling exponentially with the system size. Low-rank free-fermionic states can be efficiently and robustly tested, however. (c) We then show sample optimal bounds for quantum state tomography for free-fermionic states. Our results strongly rely on a new set of inequalities that we derive concerning the distance of free-fermionic states (which constitute a non-convex set) and the distance between their respective, efficiently tractable, correlation matrices (which form a convex set).
  • Figure 2: Numerical simulation of the performance of the tomography algorithm for pure Gaussian states. We consider different numbers of total measurement rounds $N$ as a function of the number of modes. The fit (gray dotted line) uses $\varepsilon= 2n^{3/2} N^{-1/2}$, where $\varepsilon$ denotes the trace distance error. The data is averaged over 50 randomly selected Gaussian states each.
  • Figure 3: Plot illustrating the behavior of the objective function $g(x) = \max\{(1-x)^{r+1}, |x-\lambda|\}$, which arises in Eq. \ref{['eq:maxmin']}, where $\lambda = 0.7$ and $r = 2$. The red point indicates the location of the minimum value of the function.

Theorems & Definitions (99)

  • Theorem 1: Perturbation bounds
  • Theorem 2: Lower bound on trace distance
  • Lemma 1: Closeness of quantum states in terms of correlation matrices
  • Lemma 2: Lower bounds to trace distances
  • Theorem 3: Hardness of testing bounded rank free-fermionic states
  • Theorem 4: Efficient free-fermionic testing - Informal version of Theorems \ref{['th:mix1']} and \ref{['th:mix2']}
  • Proposition 1: Tomography of pure free-fermionic states
  • Theorem 5: Tomography of mixed free-fermionic states
  • Definition 1: Majorana operators
  • Definition 2: Majorana products
  • ...and 89 more