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Non-Hermitian expander obtained with Haar distributed unitaries

Sarah Timhadjelt

TL;DR

The work develops a Hastings-style, Schwinger-Dyson–based framework to analyze random quantum channels built from Haar-distributed unitaries. By establishing an Alon–Boppana–type lower bound for the second singular value and a Hastings-type upper bound for both the second eigenvalue and the second singular value, it proves that such channels act as quantum expanders in both eigenvalue and singular-value senses, with explicit probabilistic tail decay $\mathbb{P}(\cdot) \le \exp(-c\,N^{1/12})$. The main technical engine is a careful trace method combined with iterative Schwinger-Dyson equations, encoding matrix movements via words and patterns and proving convergence of the resulting series independent of the Kraus degree $d$. The results yield concrete rates of convergence toward the limiting expansion values $\rho_d=1/\sqrt{d}$ and $\sigma_d=2\sqrt{d-1}/d$, and they extend the understanding of strong asymptotic freeness in the non-Hermitian and Hermitian expansion contexts, with implications for quantum information processing and random matrix theory.

Abstract

We consider a random quantum channel obtained by taking a selection of $d$ independent and Haar distributed $N$ dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for $d$-regular graphs. The upper bound is obtained using Schwinger-Dyson equations.

Non-Hermitian expander obtained with Haar distributed unitaries

TL;DR

The work develops a Hastings-style, Schwinger-Dyson–based framework to analyze random quantum channels built from Haar-distributed unitaries. By establishing an Alon–Boppana–type lower bound for the second singular value and a Hastings-type upper bound for both the second eigenvalue and the second singular value, it proves that such channels act as quantum expanders in both eigenvalue and singular-value senses, with explicit probabilistic tail decay . The main technical engine is a careful trace method combined with iterative Schwinger-Dyson equations, encoding matrix movements via words and patterns and proving convergence of the resulting series independent of the Kraus degree . The results yield concrete rates of convergence toward the limiting expansion values and , and they extend the understanding of strong asymptotic freeness in the non-Hermitian and Hermitian expansion contexts, with implications for quantum information processing and random matrix theory.

Abstract

We consider a random quantum channel obtained by taking a selection of independent and Haar distributed dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for -regular graphs. The upper bound is obtained using Schwinger-Dyson equations.
Paper Structure (17 sections, 21 theorems, 218 equations, 5 figures)

This paper contains 17 sections, 21 theorems, 218 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations and properties of finite dimensional operators
  3. Quantum information problematic and previous results
  4. Model, Alon-Boppana bound and main theorem
  5. Proof of Lemma \ref{['Alonbop']} and Lemma \ref{['Alonbop2']}
  6. Trace method and Schwinger-Dyson equations
  7. Strategy
  8. Overview of proofs using Schwinger-Dyson equation
  9. Schwinger-Dyson Equations
  10. Coding iterations of Schwinger Dyson equations
  11. Encoding the matrices' movements
  12. Writing the iterations of Schwinger-Dyson and convergence of series
  13. Computations of traces
  14. Rung cancellations of matrices
  15. Proof of Theorem \ref{['hastingsdetails']} and Corollary \ref{['hastingscor']}
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $d\ge 1$ be an integer and $(U(s))_{s\in [d]} \in \mathrm{G}_N$ iid Haar distributed where $\mathrm{G}_N \in \{\mathrm{U}_N, \mathrm{O}_N, \mathrm{S}_N\}$. The family $\mathbf{U}^{\otimes}=(U(s)\otimes \overline{U}(s)|_{\mathds{1}^{\perp}})_{s\in[d]}$ is strongly asymptotically free.

Figures (5)

  • Figure 1: Plot of the eigenvalues of $\mathcal{E}$ in the hermitian case for $N=40$ and $d=10$. In red the semi-circle law of radius $\lambda_{\mathrm{herm},20}:=2\frac{\sqrt{19}}{20}$.
  • Figure 2: Plot of the eigenvalues of $\mathcal{E}$ for $N=40$ and $d=10$. In red the circle of radius $\lambda_{10}:=\frac{1}{\sqrt{10}}$.
  • Figure 3: Plot of the eigenvalues of $|\mathcal{E}|:= \sqrt{\mathcal{E}^\ast\mathcal{E}}$ for $N=30$ and $d=10$. In red $x=\bar{\lambda}_{10}=2\frac{\sqrt{d-1}}{d} = \frac{3}{5}$.
  • Figure 4: Result of the algorithm after one iteration of Schwinger-Dyson equation
  • Figure 5: Tree structure for the first generation

Theorems & Definitions (48)

  • Definition 1.1: Quantum expander (eigenvalues)
  • Definition 1.2: Quantum expander (singular values)
  • Theorem 1.3: Bordenave, Collins 2018,2022
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 38 more