Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on quantum expanders

Cécilia Lancien, Pierre Youssef

Abstract

We prove that a wide class of random quantum channels with few Kraus operators, sampled as random matrices with some sparsity and moment assumptions, typically exhibit a large spectral gap, and are therefore optimal quantum expanders. In particular, our result provides a recipe to construct random quantum expanders from their classical (random or deterministic) counterparts. This considerably enlarges the list of known constructions of optimal quantum expanders, which was previously limited to few examples. Our proofs rely on recent progress in the study of the operator norm of random matrices with dependence and non-homogeneity, which we expect to have further applications in several areas of quantum information.

A note on quantum expanders

Abstract

We prove that a wide class of random quantum channels with few Kraus operators, sampled as random matrices with some sparsity and moment assumptions, typically exhibit a large spectral gap, and are therefore optimal quantum expanders. In particular, our result provides a recipe to construct random quantum expanders from their classical (random or deterministic) counterparts. This considerably enlarges the list of known constructions of optimal quantum expanders, which was previously limited to few examples. Our proofs rely on recent progress in the study of the operator norm of random matrices with dependence and non-homogeneity, which we expect to have further applications in several areas of quantum information.
Paper Structure (11 sections, 8 theorems, 107 equations)

This paper contains 11 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Quantum states and channels
  3. Quantum expanders
  4. Previously known results and our contribution
  5. Organization of the paper and notation
  6. Operator norm estimate for a random matrix model with tensor product structure
  7. Random matrix model
  8. Operator norm of random matrices with dependence and non-homogeneity
  9. Proof of Theorem \ref{['th:main-result']}
  10. Implications in terms of random quantum channels
  11. Summary and perspectives

Key Result

Theorem 1

Let $W$ be an $n\times n$ random matrix satisfying Assumptions a:indep entries, a: profil variance, a: 4 moment and a: max variance. Let $Y=\frac{1}{d}\sum_{s=1}^d W_s\otimes\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu_s$, where $W_s$, $1\leq s\leq d$, are independent copies of $W$. and what is more, where $C_\beta,C_\beta'>0$ are constants depending only on $\beta$.

Theorems & Definitions (19)

  • Example 1: i.i.d. heavy-tailed entries
  • Example 2: Gaussian matrices with variance profile
  • Example 3: Random matrices with deterministic sparsity pattern
  • Theorem 1
  • Theorem 2: vanHandelvanHandel2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 9 more