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Optimal quantum (tensor product) expanders from unitary designs

Cécilia Lancien

TL;DR

The paper shows that optimal quantum tensor-product expanders can be obtained by sampling Kraus operators from unitary $2$-designs (and, more generally, from $2k$-designs for $k$-copy expanders). By analyzing the deviation of the random channel from its design-averaged counterpart using a random-matrix approach and Weingarten calculus, it proves that in the regime $(\log n)^4≤d≤n^2$, a random unital channel with Kraus operators drawn from a $2$-design is with high probability an optimal expander, with a second largest spectral quantity bounded by $\frac{2}{\sqrt{d}}(1+o(1))$. The results extend to the $k$-copy setting, showing that sampling from a $2k$-design yields an optimal $k$-copy tensor product expander under the same growth conditions on $d$, with explicit control of the operator-norm deviations. This work thus provides a concrete derandomization route for optimal quantum expanders and suggests practical finite-design constructions (e.g., Clifford group) that achieve near-optimal spectral gaps, while outlining open questions about the necessity of $2$-designs and potential extensions to approximate designs. Overall, it advances the theory of quantum expanders by linking spectral optimality to low-length unitary designs and offering explicit, scalable sampling schemes.

Abstract

In this work we investigate how quantum expanders (i.e. quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a $2$-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form $U^{\otimes k}$, with $U$ sampled from a $2k$-design measure, then the corresponding random quantum channel is typically an optimal $k$-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).

Optimal quantum (tensor product) expanders from unitary designs

TL;DR

The paper shows that optimal quantum tensor-product expanders can be obtained by sampling Kraus operators from unitary -designs (and, more generally, from -designs for -copy expanders). By analyzing the deviation of the random channel from its design-averaged counterpart using a random-matrix approach and Weingarten calculus, it proves that in the regime , a random unital channel with Kraus operators drawn from a -design is with high probability an optimal expander, with a second largest spectral quantity bounded by . The results extend to the -copy setting, showing that sampling from a -design yields an optimal -copy tensor product expander under the same growth conditions on , with explicit control of the operator-norm deviations. This work thus provides a concrete derandomization route for optimal quantum expanders and suggests practical finite-design constructions (e.g., Clifford group) that achieve near-optimal spectral gaps, while outlining open questions about the necessity of -designs and potential extensions to approximate designs. Overall, it advances the theory of quantum expanders by linking spectral optimality to low-length unitary designs and offering explicit, scalable sampling schemes.

Abstract

In this work we investigate how quantum expanders (i.e. quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a -design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form , with sampled from a -design measure, then the corresponding random quantum channel is typically an optimal -copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).
Paper Structure (12 sections, 11 theorems, 108 equations)

This paper contains 12 sections, 11 theorems, 108 equations.

Table of Contents

  1. Preliminary facts and tools
  2. Brief reminder on quantum channels and their spectral properties
  3. Unitary designs and tensor product expanders
  4. Presentation of the problem and main results
  5. Proof strategy and main technical tool
  6. Optimal expanders from unitary $2$-designs
  7. Main technical results
  8. Implication concerning random 'mixture of unitaries' quantum channels
  9. Optimal $k$-copy tensor product expanders from unitary $2k$-designs
  10. Main technical results
  11. Implication concerning random 'mixture of tensor power unitaries' quantum channels
  12. Summary and perspectives

Key Result

Theorem 1.3

Let $Z_1,\ldots,Z_D\in\mathcal{M}_N(\mathbb{C})$ be independent centered and almost surely bounded random matrices, and set $X=\sum_{s=1}^D Z_s$. Let ${\mathbf{Cov}}(X)$ denote the covariance matrix associated to $X$, i.e. the $N^2\times N^2$ matrix such that ${\mathbf{Cov}}(X)_{ijkl}= \mathbf{E}(X_ We then have where $C<\infty$ is an absolute constant. What is more, for all $t>0$, with probabili

Theorems & Definitions (23)

  • Definition 1.1: Unitary design
  • Definition 1.2: Tensor product expander
  • Theorem 1.3: Operator norm of random matrices with dependence and non-homogeneity
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 13 more