Optimal quantum (tensor product) expanders from unitary designs

TL;DR

This paper demonstrates that quantum channels constructed from unitary designs, especially from 2-designs and their tensor powers, can be optimal quantum expanders with large spectral gaps, advancing quantum information theory.

Contribution

It proves that random quantum channels from unitary designs are typically optimal quantum expanders, including tensor product expanders from higher-order designs.

Findings

Random channels from 2-designs are high-probability optimal expanders.

Channels from $U^{ ensor k}$ with $U$ from a $2k$-design are typically optimal $k$-copy tensor product expanders.

The results connect unitary designs with the construction of quantum expanders with large spectral gaps.

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).