Classical and Quantum Tensor Product Expanders

TL;DR

This paper introduces quantum tensor product expanders, extending classical expanders to quantum systems, with applications to quantum computation and probabilistic constructions providing bounds on eigenvalues.

Contribution

It defines quantum tensor product expanders, connects classical and quantum cases, and offers probabilistic constructions with eigenvalue bounds.

Findings

Classical two-copy expanders can generate quantum expanders.

Probabilistic methods yield tight eigenvalue bounds.

Applications to the Solovay-Kitaev quantum compilation problem.

Abstract

We introduce the concept of quantum tensor product expanders. These are expanders that act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.