Efficient quantum tensor product expanders and unitary t-designs via the zigzag product

TL;DR

This paper introduces a method to construct high-dimensional quantum t-tensor product expanders with low degree using the zigzag product, improving efficiency and spectral properties for quantum information applications.

Contribution

It extends the zigzag product technique to quantum t-tensor product expanders for t > 1, enabling efficient constructions with near-optimal spectral tradeoffs.

Findings

Constructed quantum t-tensor product expanders with polylogarithmic t.

Achieved near-Ramanujan quantum expanders with optimal degree-spectral gap tradeoff.

Improved known bounds for quantum tensor product expander constructions.

Abstract

A classical t-tensor product expander is a natural way of formalising correlated walks of t particles on a regular expander graph. A quantum t-tensor product expander is a completely positive trace preserving map that is a straightforward analogue of a classical t-tensor product expander. Interest in these maps arises from the fact that iterating a quantum t-tensor product expander gives us a unitary t-design, which has many applications to quantum computation and information. We show that the zigzag product of a high dimensional quantum expander (i.e. t = 1) of moderate degree with a moderate dimensional quantum t-tensor product expander of low degree gives us a high dimensional quantum t-tensor product expander of low degree. Previously such a result was known only for quantum expanders i.e. t = 1. Using the zigzag product we give efficient constructions of quantum t-tensor product expanders in dimension D where t = polylog(D). We then show how replacing the zigzag product by the generalised zigzag product leads to almost-Ramanujan quantum tensor product expanders i.e. having near-optimal almost quadratic tradeoff between the degree and the second largest singular value. Both the products give better tradeoffs between the degree and second largest singular value than what was previously known for efficient constructions.