Quantum expanders from any classical Cayley graph expander

TL;DR

This paper presents a method to convert classical Cayley graph walks into quantum operations, enabling the construction of efficient quantum expanders from classical expanders on groups like the symmetric group.

Contribution

It introduces a straightforward recipe for translating classical Cayley graph walks into quantum operations, preserving key properties and enabling efficient quantum expander construction.

Findings

Classical Cayley graph properties carry over to quantum operations.

Quantum expanders can be efficiently constructed from classical expanders.

The method applies to groups like the symmetric group.

Abstract

We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders.