Quantum expanders from any classical Cayley graph expander

Understand

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.