Efficient Quantum Tensor Product Expanders and k-designs

TL;DR

This paper presents an efficient method to construct quantum k-tensor product expanders and approximate unitary k-designs for up to O(n/log n) qubits, advancing quantum randomness and complexity theory.

Contribution

It introduces the first efficient construction of quantum k-tensor product expanders for k>2, utilizing classical expanders and quantum Fourier transform.

Findings

Constructed constant-degree, constant-gap quantum k-tensor product expanders.

Achieved efficient approximate unitary k-designs for k=O(n/log n).

Extended the range of known efficient quantum designs beyond k=2.

Abstract

Quantum expanders are a quantum analogue of expanders, and k-tensor product expanders are a generalisation to graphs that randomise k correlated walkers. Here we give an efficient construction of constant-degree, constant-gap quantum k-tensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k=O(n/log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of an approximate unitary k-design, which is a quantum analogue of an approximate k-wise independent function, on n qubits for any k=O(n/log n). Previously, no efficient constructions were known for k>2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].