Efficient Quantum Tensor Product Expanders and k-designs
Aram W. Harrow, Richard A. Low
TL;DR
The paper addresses the challenge of constructing efficient constant-degree, constant-gap quantum k-TPEs and their use to produce approximate unitary k-designs on n qubits for k = O(n/log n). It introduces a novel approach that mixes a fast classical 2k-TPE with a quantum Fourier transform, and provides a rigorous projector-based analysis (utilizing Möbius inversion and Schur-Weyl duality) to bound the quantum gap. The main contribution is the first efficient construction of a quantum k-TPE for any k ≥ 2 with constant degree and gap, yielding efficient approximate unitary k-designs under a dimension constraint N = Ω((2k)^{8k}). This advances derandomization and pseudo-random unitary generation in quantum information, offering practical ensembles with Haar-like moments for quantum tasks.
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].