Quantum-driven sampling of the quasi-uniform distribution via quantum walks

TL;DR

This paper explores how discrete-time quantum walks can be used to generate nearly uniform random samples without external randomness, leveraging quantum properties and ergodic conditions to ensure convergence.

Contribution

It introduces a quantum walk-based sampling method that converges to a uniform distribution, with analysis of parameter choices to reduce correlations between samples.

Findings

Quantum walks can produce nearly uniform samples asymptotically.

Proper parameter selection reduces correlations between outcomes.

Convergence is demonstrated through ergodic theorem analysis.

Abstract

We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to produce the sequence of random numbers. We show that when the quantum walk parameters, such as the coin operator and initial state, satisfy the conditions of the ergodic theorem for random walks on finite groups, the resulting sequence converges asymptotically to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time. By analyzing the iterated convolution of the quantum walk transition probability and exploiting the ergodic theorem, we demonstrate convergence of the marginal distributions toward the uniform distribution in the asymptotic limit.