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

TL;DR

This work investigates using discrete-time quantum walks (DTQWs) on a cycle to sample from an almost-uniform distribution without external randomness. It introduces a measure-and-reset protocol that evolves the walker for a fixed number of steps $m$, measures the position, and reinitializes at the measured site, yielding a sequence whose marginal distribution is $\mathbb{P}(X_n)=\mu^{\star n}(x_n)$, where $\mu(x)$ is the one-step transition probability. Under the ergodic theorem, if the support of $\mu$ is not contained in a coset of a proper subgroup, the marginals converge to the uniform distribution $1/N$ as $n\to\infty$, and the Diaconis–Shahshahani bound quantifies the entropy convergence towards $\log_2 N$. While correlations between successive outputs are present due to reinitialization, they can be mitigated by choosing the evolution time $m$, enabling the protocol to serve as a practical, randomness-source-free quantum RNG building block with potential integration into quantum hardware and networks.

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.

Figures (3)

• Figure 1: Shannon entropy of the spatial probability distribution on an $N=25$ cycle as a function of timesteps $T$, both for the direct sampling and for the Cesàro distribution obtained with Hadamard coin (see the main text for details).
• Figure 2: Autocorrelation between a string generated with the convolution protocol and itself, shifted by one position, on an $N=25$ cycle with $m=100$ (a) and $m=10$ (b). Lighter colours correspond to higher values.
• Figure 3: Transition probability $\mu(x)$ for two different values of timesteps $m$, on a $N=25$ cycle.

Theorems & Definitions (5)

• Theorem 3.1: Ergodic theorem
• Theorem A.1
• proof
• Corollary A.1.1
• proof