Symmetric quantum walks on Hamming graphs and their limit distributions

TL;DR

This work develops a spectral theory for symmetric quantum walks on Hamming graphs $H(d,n)$ by tying distance-based transitions to the Bose–Mesner algebra of the association scheme and using Szegedy’s coin construction from an underlying random walk. Eigenvalues of the quantum-walk evolution operator are shown to be zeros of certain self-reciprocal polynomials, and the wave vectors admit a spectral representation expressed through Krawtchouk polynomials. The authors derive explicit limit distributions for several quantum walks, including the hypercube ($n=2$) and prime-$n$ cases, revealing a characteristic arcsine-type component in long-time behavior. This links quantum walk dynamics to orthogonal-polynomial spectral theory and finite-geometry methods, yielding exact, computable long-time statistics useful for quantum information and computation contexts.

Abstract

We study a class of symmetric quantum walks on Hamming graphs, where the distance between vertices specifies the transition probability. A special model is the simple quantum walk on the hypercube, which has been discussed in the literature. Eigenvalues of the unitary operator of the quantum walks are zeros of certain self-reciprocal polynomials. We obtain a spectral representation of the wave vector, where our systematic treatment relies on the coin space isomorphic to the state space and the commutative association scheme. The limit distributions of several quantum walks are obtained.

Theorems & Definitions (31)

• Example 2.1: The simple random walk
• Example 2.2: The independent random walk
• Example 2.3: The non-local random walk with cardinality $m\in\{2,\ldots,d\}$
• Example 2.4: The mixture of i.i.d. updates for each coordinate
• Lemma 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof