arXiv:2509.26243 · Symmetry Integrability and Geometry Methods and Applications

Symmetric Quantum Walks on Hamming Graphs and Their Limit Distributions

Robert Griffiths, Shuhei Mano

Published Jul 3, 2026• 0 citations

Abstract

We study a class of symmetric coined 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 Grover coin is extended to the reflection about a vector in an invariant subspace of the Terwilliger algebra. The limit distributions of several quantum walks are obtained.

DOI: 10.3842/sigma.2026.063

Related papers

Probability distributions for Markov chain based quantum walks

Radhakrishnan Balu, Chaobin Liu, Salvador Elías Venegas-Andraca

Abstract We analyze the probability distributions of the quantum walks induced from Markov chains by Szegedy (2004). The first part of this paper is devoted to the quantum walks induced from finite state Markov chains. It is shown that the probability distribution on the states of the underlying Markov chain is always convergent in the Cesaro sense. In particular, we deduce that the limiting distribution is uniform if the transition matrix is symmetric. In the case of a non-symmetric Markov chain, we exemplify that the limiting distribution of the quantum walk is not necessarily identical with the stationary distribution of the underlying irreducible Markov chain. The Szegedy scheme can be extended to infinite state Markov chains (random walks). In the second part, we formulate the quantum walk induced from a lazy random walk on the line. We then obtain the weak limit of the quantum walk. It is noted that the current quantum walk appears to spread faster than its counterpart-quantum walk on the line driven by the Grover coin discussed in literature. The paper closes with an outlook on possible future directions.

arXiv:1703.04131Journal of Physics A Mathematical and TheoreticalNov 10, 201713 citations

Quantum random walks on d-regular graphs with Haar-random coin operators

Alice C. Quillen

With unitary coin operator that is a random matrix drawn from a uniform distribution with respect to the Haar measure, we construct a variant of a discrete quantum random walk using a d-regular undirected connected simple graph. With each step of the walk, the coin operator random matrix is drawn independently from the distribution. The expectation value over the distribution of random unitaries for the associated quantum channel gives a depolarization channel for the coin subspace and resembles the associated classical random walk where the direction a walker steps depends upon the outcome of a fair coin or balanced d-sided dice. Remarkably, despite the fact that the averaged channel depolarizes the coin subspace, measurements in the vertex subspace can be designed that would reveal information about the initial quantum state, even after many iterations of the channel. We illustrate with examples of quantum walks on Cayley graphs of Abelian groups, such as the cycle and hypercube graphs. For Cayley graphs of Abelian groups, the averaged channel is dephasing in the Fourier basis. These quantum walks are examples of bipartite strongly interacting systems, where one subsystem is strongly perturbed, yet information about the initial state in the other subsystem is potentially measurable forever. Due to its decoherence, a quantum random walk with a Haar-random coin would not be useful in search algorithms but could aid in understanding quantum systems with strongly perturbed subsystems.

quant-pharXiv:2607.04584Physical Review AJul 7, 20260 citations

Classification of coined quantum walks on the line and comparison to correlated classical random walks

Lukas Hantzko, Lennart Binkowski

Abstract We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead to symmetric quantum walks for arbitrary coins and provide a bijective parametrisation of all symmetric quantum walks modulo distributional equivalence. Extending beyond the symmetric case, we also give a surjective parametrisation of all coined quantum walks under the same equivalence relation, as well as a bijective parametrisation modulo equivalence of their limiting distributions. Furthermore, we derive corrected closed-form expressions for the walk amplitudes, resolving inaccuracies in previous literature, and generalise the approach to the correlated classical random walk. This unified framework enables a direct comparison between quantum and classical dynamics. Additionally, we discuss the asymptotic scaling of variances for both models, identifying quadratic spreading as a hallmark of non-trivial quantum walks and contrasting it with the linear behaviour of classical walks, except at the extremal points of maximal correlation. Finally, we compare the limiting distributions arising from quantum walks with those of the classical case. Graphical abstract - compact classification results for symmetric and general quantum walks, as well as classically correlated random walks - quantum walks spread ballistically while classically correlated random walks approach Gaussian limiting distribution - quantum coin parameter directly influence limiting distribution’s skewness

arXiv:2507.23524The European Physical Journal DJul 1, 20260 citations

Quantum graph walks I: mapping to quantum walks

Yusuke Higuchi, Norio Konno, Iwao Sato +1 more

We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph.

arXiv:1211.0803Nov 5, 201222 citations

Self-avoiding quantum walks: realisations in subspaces and limit theorems

Takuya Machida, C. M. Chandrashekar, Norio Konno +1 more

While completely self-avoiding quantum walks have the distinct property of leading to a trivial unidirectional transport of a quantum state, an interesting and non-trivial dynamics can be constructed by restricting the self-avoidance to a subspace of the complete Hilbert space. Here, we present a comprehensive study of three two-dimensional quantum walks, which are self-avoiding in coin space, in position space and in both, coin and position space. We discuss the properties of these walks and show that all result in delocalisation of the probability distribution for initial states which are strongly localised for a walk with a standard Grover coin operation. We also present analytical results for the evolution in the form of limit distributions for the self-avoiding walks in coin space and in both, coin and position space.

arXiv:1307.6288Jul 24, 20130 citations

Influence of coin symmetry on infinite hitting times in quantum walks

Prithviraj Prabhu, Todd A. Brun

Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other vertex in finite time, provided they are connected. Discrete-time quantum walks on finite connected graphs, however, can have infinite hitting times. This phenomenon is related to graph symmetry, as previously characterized by the group of direction-preserving graph automorphisms that trivially affect the coin Hilbert space. If a graph is symmetric enough (in a particular sense), then the associated quantum walk unitary will contain eigenvectors that do not overlap a set of target vertices for any coin-flip operator. These eigenvectors span the infinite-hitting-time (IHT) subspace. Quantum states in the IHT subspace never reach the target vertices, leading to infinite hitting times. However, this is not the whole story: the graph of the three-dimensional cube does not satisfy this symmetry constraint, yet quantum walks on this graph with certain symmetric coins can exhibit infinite hitting times. We study the effect of coin symmetry by analyzing the group of coin-permutation symmetries (CPSs): graph automorphisms that act nontrivially on the coin Hilbert space but leave the coin operator invariant. Unitaries using highly symmetric coins with large CPS groups, such as the permutation-invariant Grover coin, are associated with higher probabilities of never arriving, as a result of their larger IHT subspaces.

arXiv:2107.02383Physical Review AMar 11, 20222 citations