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arXiv:2603.14269

On aggregation-quantization permutability problem for discrete-time Markov chains

Adam Doliwa, Artur Siemaszko, Adam Zalewski

quant-phmath-phmath.COmath.DSSubmitted Mar 15, 2026• Announced Mar 17, 2026• 0 citations• Revised Oct 27, 2026 (v3)

v1 · Mar 15, 2026 v2 · Jun 30, 2026 v3 · Oct 27, 2026

Abstract

Given random walk on a graph, the corresponding discrete-time quantum walk can be constructed using the method proposed by Szegedy. On the other hand, given a partition of the set of states of a Markov chain, one can study the corresponding aggregated process. We extend the aggregation technique to the level of quantum Markov chains. We provide conditions under which application of these two operations - Szegedy's quantization and aggregation - give the same result. These consist of the strong lumpability of the underlying Markov chain supplemented by nonlinear relations, generalizing Kolmogorov's cycle condition, that account for the coefficients of both the original and the aggregated transition matrices. In particular, we show that the conditions are satisfied in the case of the random walk on graphs equipped with equitable partitions. We present several examples, which include the classical/quantum walks on Platonic solids. We discuss also relation of discrete-time classical/quantum walks on $N$-dimensional hypercube and the Ehrenfests urn model with $N$ particles. We apply our technique for of discrete-time walks on Cayley graphs of free groups. We also compare our results with those obtained using Cantero-Moral-Velazquez uniformization of unitary matrices.

Comments: 22 pages, 10 figures; v2 presentation improved; v3 formulation of Proposition 3.1 corrected, while the main idea of the work remains unchanged

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Given random walk on a graph, the corresponding discrete-time quantum walk can be constructed using the method proposed by Szegedy. On the other hand, given a partition of the set of states of a Markov chain, one can study the corresponding aggregated process. We extend the aggregation technique to the level of quantum Markov chains. We provide conditions under which application of these two operations - Szegedy's quantization and aggregation - give the same result. These consist of the strong lumpability of the underlying Markov chain supplemented by nonlinear relations, generalizing Kolmogorov's cycle condition, that account for the coefficients of both the original and the aggregated transition matrices. In particular, we show that the conditions are satisfied in the case of the random walk on graphs equipped with equitable partitions. We present several examples, which include the classical/quantum walks on Platonic solids. We discuss also relation of discrete-time classical/quantum walks on

N

-dimensional hypercube and the Ehrenfests urn model with

N

particles. We apply our technique for of discrete-time walks on Cayley graphs of free groups. We also compare our results with those obtained using Cantero-Moral-Velazquez uniformization of unitary matrices.

A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.

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.

Abstract We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy's quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment [-1,1], we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szegö map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so called Cantero--Grünbaum--Moral--Velázquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period. 2010 Mathematics Subject Classification. 81P68, 05C81, 42C05, 94A11.

In this paper we discuss the quantum Markov chains of unitary quantum walks.The construction of the quantum Markov chain associated with a unitary quantum walk serves as a good example in the view point of quantum Markov chains.On the other hand, the method of quantum Markov chain provides a tool for the investigation of dynamical properties of the unitary quantum walks.We discuss the reducibility/irreducibility and the recurrence/transience properties.Comparing with the results known in other literature and in classical random walks, we will see some similarity as well as some differences.

Quantum walks are promising tools based on classical random walks, with plenty of applications such as many variants of optimization. Here we introduce the semiclassical walks in discrete time, which are algorithms that combines classical and quantum dynamics. Specifically, a semiclassical walk can be understood as a classical walk where the transition matrix encodes the quantum evolution. We have applied this algorithm to Szegedy’s quantum walk, which can be applied to any arbitrary weighted graph. We first have solved the problem analytically on regular 1D cycles to show the performance of the semiclassical walks. Next, we have simulated our algorithm in a general inhomogeneous symmetric graph, finding that the inhomogeneity drives a symmetry breaking on the graph. Moreover, we show that this phenomenon is useful for the problem of ranking nodes in symmetric graphs, where the classical PageRank fails. We have demonstrated experimentally that the semiclassical walks can be applied on real quantum computers using the platform IBM Quantum.