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

Adam Doliwa, Artur Siemaszko, Adam Zalewski

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. 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.

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

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. 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 -dimensional hypercube and the Ehrenfests urn model with 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.
Paper Structure (17 sections, 4 theorems, 95 equations, 12 figures)

This paper contains 17 sections, 4 theorems, 95 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lumpability of Markov chains
  4. Szegedy's quantization of Markov chains
  5. Cantero--Moral--Velázquez (CMV) and Jacobi matrices
  6. Quantum aggregation
  7. Formulation of the problem
  8. Solution of the problem
  9. Quantum walks on Platonic solids and their reductions
  10. Tetrahedron
  11. Octahedron
  12. Icosahedron
  13. Dodecahedron
  14. Another lumping of the random walk on hexahedron graph
  15. Quantum walk on the hypercube graph, and the quantum Ehrenfests model
  16. ...and 2 more sections

Key Result

Proposition 2.1

The CMV basis of the quantum evolution operator for Szegedy's quantization of the random walk on the half-line with the cyclic vector $e_0 = | \phi_0 \rangle$ has real Verblunsky coefficients related to the random walk transition probabilities by the formulas eq:rk and

Figures (12)

  • Figure 1: The hexahedron graph
  • Figure 2: Lumped random walk on the hexahedron graph
  • Figure 3: discrete-time random walk on the half-line
  • Figure 4: Self-consistency of equations \ref{['eq:sPuv']}
  • Figure 5: Self-consistency of equations \ref{['eq:sPij']}
  • ...and 7 more figures

Theorems & Definitions (26)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark
  • Remark
  • Proposition 2.1
  • Remark
  • Corollary 2.2
  • Example 2.4
  • Example 2.5
  • ...and 16 more