Quantum Markov Chains Associated with Open Quantum Random Walks

Abstract

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

Comparison of quantum-walk implementations on noisy intermediate-scale quantum computers

“…Precisely, a generalized CNOT gate with n c control qubits requires additional n c − 1 ancilla qubits for the implementation (refer to Appendix A). For example, considering IBMQ's 15-qubit Melbourne machine, we can implement a quantum walk on a cycle with at most 2 8 = 256 states.…”

“…Quantum walks have the potential to speed up classical algorithms that are based on random walks [2][3][4]. There have been many systematic studies on this subject area and many of them can lead to further in-depth analysis of more advanced quantum algorithms, such as quantum Metropolis, quantum Markov chains, or quantum Monte Carlo methods [2,[5][6][7][8]]. An early work from Aharonov et al [9] proves that, in the context of quantum walks on graphs, the walker's propagation in the quantum case is quadratically faster than the classical random walk.…”

Quantum Markov Chains on the Comb graphs: Ising model

“…In [23] it has been proposed a novel approach to investigate quantum cryptography problems by means of QMC [24] where quantum effects are entirely encoded into super-operators labelling transitions, and the nodes of its transition graph carry only classical information and thus they are discrete. Recently, QMC have been applied [18,19,20] to the investigations of so-called "open quantum random walks" [13,16,26].…”

Hypergroup structures of open quantum random walks on distance sets

“…In [2], Attal-Guillotin-Plantard-Sabot have established the central limit theorem for OQRW. In [8], an OQRW is associated with a quantum Markov chain in the sense of [1], and investigated the irreducibility and reducibility.…”