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Time-Inhomogeneous Quantum Markov Chains

Chia-Han Chou

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

In quantum computation theory, quantum Markov chains and quantum walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. More recently, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. We study time-inhomogeneous quantum Markov chains with decoherence on finite spaces and discrete infinite spaces and their large scale equilibrium properties. In this thesis, we prove the convergence of decoherent time-inhomogeneous quantum Markov chain on finite state spaces, and a representation theorem for time-inhomogeneous quantum walk on discrete infinite state space. Additionally, the convergence of the distributions of the decoherent quantum walks are numerically estimated as an application of the representation theorem, and the convergence in distribution of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are statistically analyzed.

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In quantum computation theory, quantum Markov chains and quantum walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. More recently, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. We study time-inhomogeneous quantum Markov chains with decoherence on finite spaces and discrete infinite spaces and their large scale equilibrium properties. In this thesis, we prove the convergence of decoherent time-inhomogeneous quantum Markov chain on finite state spaces, and a representation theorem for time-inhomogeneous quantum walk on discrete infinite state space. Additionally, the convergence of the distributions of the decoherent quantum walks are numerically estimated as an application of the representation theorem, and the convergence in distribution of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are statistically analyzed.
In quantum computation theory, quantum random walks have been utilized by many quantum search algorithms which provide improved performance over their classical counterparts. However, due to the importance of the quantum decoherence phenomenon, decoherent quantum walks and their applications have been studied on a wide variety of structures. Recently, a unified time-inhomogeneous coin-turning random walk with rescaled limiting distributions, Bernoulli, uniform, arcsine and semi-circle laws as parameter varies have been obtained. In this paper we study the quantum analogue of these models. We obtained a representation theorem for time-inhomogeneous quantum walk on discrete infinite state space. Additionally, the convergence of the distributions of the decoherent quantum walks are numerically estimated as an application of the representation theorem, and the convergence in distribution of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are statistically analyzed.
We introduce and study time-inhomogeneous quantum Markov chains with parameter ζ≥0ζ\ge 0ζ≥0 and decoherence parameter 0≤p≤10 \leq p \leq 10≤p≤1 on finite spaces and their large scale equilibrium properties. Here ζζζ resembles the inverse temperature in the annealing random process and ppp is the decoherence strength of the quantum system. Numerical evaluations show that if $ ζ$ is small, then quantum Markov chain is ergodic for all 0<p≤10 < p \le 10<p≤1 and if $ ζ$ is large, then it has multiple limiting distributions for all 0<p≤10 < p \le 10<p≤1. In this paper, we prove the ergodic property in the high temperature region 0≤ζ≤10 \le ζ\le 10≤ζ≤1. We expect that the phase transition occurs at the critical point ζc=1ζ_c=1ζc​=1. For coherence case p=0p=0p=0, a critical behavior of periodicity also appears at critical point ζo=2ζ_o=2ζo​=2.
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.
The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. Unlike classical continuous-time Markov chains, the CTQMCs studied here do not satisfy the classical Chapman-Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ``quantum Markov chain'' is used. In this paper, we study a large class of p-adic Schrödinger equations and the associated CTQMCs by comparing them with p-adic heat equations and the associated continuous-time Markov chains (CTMCs). The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.