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arXiv:1502.01680

From open quantum walks to unitary quantum walks

Chaobin Liu

Published Feb 5, 2015• 4 citations
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Abstract

We present an idea to convert to a unitary quantum walk any open quantum walk which is defined on lattices as well as on finite graphs. This approach generalizes to the domain of open quantum walks (or quantum Markov chains) the framework introduced by Szegedy for quantizing Markov chains. For the unitary quantum walks formulated in this article, we define the probability and the mean probability of finding the walk at a node, then derive the asymptotic mean probability.

DOI
10.48550/arxiv.1502.01680

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We present an idea to convert to a unitary quantum walk any open quantum walk which is defined on lattices as well as on finite graphs. This approach generalizes to the domain of open quantum walks (or quantum Markov chains) the framework introduced by Szegedy for quantizing Markov chains. For the unitary quantum walks formulated in this article, we define the probability and the mean probability of finding the walk at a node, then derive the asymptotic mean probability.
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.
We make use of the Open Quantum Random Walk setting due to S. Attal, F. Petruccione, C. Sabot and I. Sinayskiy in order to discuss hitting times and a quantum version of the Mean Hitting Time Formula from classical probability theory. We study an open quantum notion of hitting probability on a finite collection of sites and with this we are able to describe the problem in terms of linear maps and its matrix representations. After setting an open quantum version of the fundamental matrix for ergodic Markov chains we are able to prove our main result and as consequence a version of the Random Target Lemma. We also study a mean hitting time formula in terms of the minimal polynomial associated to the matrix representation of the quantum walk. We discuss applications of the results to open quantum dynamics on graphs together with open questions.
We consider an abstract quantum walk defined by a unitary evolution operator UUU, which acts on a Hilbert space decomposed into a direct sum of Hilbert spaces {Hv}v∈V\{\mathcal{H}_v \}_{v \in V}{Hv​}v∈V​. We show that such UUU naturally defines a directed graph GUG_UGU​ and the probability of finding a quantum walker on GUG_UGU​. The asymptotic property of an abstract quantum walker is governed by the generator HHH of UUU such that Un=einHU^n = e^{inH}Un=einH. We derive the generator of an evolution of the form U=S(2dA∗dA−1)U = S(2d_A^* d_A -1)U=S(2dA∗​dA​−1), a generalization of the Szegedy evolution operator. Here dAd_AdA​ is a boundary operator and SSS a shift operator.
Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.
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.