Quantum stochastic walks: A generalization of classical random walks and quantum walks

Abstract

We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on a line, the QW to CRW transition and transitions to genearlized QSWs that go beyond the CRW and QW. QSWs provide a new framework to the study of quantum algorithms as well as of quantum walks with environmental effects.Many classical algorithms, such as most Markov-chain Monte Carlo algorithms, are based on classical random walks (CRW), a probabilistic motion through the vertices of a graph. The quantum walk (QW) model is a unitary analogue of the CRW that is generally used to study and develop quantum algorithms [1,2,3]. The quantum mechanical nature of the QW yields different distributions for the position of the walker, as a QW allows for superposition and interference effects [4]. Algorithms based on QWs exhibit an exponential speedup over their classical counterparts have been developed [5,6,7]. QWs have inspired the development of an intuitive approach to quantum algorithm design [8], some based on scattering theory [9]. They have recently been shown to be capable of performing universal quantum computation [10].The transition from the QW into the classical regime has been studied by introducing decoherence to specific models of the discrete-time QW [11,12,13,14]. Decoherence has also been been studied as non-unitary effects on continuous-time QW in the context of quantum transport, such as environmentally-assisted energy transfer in photosynthetic complexes [15,16,17,18,19] and state transfer in superconducting qubits [20,21]. For the purposes of experimental implementation, the vertices of the graph in a walk can be implemented using a qubit per vertex (an inefficient or unary mapping) or by employing a quantum state per vertex (the binary or efficient mapping). The choice of mapping impacts the simulation efficiency and their robustness under decoherence [22,23,24]. The previous proposed approaches for exploring decoherence in quantum walks have added environmental-effects to a QW based on computational or physical models such as pure dephasing [17] but have not considered walks where the environmental effects are constructed axiomatically from the underlying graph.In this work, we define the quantum stochastic walk (QSW) using a set of axioms that incorporate unitary and non-unitary effects. A CRW is a type of classical stochastic processes. From the point of view of the theory of open quantum systems, the generalization of a classical stochastic process to the quantum regime is known to be a quantum stochastic process [16,25,26,27,28,29] ...

Non-equilibrium transition from dissipative quantum walk to classical random walk

“…Nevertheless there are many other mechanism of decoherence, see for example [31]. To our knowledge this pioneer problem was well posed in van Kampen's paper [12], but many other related approaches have also been presented in the literature [15,40]. The propagation of photons in waveguide lattices have been studied in recent years [24,25], and they are possible scenarios where our present results can be applied.…”

Thermodynamic formalism for dissipative quantum walks

“…If only the coherent term was present the QSW would correspond to a standard CTQW. On the other hand, it can be shown that if only the incoherent term was present the QSW would correspond to a classical random walk for the diagonal elements of ρ(t) [22]. A standard stochastic method to study the dynamics described by eq.…”

“…QSWs are a class of stochastic processes introduced in [22] with the purpose of generalizing both continuous time quantum walks, and classical random walks [29] into a single framework. A QSW is specified by a quantum master equation in Lindblad form [30] describing the evolution of the system's density matriẋ…”

“…The rest of the paper is divided as follows: in the second section we define the notion of QSW as introduced in [22]; in the third section we provide a brief introduction to the LD approach; in the fourth section we apply the thermodynamic formalism to describe the dynamics of a QSW on a directed graphs; while the last section is devoted to discussions and conclusions.…”

“…In this work, following [22], we will refer to dissipative CTQWs as Quantum Stochastic Walks (QSWs). Using the quantum jump approach [23] we study the statistical properties of the ensemble of trajectories originated by QSWs.…”