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arXiv:0911.0042 · Physical Review A

Equivalence between discrete quantum walk models in arbitrary topologies

Fabiano M. Andrade, M. G. E. da Luz

Published Nov 2, 2009• 25 citations
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Abstract

Coin and scattering are the two major formulations for discrete quantum walks models, each believed to have its own advantages in different applications. Although they are related in some cases, it was an open question their equivalence in arbitrary topologies. Here we present a general construction for the two models for any graph and also for position dependent transition amplitudes. We then prove constructively their unitary equivalence. Defining appropriate projector operators, we moreover show how to obtain the probabilities for one model from the evolution of the other.

DOI
10.1103/physreva.80.052301

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Coin and scattering are the two major formulations for discrete quantum walks models, each believed to have its own advantages in different applications. Although they are related in some cases, it was an open question their equivalence in arbitrary topologies. Here we present a general construction for the two models for any graph and also for position dependent transition amplitudes. We then prove constructively their unitary equivalence. Defining appropriate projector operators, we moreover show how to obtain the probabilities for one model from the evolution of the other.
The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position-dependent transition amplitudes and any topology (Andrade et al 2009 Phys. Rev. A 80 052301). Although the proof explicitly describes the mapping obtention, its high technicality may hinder relevant physical aspects involved in the equivalence. Discussing concrete examples—the most general constructions for the line, square and honeycomb lattices—here we unveil the similarities and differences of these two versions of quantum walks. We moreover show how to derive the dynamics of one from the other by means of proper projections. We perform calculations for different probability amplitudes such as Hadamard, Grover, discrete Fourier transform and the uncommon in the area (but interesting) discrete Hartley transform, comparing the evolutions. Our study illustrates the models' interplay, an important issue for implementations and applications of such systems.
Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the two processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The first result establishes procedure for a stochastic matrix sequence to induce a random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. The second result establishes a similar procedure in the opposite direction. Given any random walk, a time-dependent quantum walk with the exact same vertex probability distribution is constructed. Interestingly, the matrices constructed by the first procedure allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial-time) sampling of quantum graph trajectories. Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.
In this paper, we consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the graph’s edges, where a scattering process at the vertices is governed by the scattering matrices assigned to each vertex. We show the Scattering Quantum Walks encompass several known Quantum Walks. We further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one defined on the edges, the other on the vertices of the graph. We show these walks give rise to proper quantum channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.
Quantum and random walks were proven to be equivalent on finite graphs by demonstrating how to construct a time-dependent random walk sharing the exact same evolution of vertex probability of any given discrete-time coined quantum walk. Such equivalence stipulated a deep connection between the processes that is far stronger than simply considering quantum walks as quantum analogues of random walks. This article expands on the connection between quantum and random walks by demonstrating a procedure that constructs a time-dependent quantum walk matching the evolution of vertex probability of any given random walk in a unitary way. It is a trivial fact that a quantum walk measured at all time steps of its evolution degrades to a random walk. More interestingly, the method presented describes a quantum walk that matches a random walk without measurement operations, such that the unitary evolution of the quantum walk captures the probability evolution of the random walk. The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For the homogeneous random walk case, the properties of unitary evolution imply that the quantum walk described is time-dependent since homogeneous quantum walks do not converge for arbitrary initial conditions
In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step toward this objective, the following question is being addressed: Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps? This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general SU(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric SU(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution.
Quantum and random walks have been shown to be equivalent in the following sense: a time-dependent random walk can be constructed such that its vertex distribution at all time instants is identical to the vertex distribution of any discrete-time coined quantum walk on a finite graph. This equivalence establishes a deep connection between the two processes, far stronger than simply considering quantum walks as quantum analogues of classical random walks. The present work strengthens this connection by providing a construction that establishes this equivalence in the reverse direction: a unitary time-dependent quantum walk can be constructed such that its vertex distribution is identical to the vertex distribution of any random walk on a finite graph at all time instants. The construction shown here describes a quantum walk that matches a random walk without measurements at all time steps (an otherwise trivial statement): measurement is performed in a quantum walk that evolved unitarily until a given time ttt such that its vertex distribution is identical to the random walk at time ttt. The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For homogeneous random walks, unitary evolution implies time dependency for the quantum walk, since homogeneous quantum walks do not converge under arbitrary initial conditions, while a broad class of random walks does. Thus, the absence of convergence demonstrated for quantum walks in its debut comes from both time-homogeneity and unitarity, rather than unitarity alone, and our results shed light on the power of quantum walks to generate samples for arbitrary probability distributions. Finally, the construction here proposed is used to simulate quantum walks that match uniform random walks on the cycle and the torus.