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arXiv:1304.3690 · Journal of Physics A Mathematical and Theoretical

Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models

B. F. Venancio, F M Andrade, M G E da Luz

Published Apr 9, 2013• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/46/16/165302

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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.
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.
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which is also a quantum walk. Since the effective walking graph of the projected walk is not necessarily simpler than the original, this may bring new insights into the dynamics of some kinds of quantum walks using known results from thoroughly studied cases like Euclidean lattices. We use abstract treatment of the walking space and walker displacements in aim for a generality of the presented statements. Using this approach we also identify some pathological cases in which the projection mapping breaks down. For walks on lattices, the operation typically results in quantum walks with hyper-dimensional coin spaces. Such walks can, conversely, be viewed as projections of walks on inaccessible, larger spaces, and their properties can be inferred from the parental walk. We show that this is is the case for a lazy quantum walk, a walk with large coherent jumps and a walk on a circle with a twisted boundary condition. We also discuss the relation of this theory to the time-multiplexing optical implementations of quantum walks. Moreover, this manifestly irreversible operation can, in some cases and with a minor adjustment, be undone, and a quantum walk can be reconstructed from a set of its projections.
In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the currently widespread discrete-time quantum walk on a line. Here the walker at each time step can either remain in place or move in a fixed direction, e.g., rightward or upward. While both formulations are essentially equivalent, the present approach leads us to consider discrete Fourier transforms, which eventually results in obtaining explicit expressions for the wave functions in terms of finite sums and allows the use of efficient algorithms based on the fast Fourier transform. The wave functions here obtained govern the probability of finding the particle at any given location but determine as well the exit-time probability of the walker from a fixed interval, which is also analyzed.
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
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.
A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that it is equivalent to introducing spacetime-dependent local unitaries -whilst keeping the lattice fixed. By exploiting this duality between changes in geometry, and changes in local unitaries, we show that the spacetime-dependent QW simulates the Dirac equation in (2 + 1)-dimensional curved spacetime. Interestingly, the duality crucially relies on the non linear-independence of the three preferred directions of the honeycomb and triangular lattices: The same construction would fail for the square lattice. At the practical level, this result opens the possibility to simulate field theories on curved manifolds, via the quantum walk on different kinds of lattices.