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Register a new userUnidirectional quantum walks: evolution and exit times
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Authors
Miquel Montero
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In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the nowadays 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 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.
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We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, and the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The results we obtain extend those for classical Markov chains.
Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t to infty$ of all joint moments of two components of walkers pseudovelocity, $X_t/t$ and $Y_t/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer-simulations is also shown.
Quantum walks constitute important tools in different applications, especially in quantum algorithms. To a great extent their usefulness is due to unusual diffusive features, allowing much faster spreading than their classical counterparts. Such behavior, although frequently credited to intrinsic quantum interference, usually is not completely characterized. Using a recently developed Greens function approach [Phys. Rev. A {bf 84}, 042343 (2011)], here it is described -- in a rather general way -- the problem dynamics in terms of a true sum over paths history a la Feynman. It allows one to explicit identify interference effects and also to explain the emergence of superdiffusivity. The present analysis has the potential to help in designing quantum walks with distinct transport properties.
The phenomenon of localization usually happens due to the existence of disorder in a medium. Nevertheless, certain quantum systems allow dynamical localization solely due to the nature of internal interactions. We study a discrete time quantum walker which exhibits disorder free localization. The quantum walker moves on a one-dimensional lattice and interacts with on-site spins by coherently rotating them around a given axis at each step. Since the spins do not have dynamics of their own, the system poses the local spin components along the rotation axis as an extensive number of conserved moments. When the interaction is weak, the spread of the walker shows subdiffusive behaviour having downscaled ballistic tails in the evolving probability distribution at intermediate time scales. However, as the interaction gets stronger the walker gets exponentially localized in the complete absence of disorder in both lattice and initial state. Using a matrix-product-state ansatz, we investigate the relaxation and entanglement dynamics of the on-site spins due to their coupling with the quantum walker. Surprisingly, we find that even in the delocalized regime, entanglement growth and relaxation occur slowly unlike marjority of the other models displaying a localization transition.
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