Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete-time quantum walks.

Quantum walks: a comprehensive review

Photonics is a promising platform for implementing universal quantum information processing. Its main challenges include precise control of massive circuits of linear optical components and effective implementation of entangling operations on photons. By using large-scale silicon photonic circuits to implement an extension of the linear combination of quantum operators scheme, we realize a fully programmable two-qubit quantum processor, enabling universal two-qubit quantum information processing in optics. The quantum processor is fabricated with mature CMOS-compatible processing and comprises more than 200 photonic components. We programmed the device to implement 98 different two-qubit unitary operations (with an average quantum process fidelity of 93.2 ± 4.5%), a two-qubit quantum approximate optimization algorithm, and efficient simulation of Szegedy directed quantum walks. This fosters further use of the linear-combination architecture with silicon photonics for future photonic quantum processors.

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Large-scale silicon quantum photonics implementing arbitrary two-qubit processing

Dynamical systems: stability, symbolic dynamics and chaos

Quantum computation, one of the latest joint ventures between physics and the theory of computation, is a scientific field whose main goals include the development of hardware and algorithms based on the quantum mechanical properties of those physical systems used to implement such algorithms.Solving difficult tasks (for example, the Satisfiability Problem and other NP-complete problems) requires the development of sophisticated algorithms, many ofwhich employ stochastic processes as their mathematical basis.Discrete random walks are a popular choice among those stochastic processes.Inspired on the success of discrete random walks in algorithm development, quantum walks, an emerging field of quantum computation, is a generalization of random walks into the quantum mechanical world.The purpose of this lecture is to provide a concise yet comprehensive introduction to quantum walks.Table of Contents: Introduction / Quantum Mechanics / Theory of Computation / Classical Random Walks / Quantum Walks / Computer Science and Quantum Walks / Conclusions

Quantum Walks for Computer Scientists

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. 
In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.

We study a generalized Hadamard walk in one dimension with three inner states. The particle governed by the three-state quantum walk moves, in superposition, both to the left and to the right according to the inner state. In addition to these two degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle. In contrast with the Hadamard walk with two inner states on a line, the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states. This implies that the particle is trapped near the origin after a long time with high probability.

https://arxiv.org/pdf/quant-ph/0507207.pdf

One-dimensional three-state quantum walk

We treat three types of measures of the quantum walk (QW) with the spatial perturbation at the origin, which was introduced by Konno (Quantum Inf Proc 9:405, 2010): time averaged limit measure, weak limit measure, and stationary measure. From the first two measures, we see a coexistence of the ballistic and localized behaviors in the walk as a sequential result following (Konno in Quantum Inf Proc 9:405, 2010; Quantum Inf Proc 8:387---399, 2009). We propose a universality class of QWs with respect to weak limit measure. It is shown that typical spatial homogeneous QWs with ballistic spreading belong to the universality class. We find that the walk treated here with one defect also belongs to the class. We mainly consider the walk starting from the origin. However when we remove this restriction, we obtain a stationary measure of the walk. As a consequence, by choosing parameters in the stationary measure, we get the uniform measure as a stationary measure of the Hadamard walk and a time averaged limit measure of the walk with one defect respectively.

Limit measures of inhomogeneous discrete-time quantum walks in one dimension

Spectral and asymptotic properties of Grover walks on crystal lattices

We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization of the corresponding QW. We find the following two fundamental derivations of localization of the QW. The first one is the set of all the $\ell^2$ summable eigenvectors of the corresponding RW. The second one is the orthogonal complement, whose eigenvalues are $\pm 1$, of the subspace induced by the RW. In particular, as a consequence, for an infinite half line, we show that localization of the QW can be ensured by the positive recurrence of the corresponding RWs, and also that the existence of only one self loop affects localization properties.

Localization of quantum walks induced by recurrence properties of random walks

We give an explicit formula of the generator of an abstract Szegedy evolution operator in terms of the discriminant operator of the evolution. We also characterize the asymptotic behavior of a quantum walker through the spectral property of the discriminant operator by using the discrete analog of the RAGE theorem.

Etsuo Segawa

Papers

One-dimensional three-state quantum walk

Limit measures of inhomogeneous discrete-time quantum walks in one dimension

Spectral and asymptotic properties of Grover walks on crystal lattices

Localization of quantum walks induced by recurrence properties of random walks

Generator of an abstract quantum walk