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

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.

A quantum walk is a time-homogeneous quantum-mechanical process on a graph defined by analogy to classical random walk. The quantum walker is a particle that moves from a given vertex to adjacent vertices in quantum superposition. We consider a generalization to interacting systems with more than one walker, such as the Bose-Hubbard model and systems of fermions or distinguishable particles with nearest-neighbor interactions, and show that multiparticle quantum walk is capable of universal quantum computation. Our construction could, in principle, be used as an architecture for building a scalable quantum computer with no need for time-dependent control.

Universal Computation by Multiparticle Quantum Walk

Full control over the dynamics of interacting, indistinguishable quantum particles is an important prerequisite for the experimental study of strongly correlated quantum matter and the implementation of high-fidelity quantum information processing. We demonstrate such control over the quantum walk-the quantum mechanical analog of the classical random walk-in the regime where dynamics are dominated by interparticle interactions. Using interacting bosonic atoms in an optical lattice, we directly observed fundamental effects such as the emergence of correlations in two-particle quantum walks, as well as strongly correlated Bloch oscillations in tilted optical lattices. Our approach can be scaled to larger systems, greatly extending the class of problems accessible via quantum walks.

https://science.sciencemag.org/content/sci/347/6227/1229.full.pdf

Strongly correlated quantum walks in optical lattices

This paper combines background information about the IBM Journal of Research and Development with guidelines for the preparation of Journal manuscripts. The purpose is to acquaint authors with the Journal as a primary, professional publication and to present suggestions to ease the work of author and editor in preparing clear, concise, and useful manuscripts.

IBM journal of research and development: information for authors

We show that the presence of an interaction in the quantum walk of two atoms leads to the formation of a stable compound, a molecular state The wave function of the molecule decays exponentially in the relative position of the two atoms; hence it constitutes a true bound state Furthermore, for a certain class of interactions, we develop an effective theory and find that the dynamics of the molecule is described by a quantum walk in its own right We propose a setup for the experimental realization as well as sketch the possibility to observe quasi-particle effects in quantum many-body systems

https://iopscience.iop.org/article/10.1088/1367-2630/14/7/073050/pdf

Molecular binding in interacting quantum walks

Quantum walks have by now been realized in a large variety of different physical settings. In some of these, particularly with trapped ions, the walk is implemented in phase space, where the corresponding position states are not orthogonal. We develop a general description of such a quantum walk and show how to map it into a standard one with orthogonal states, thereby making available all the tools developed for the latter. This enables a variety of experiments, which can be implemented with smaller step sizes and more steps. Tuning the nonorthogonality allows for an easy preparation of extended states such as momentum eigenstates, which travel at a well-defined speed with low dispersion. We introduce a method to adjust their velocity by momentum shifts, which allows us to experimentally probe the dispersion relation, providing a benchmarking tool for the quantum walk, and to investigate intriguing effects such as the analog of Bloch oscillations.

Quantum Walks with Nonorthogonal Position States

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Molecular binding in interacting quantum walks

Quantum Walks with Nonorthogonal Position States