Quantum walk on distinguishable non-interacting many-particles and indistinguishable two-particle

We present an investigation of many-particle quantum walks in systems of non-interacting distinguishable particles. Along with a redistribution of the many-particle density profile we show that the collective evolution of the many-particle system resembles the single-particle quantum walk evolution when the number of steps is greater than the number of particles in the system. For non-uniform initial states we show that the quantum walks can be effectively used to separate the basis states of the particle in position space and grouping like state together. We also discuss a two-particle quantum walk on a two- dimensional lattice and demonstrate an evolution leading to the localization of both particles at the center of the lattice. Finally we discuss the outcome of a quantum walk of two indistinguishable particles interacting at some point during the evolution.Comment: 8 pages, 7 figures, To appear in special issue: "quantum walks" to be published in Quantum Information Processin

Quantum Diffusions and Appell Systems

Within the algebraic framework of Hopf algebras, random walks and associated diffusion equations (master equations) are constructed and studied for two basic operator algebras of Quantum Mechanics i.e the Heisenberg-Weyl algebra (hw) and its q-deformed version hw_q. This is done by means of functionals determined by the associated coherent state density operators. The ensuing master equations admit solutions given by hw and hw_q-valued Appell systems.Comment: Latex 12 pages, no figures. Submitted to Journal of Computational and Applied Mathematics. Special Issue of Proccedings of Fifth Inter. Symp. on Orthogonal Polynomaials, Special Functions and their Application

Limits of quantum speedup in photosynthetic light harvesting

It has been suggested that excitation transport in photosynthetic light harvesting complexes features speedups analogous to those found in quantum algorithms. Here we compare the dynamics in these light harvesting systems to the dynamics of quantum walks, in order to elucidate the limits of such quantum speedups. For the Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria, we show that while there is indeed speedup at short times, this is short lived (70 fs) despite longer lived (ps) quantum coherence. Remarkably, this time scale is independent of the details of the decoherence model. More generally, we show that the distinguishing features of light-harvesting complexes not only limit the extent of quantum speedup but also reduce rates of diffusive transport. These results suggest that quantum coherent effects in biological systems are optimized for efficiency or robustness rather than the more elusive goal of quantum speedup.Comment: 9 pages, 6 figures. To appear in New Journal Physics, special issue on "Quantum Effects and Noise in Biomolecules." Updated to accepted versio

Quantum routing of information using chiral quantum walks

We address routing of classical and quantum information over quantum network, and show how to exploit chirality to achieve nearly optimal and robust transport. In particular, we prove how continuous time chiral quantum walks over a minimal graph may be used to model directional transfer and routing of information over a network. At first, we show how classical information, encoded onto an excitation localized at one vertex of a simple graph, may be sent to any other chosen location with nearly unit fidelity by tuning a single phase. Then, we prove that high-fidelity transport is also possible for coherent superpositions of states, i.e. for routing of quantum information. Furthermore, we show that by tuning the phase parameter one obtains universal quantum routing, i.e. indipendent on the input state. In our scheme, chirality is governed by a single phase, and the routing probability is robust against fluctuations of this parameter. Finally, we address characterization of quantum routers and show how to exploit the self energies of the graph to achieve high precision in estimating the phase parameter.Comment: This paper has been submitted to the Jonathan P. Dowling Memorial Special Issue of AVS QUANTUM SCIENCE (https://publishing.aip.org/publications/journals/special-topics/aqs/

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.Comment: Paper accepted for publication in Quantum Information Processing Journa

Birth and death processes and quantum spin chains

This papers underscores the intimate connection between the quantum walks generated by certain spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the Karlin-McGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.Comment: 14 page