21 research outputs found
Decoherence on Staggered Quantum Walks
Decoherence phenomenon has been widely studied in different types of quantum
walks. In this work we show how to model decoherence inspired by percolation on
staggered quantum walks. Two models of unitary noise are described: breaking
polygons and breaking vertices. The evolution operators subject to these noises
are obtained and the equivalence to the coined quantum walk model is presented.
Further, we numerically analyze the effect of these decoherence models on the
two-dimensional grid of -cliques. We examine how these perturbations affect
the quantum walk based search algorithm in this graph and how expanding the
tessellations intersection can make it more robust against decoherence.Comment: 17 pages, 14 figure
Spatial search in a honeycomb network
The spatial search problem consists in minimizing the number of steps
required to find a given site in a network, under the restriction that only
oracle queries or translations to neighboring sites are allowed. In this paper,
a quantum algorithm for the spatial search problem on a honeycomb lattice with
sites and torus-like boundary conditions. The search algorithm is based on
a modified quantum walk on a hexagonal lattice and the general framework
proposed by Ambainis, Kempe and Rivosh is used to show that the time complexity
of this quantum search algorithm is .Comment: 10 pages, 2 figures; Minor typos corrected, one Reference added.
accepted in Math. Structures in Computer Science, special volume on Quantum
Computin
Quantum Search Algorithms on Hierarchical Networks
The "abstract search algorithm" is a well known quantum method to find a
marked vertex in a graph. It has been applied with success to searching
algorithms for the hypercube and the two-dimensional grid. In this work we
provide an example for which that method fails to provide the best algorithm in
terms of time complexity. We analyze search algorithms in degree-3 hierarchical
networks using quantum walks driven by non-groverian coins. Our conclusions are
based on numerical simulations, but the hierarchical structures of the graphs
seems to allow analytical results.Comment: IEEE Information Theory Workshop 201
Mixing Times in Quantum Walks on Two-Dimensional Grids
Mixing properties of discrete-time quantum walks on two-dimensional grids
with torus-like boundary conditions are analyzed, focusing on their connection
to the complexity of the corresponding abstract search algorithm. In
particular, an exact expression for the stationary distribution of the coherent
walk over odd-sided lattices is obtained after solving the eigenproblem for the
evolution operator for this particular graph. The limiting distribution and
mixing time of a quantum walk with a coin operator modified as in the abstract
search algorithm are obtained numerically. On the basis of these results, the
relation between the mixing time of the modified walk and the running time of
the corresponding abstract search algorithm is discussed.Comment: 11 page
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