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@article{Loke_2017,

title = {Efficient quantum circuits for continuous-time quantum walks on composite graphs},

author = {T Loke and J B Wang},

url = {https://doi.org/10.1088/1751-8121/aa53a9},

doi = {10.1088/1751-8121/aa53a9},

year  = {2017},

date = {2017-01-01},

urldate = {2017-01-01},

journal = {Journal of Physics A: Mathematical and Theoretical (IF 2.132)},

volume = {50},

number = {5},

pages = {055303},

publisher = {IOP Publishing},

abstract = {In this paper, we investigate the simulation of continuous-time quantum walks on specific classes of graphs, for which it is possible to fast-forward the time-evolution operator to achieve constant-time simulation complexity and to perform the simulation exactly, i.e. , while maintaining efficiency. In particular, we discuss two classes of composite graphs, commuting graphs and Cartesian product of graphs, that contain classes of graphs which can be simulated in this fashion. This allows us to identify new families of graphs that we can efficiently simulate in a quantum circuit framework, providing practical and explicit means to explore quantum-walk based algorithms in laboratories.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{doi:10.1098/rsos.160906,

title = {Efficient quantum circuits for dense circulant and circulant like operators},

author = {S S Zhou and J B Wang},

url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsos.160906},

doi = {10.1098/rsos.160906},

year  = {2017},

date = {2017-01-01},

journal = {Royal Society Open Science},

volume = {4},

number = {5},

pages = {160906},

abstract = {Circulant matrices are an important family of operators, which have a wide range of applications in science and engineering-related fields. They are, in general, non-sparse and non-unitary. In this paper, we present efficient quantum circuits to implement circulant operators using fewer resources and with lower complexity than existing methods. Moreover, our quantum circuits can be readily extended to the implementation of Toeplitz, Hankel and block circulant matrices. Efficient quantum algorithms to implement the inverses and products of circulant operators are also provided, and an example application in solving the equation of motion for cyclic systems is discussed.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{LOKE201764,

title = {Efficient quantum circuits for Szegedy quantum walks},

author = {T Loke and J B Wang},

url = {http://www.sciencedirect.com/science/article/pii/S0003491617301124},

doi = {https://doi.org/10.1016/j.aop.2017.04.006},

issn = {0003-4916},

year  = {2017},

date = {2017-01-01},

urldate = {2017-01-01},

journal = {Annals of Physics (IF 2.730)},

volume = {382},

pages = {64 - 84},

abstract = {A major advantage in using Szegedy’s formalism over discrete-time and continuous-time quantum walks lies in its ability to define a unitary quantum walk by quantizing a Markov chain on a directed or weighted graph. In this paper, we present a general scheme to construct efficient quantum circuits for Szegedy quantum walks that correspond to classical Markov chains possessing transformational symmetry in the columns of the transition matrix. In particular, the transformational symmetry criteria do not necessarily depend on the sparsity of the transition matrix, so this scheme can be applied to non-sparse Markov chains. Two classes of Markov chains that are amenable to this construction are cyclic permutations and complete bipartite graphs, for which we provide explicit efficient quantum circuit implementations. We also prove that our scheme can be applied to Markov chains formed by a tensor product. We also briefly discuss the implementation of Markov chains based on weighted interdependent networks. In addition, we apply this scheme to construct efficient quantum circuits simulating the Szegedy walks used in the quantum Pagerank algorithm for some classes of non-trivial graphs, providing a necessary tool for experimental demonstration of the quantum Pagerank algorithm.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{FALLOON2017162,

title = {QSWalk: A Mathematica package for quantum stochastic walks on arbitrary graphs},

author = {Peter E Falloon and Jeremy Rodriguez and Jingbo B Wang},

url = {http://www.sciencedirect.com/science/article/pii/S0010465517301029},

doi = {https://doi.org/10.1016/j.cpc.2017.03.014},

issn = {0010-4655},

year  = {2017},

date = {2017-01-01},

urldate = {2017-01-01},

journal = {Computer Physics Communications (IF 4.390)},

volume = {217},

pages = {162 - 170},

abstract = {We present a Mathematica package, QSWalk, to simulate the time evaluation of Quantum Stochastic Walks (QSWs) on arbitrary directed and weighted graphs. QSWs are a generalization of continuous time quantum walks that incorporate both coherent and incoherent dynamics and as such, include both quantum walks and classical random walks as special cases. The incoherent component allows for quantum walks along directed graph edges. The dynamics of QSWs are expressed using the Lindblad formalism, originally developed for open quantum systems, which frames the problem in the language of density matrices. For a QSW on a graph of N vertices, we have a sparse superoperator in an N2-dimensional space, which can be solved efficiently using the built-in MatrixExp function in Mathematica. We illustrate the use of the QSWalk package through several example case studies. Program summary Program Title: QSWalk.m Program Files doi: http://dx.doi.org/10.17632/8rwd3j9zhk.1 Licensing provisions: GNU General Public License 3 (GPL) Programming language: Mathematica. Nature of problem: The QSWalk package provides a method for simulating quantum stochastic walks on arbitrary (directed/undirected, weighted/unweighted) graphs. Solution method: For an N-vertex graph, the solution of a quantum stochastic walk can be expressed as an N2×N2 sparse matrix exponential. The QSWalk package makes use of Mathematica’s sparse linear algebra routines to solve this efficiently. Restrictions: The size of graphs that can be treated is constrained by available memory.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}