
J. A. Izaac and J. B. Wang. Systematic dimensionality reduction for continuoustime quantum walks of interacting fermions. Physical Review E, September 2017. doi:10.1103/PhysRevE.96.032136 [ Link ] [ Abstract ]
To extend the continuoustime quantum walk (CTQW) to simulate P distinguishable particles on a graph G composed of N vertices, the Hamiltonian of the system is expanded to act on an NPdimensional Hilbert space, in effect, simulating the multiparticle CTQW on graph G via a singleparticle CTQW propagating on the Cartesian graph product $G^\square P$. The properties of the Cartesian graph product have been well studied, and classical simulation of multiparticle CTQWs are common in the literature. However, the above approach is generally applied as is when simulating indistinguishable particles, with the particle statistics then applied to the propagated NP state vector to determine walker probabilities. We address the following question: How can we modify the underlying graph structure $G^\square P$ in order to simulate multiple interacting fermionic CTQWs with a reduction in the size of the state space? In this paper, we present an algorithm for systematically removing \textquotedblleftredundant\textquotedblright and forbidden quantum states from consideration, which provides a significant reduction in the effective dimension of the Hilbert space of the fermionic CTQW. As a result, as the number of interacting fermions in the system increases, the classical computational resources required no longer increases exponentially for fixed N.

J. A. Izaac, J. B. Wang, P. C. Abbott, and X. S. Ma. Quantum centrality testing on directed graphs via PTsymmetric quantum walks. Physical Review A, September 2017. doi:10.1103/PhysRevA.96.032305 [ Link ] [ Abstract ]
Various quantumwalkbased algorithms have been proposed to analyze and rank the centrality of graph vertices. However, issues arise when working with directed graphs: the resulting nonHermitian Hamiltonian leads to nonunitary dynamics, and the total probability of the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using PTsymmetric quantum walks, allowing probabilityconserving nonunitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set, and can be extended to cover interdependent networks of directed graphs. Previous work has shown centrality measures based on the continuoustime quantum walk provide an eigenvectorlike quantum centrality; using the PTsymmetric framework, we extend these centrality algorithms to directed graphs with a significantly reduced Hilbert space compared to previous proposals. In certain cases, this centrality measure provides an advantage over classical algorithms used in network analysis, for example, by breaking vertex rank degeneracy. Finally, we perform a statistical analysis over ensembles of random graphs, and show strong agreement with the classical PageRank measure on directed acyclic graphs.

Aaron C. H. Hurst, Joshua A. Izaac, Fouzia Altaf, Vincent Baltz, and Peter J. Metaxas. Reconfigurable magnetic domain wall pinning using vortexgenerated magnetic fields. Applied Physics Letters, 110(18):182404, May 2017. doi:10.1063/1.4982237 [ Link ] [ Abstract ]
Although often important for domain wall device applications, reproducible fabrication of pinning sites at the nanoscale remains challenging. Here, we demonstrate that the stray magnetic field generated beneath magnetic vortex cores can be used to generate localized pinning sites for magnetic domain walls in an underlying, perpendicularly magnetized nanostrip. Moreover, we show that the pinning strength can be tuned by switching the vortex core polarity: switching the core polarity so that it is aligned with the magnetization of the expanding domain (rather than against it) can reduce the vortexmediated wall depinning field by between 40\% and 90\%, depending on the system geometry. Significant reductions in the depinning field are also demonstrated in narrow strips by shifting the core away from the strips' centers.

S. S. Zhou, T. Loke, J. A. Izaac, and J. B. Wang. Quantum fourier transform in computational basis. Quantum Information Processing, 16(3):82, March 2017. doi:10.1007/s1112801715150 [ Link ] [ Abstract ]
The conventional Quantum Fourier Transform, with exponential speedup compared to the classical Fast Fourier Transform, has played an important role in quantum computation as a vital part of many quantum algorithms (most prominently, the Shor's factoring algorithm). However, situations arise where it is not sufficient to encode the Fourier coefficients within the quantum amplitudes, for example in the implementation of control operations that depend on Fourier coefficients. In this paper, we detail a new quantum algorithm to encode the Fourier coefficients in the computational basis, with success probability $1\delta$ and desired precision $\epsilon$. Its time complexity $O((\log N)^2\log(N/\delta)/\epsilon)\big)$ depends polynomially on $\log(N)$, where $N$ is the problem size, and linearly on $\log(1/\delta)$ and $1/\epsilon$. We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians.

Josh A. Izaac, Xiang Zhan, Zhihao Bian, Kunkun Wang, Jian Li, Jingbo B. Wang, and Peng Xue. Centrality measure based on continuoustime quantum walks and experimental realization. Physical Review A, 95(3):032318, March 2017. doi:10.1103/PhysRevA.95.032318 [ Link ] [ Abstract ]
Network centrality has important implications well beyond its role in physical and information transport analysis; as such, various quantumwalkbased algorithms have been proposed for measuring network vertex centrality. In this work, we propose a continuoustime quantum walk algorithm for determining vertex centrality, and show that it generalizes to arbitrary graphs via a statistical analysis of randomly generated scalefree and Erd\HosR\'enyi networks. As a proof of concept, the algorithm is detailed on a fourvertex star graph and physically implemented via linear optics, using spatial and polarization degrees of freedoms of single photons. This paper reports a successful physical demonstration of a quantum centrality algorithm.

J. A. Izaac, J. B. Wang, P. C. Abbott, and X. S. Ma. Quantum centrality testing on directed graphs via PTsymmetric quantum walks. arXiv:1607.02673 [quantph], July 2016. [ Link ] [ Abstract ]
Various quantumwalk based algorithms have been proposed to analyse and rank the centrality of graph vertices. However, issues arise when working with directed graphs  the resulting nonHermitian Hamiltonian leads to nonunitary dynamics, and the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using PTsymmetric quantum walks, allowing probability conserving nonunitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set. It can be extended to cover interdependent networks of directed graphs. Finally, we use this framework to implement a quantum algorithm for centrality testing on directed graphs with a significantly reduced Hilbert space compared to previous proposals. We demonstrate that, in certain cases, this centrality measure provides an advantage over classical algorithms used in network analysis, for example by breaking vertex rank degeneracy.

A. Mahasinghe, J. A. Izaac, J. B. Wang, and J. K. Wijerathna. Phasemodified CTQW unable to distinguish strongly regular graphs efficiently. Journal of Physics A: Mathematical and Theoretical, 48(26):265301, July 2015. doi:10.1088/17518113/48/26/265301 [ Link ] [ Abstract ]
Various quantum walkbased algorithms have been developed, aiming to distinguish nonisomorphic graphs with polynomial scaling, within both the discretetime quantum walk (DTQW) and continuoustime quantum walk (CTQW) frameworks. Whilst both the singleparticle DTQW and CTQW have failed to distinguish nonisomorphic strongly regular graph families (prompting the move to multiparticle graph isomorphism (GI) algorithms), the singleparticle DTQW has been successfully modified by the introduction of a phase factor to distinguish a wide range of graphs in polynomial time. In this paper, we prove that an analogous phase modification to the single particle CTQW does not have the same distinguishing power as its discretetime counterpart, in particular it cannot distinguish strongly regular graphs with the same family parameters with the same efficiency.

Josh A. Izaac and Jingbo B. Wang. pyCTQW: a continuoustime quantum walk simulator on distributed memory computers. Computer Physics Communications, 186(0):81–92, January 2015. doi:10.1016/j.cpc.2014.09.011 [ Link ] [ Abstract ]
In the general field of quantum information and computation, quantum walks are playing an increasingly important role in constructing physical models and quantum algorithms. We have recently developed a distributed memory software package pyCTQW, with an objectoriented Python interface, that allows efficient simulation of large multiparticle CTQW (continuoustime quantum walk)based systems. In this paper, we present an introduction to the Python and Fortran interfaces of pyCTQW, discuss various numerical methods of calculating the matrix exponential, and demonstrate the performance behavior of pyCTQW on a distributed memory cluster. In particular, the Chebyshev and Krylovsubspace methods for calculating the quantum walk propagation are provided, as well as methods for visualization and data analysis.

J. A. Izaac, J. B. Wang, and Z. J. Li. Continuoustime quantum walks with defects and disorder. Physical Review A, 88(4):042334, October 2013. doi:10.1103/PhysRevA.88.042334 [ Link ] [ Abstract ]
With the advent of physical implementations of quantum walks, a general theoretical and efficient numerical framework is required for the study of their interactions with defects and disorder. In this paper, we derive analytic expressions for the eigenstates of a onedimensional continuoustime quantum walk interacting with a single defect, before investigating the effects of multiple diagonal defects and disorder, with emphasis on its transmission and reflection properties. Complex resonance behavior is demonstrated, showing alternating bands of zero and perfect transmission for various defect parameters. Furthermore, we provide an efficient numerical method to characterize quantum walks in the presence of diagonal disorder, paving the way for selective control of quantum walks via the optimization of positiondependent defects. The numerical method can be readily extended to higher dimensions and multiple interacting walkers.

Z. J. Li, J. A. Izaac, and J. B. Wang. Positiondefectinduced reflection, trapping, transmission, and resonance in quantum walks. Physical Review A, 87(1):012314, January 2013. doi:10.1103/PhysRevA.87.012314 [ Link ] [ Abstract ]
We investigate the scattering properties of quantum walks by considering single and double position defects on a onedimensional line. This corresponds to introducing, at designated positions, delta potential defects for continuoustime quantum walks and phasedefect Hadamard coins for discrete time quantum walks. The delta potential defects can be readily considered as potential barriers in discrete position space, which affect the time evolution of the system in a similar way as the quantum wavepacket dynamics in a continuous position space governed by Schr\"odinger's equation. Although there is no direct analogy of potential barriers in the theoretical formulation of discrete time quantum walks, in this paper we show that the phase defects in the coin space can be utilized to provide similar scattering effects. This study provides means of controlling the scattering properties of quantum walks by introducing designated positiondependent defects.