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UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Mean hitting times of open quantum walks in terms of generalized inverses
Banff International Research Station for Mathematical Innovation and Discovery
Lardizabal, Carlos — 2019-10-20 In this talk we discuss the model of quantum Markov chains, due to S. Gudder, and most particularly the subset of open quantum walks, due to S. Attal et al., acting on finite graphs. As an iterative process, we use a monitoring procedure to determine the mean time for a quantum walker to visit some chosen vertex for the first time. We are interested in ways of calculating such hitting times besides making direct use of its definition and here we notice algebraic similarities and differences with the classical case. The case of unitary quantum walks remains an interesting open problem for which a solution could have potential applications to the associated theory of Schur functions. Non UBC Unreviewed Author affiliation: Federal University of Rio Grande do Sul Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Detecting two-dimensional topological states using quantum walks
Banff International Research Station for Mathematical Innovation and Discovery
Feder, David — 2019-10-21 We show that the evolution of two-component particles under a continuous-time quantum walk can reveal topological phases. A kink in the mean width of the walker distribution signals the closing of the energy gap, a prerequisite for a quantum phase transition between topological phases. For realistic and experimentally motivated Hamiltonians, the distribution in topologically non-trivial phases displays characteristic rings in the vicinity of the origin that are absent in topologically trivial phases. The results are expected to have immediate application to systems of ultracold atoms and photonic lattices, and should aid in the realization of topological states suitable for quantum computation. Non UBC Unreviewed Author affiliation: University of Calgary Researcher http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Quadratic speedup for finding marked vertices by quantum walks
Banff International Research Station for Mathematical Innovation and Discovery
Ambainis, Andris — 2019-10-22 A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk. <BR> Joint work with András Gilyén, Stacey Jeffery, Martins Kokainis, arxiv preprint 1903.07493. Non UBC Unreviewed Author affiliation: University of Latvia Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Finding a marked node on any graph by continuous time quantum walk
Banff International Research Station for Mathematical Innovation and Discovery
Roland, Jérémie — 2019-10-22 We provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any ergodic, reversible Markov chain P, in a time that is quadratically faster than the corresponding classical random walk on P. In the scenario where multiple nodes are marked, the running time of our algorithm scales as the square root of a quantity known as the extended hitting time. This solves an open problem concerning the difference between the running time of spatial search by discrete-time and continuous time quantum walk. We also show that the widely used Childs and Goldstone algorithm for spatial search by continuous-time quantum walk is quite restrictive: we identify limitations in its applicability whenever P is not state-transitive. We subsequently improve and extend this algorithm to be applicable for any P. Our generalizations imply that most hitherto published results on the performance of quantum spatial search in the Childs and Goldstone framework on specific graphs are particular cases of our result. However, we prove that the running time of the Childs and Goldstone algorithm and its subsequent improvement is suboptimal: our spatial search algorithm outperforms it. Our results can be adapted to a number of Markov chain-based quantum algorithms and will lead to exploring other connections between discrete-time and continuous-time quantum walks. <BR> Joint work with Shantanav Chakraborty and Leonardo Novo. Non UBC Unreviewed Author affiliation: Université libre de Bruxelles Researcher http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Open problems in continuous-time quantum walks
Banff International Research Station for Mathematical Innovation and Discovery
Coutinho, Gabriel — 2019-10-23 I will present several open problems in the context of continuous-time quantum walks in graphs. To each problem, I will try to outline their technical nature, and I will speculate which techniques could be applied to solve them. Non UBC Unreviewed Author affiliation: Federal University of Minas Gerais Researcher http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Average mixing matrix of quantum walks
Banff International Research Station for Mathematical Innovation and Discovery
Guo, Krystal — 2019-10-23 A quantum walk is governed by its transition matrix which is unitary; since this process is necessarily non-ergodic and one cannot speak of a stationary distribution, we study the average behaviour of the quantum walk. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. There has been a considerable amount of success in approaching questions about continuous-time quantum walks with tools in linear algebra and algebraic graph theory and we will discuss several recent works in this area, based on joint work with Chris Godsil, Gabriel Coutinho, Harmony Zhan and John Sinkovic. Non UBC Unreviewed Author affiliation: Universite libre de Bruxelles Postdoctoral http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Fractional revival of threshold graphs under Laplacian dynamics
Banff International Research Station for Mathematical Innovation and Discovery
Zhang, Xiaohong — 2019-10-23 Let $X$ be a graph, and denote its Laplacian matrix by $L$. Let $U(t) = e^{itL}$. Then $U(t)$ is a complex symmetric unitary matrix. We say that $X$ admits Laplacian fractional revival between vertices $j$ and $k$ at time $t = t_0$, if $U(t_0)e_j = \alpha e_j + \beta e_k$ for some complex numbers $\alpha$ and $\beta$ with $\beta\neq0$. In the special case where $\alpha=0$, we say there is perfect state transfer between vertices $j$ and $k$ at time $t = t_0$. Assume that a graph $X$ admits Laplacian fractional revival at time $t=t_0$ between vertices $1$ and $2$. We prove that for the spectral decomposition $L=\sum_{r=0}^q\theta_rE_r$ of $L$, for each $r=0,1,\ldots, q$, either $E_re_1=E_re_2$, or $E_re_1=-E_re_2$, depending on whether $e^{it_0\theta_r}$ equals to 1 or not. That is to say, vertices 1 and 2 are strongly cospectral with respect to $L$. We give a characterization of the parameters of threshold graphs that allow for Laplacian fractional revival between two vertices; those graphs can be used to generate more graphs with Laplacian fractional revival. We also characterize threshold graphs that admit Laplacian fractional revival between a subset of more than two vertices. Non UBC Unreviewed Author affiliation: University of Manitoba Graduate http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Quantum walks, orthogonal polynomials, and spectral graph theory
Banff International Research Station for Mathematical Innovation and Discovery
Zhan, Hanmeng — 2019-10-23 Some quantum walks can be modeled using weighted graphs, where each vertex represents a qubit, each weighted edge indicates the coupling strength between two qubits, and each weighted loop indicates the strength of the magnetic field of a qubit. In this talk, I will discuss two analytic approaches to quantum walks: orthogonal polynomials, which have been applied mostly to weighted paths, and spectral graphs theory, which has been applied mostly to simple unweighted graphs. I will also talk about some interesting relations between quantum walks on weights paths and quantum walks on simple unweighted graphs. <BR> Part of this talk is joint work with Gabriel Coutinho, Luv Vinet and Alexei Zhedanov. Non UBC Unreviewed Author affiliation: University of Waterloo Graduate http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Multivariate Krawtchouk polynomials and quantum walks on the associated graphs
Banff International Research Station for Mathematical Innovation and Discovery
Miki, Hiroshi — 2019-10-23 It is known that Krawtchouk polynomials play a central role in the analysis of (continuous-time) quantum walk on hypercube. Such walk also gives the spin lattice Hamiltonian where perfect state transfer occurs. In this talk (or presentation), we will start with multivariate Krawtchouk polynomials. We will introduce the associated graph and show that some spin lattice Hamiltonian in multi-dimension can be described as projections of quantum walk on this graph. In the model, fractional revival in addition to perfect state transfer is shown to take place. <BR> This is based on the joint-work with Satoshi Tsujimoto and Luc Vinet. Non UBC Unreviewed Author affiliation: Meteorological College Researcher http://creativecommons.org/licenses/by-nc-nd/4.0/
UBC cIRcle BIRS Workshop Lecture Videos Translation missing: fr.blacklight.search.logo
Time-band limiting: searching for a miracle, armed with a new motivation
Banff International Research Station for Mathematical Innovation and Discovery
Grunbaum, Francisco Alberto — 2019-10-24 The subject starts with C. Shannon questions at Bell Labs and the amazing answers found in the 60's by Slepian, Landau and Pollak. The crux of the matter is that in some naturally appearing problems in signal processing a certain integral operator admits an explicit commuting second order differential one, or a full matrix admits a commuting tridiagonal one. These operators are parametrized by the duration of the signal and its bandwidth. I have been trying to understand and extend this miracle for a long time, mainly in "non translation invariant cases". All the examples from Bell Labs involve Fourier analysis in different setups. I have managed to extend this to a few other cases. My "new motivation" comes from looking at two papers: "Maximal violations of Bell inequalities by position measurements" , J. Kiukas and R. Werner 2009 and "Properties of the entanglement Hamiltonian for finite free-fermion chains", V. Eisler and I. Peschel 2018. Both of these papers exploit the commutativity property in some physically important cases. My hope is that some members of the audience may point out a few more cases where this very exceptional mathematical phenomenon plays a physically relevant role. Non UBC Unreviewed Author affiliation: University of California, Berkeley Faculty http://creativecommons.org/licenses/by-nc-nd/4.0/

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