Discrete-time quantum walks and graph structures

Godsil, Chris; Zhan, Hanmeng

doi:10.1016/j.jcta.2019.05.003

Cited by 22 publications

(13 citation statements)

References 23 publications

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“…does not converge, its time average converges, and the limit can be expressed using the spectral idempotents of U. This was observed in Aharonov, Ambainis, Kempe, and Vazirani [1], and we will state the result following the notation in Godsil and Zhan [10].…”

Section: Quantum Walks With Oraclesmentioning

confidence: 81%

The average search probabilities of discrete-time quantum walks

Zhan¹

2021

Preprint

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We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an augmented graph. We then consider the average search probability on a distance regular graph, and find a formula in terms of the adjacency matrix of its vertexdeleted subgraph. In particular, for any family of • complete graphs, or• strongly regular graphs, or• distance regular graphs of a fixed parameter d, varying valency k and varying size n, such that k d−1 /n vanishes as k increases, the average search probability approaches 1/4 as the valency goes to infinity. We also present a more relaxed criterion, in terms of the intersection array, for this limit to be approached by distance regular graphs.

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Section: Quantum Walks With Oraclesmentioning

confidence: 81%

The average search probabilities of discrete-time quantum walks

Zhan¹

2021

Preprint

Self Cite

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“…This is in sharp contrast to the Schrödinger equation, which is often characterized as formally being simply a heat equation with an imaginary scale factor k, without any elaboration as to what, if anything, that might mean in a physical sense. That being said, there is also a growing body of research devoted to quantum mechanics with discrete time on quantum graphs that does frequently makes use of that formal diffusion-Schrödinger relationship [22][23][24]. These quantum walk approaches are essentially discrete extensions of the aforementioned pathintegral formulation of Feynman, Dirac, et al, so that along these discrete quantum-walk steps, a Lagrangian or Hamiltonian is integrated, so as to derive the particle's complexvalued phase factor (or alternatively, the overall phase evolution occurs by way of discrete phase shifts obtained by flipping a "quantum coin").…”

Section: Discussionmentioning

confidence: 99%

Brownian-Huygens Propagation: Modeling Wave Functions with Discrete Particle-Antiparticle Random Walks

Hrgovčić

2022

Int J Theor Phys

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We present a simple method of discretely modeling solutions of the classical wave and Klein-Gordon equations using variations of random walks on a graph. Consider a collection of particles executing random walks on an undirected bipartite graph embedded in $$\mathbb {R}^{D}$$ R D at discrete times $$\mathbb {Z}$$ Z , and assume those walks are “heat-like”, in the sense that a (conserved) density of particles obeys the heat equation in the continuum limit. If the particles possess a binary degree of freedom (so that they may be said to be either particles or “antiparticles”, i.e., “positive” or “negative), then there exist closely related branching random walks on the same graph that are “wave-like”, in the sense that their (also conserved) net density obeys the D-dimensional classical wave equation. Such wave-like paths can be generated even on random graphs. The transformation by which the heat-like random walks become wave-like branching random walks is as follows: at every time step, any “incoming” particle arriving at any node X of the graph along some edge eX creates a particle-antiparticle pair at that node, with the stipulation that the newly created particle with the opposite sign of the incoming particle must initially step (in “Huygens” back-propagation fashion) along eX in the reverse direction, while the other two take a step (in “Brownian” fashion) along any edge of X (including possibly eX) with equal likelihood, and chosen independently. An additional degree of freedom (resulting in “bra” and “ket” particles) leads to quasi-probability densities proportional to the square of the wave functions.

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“…An example of this is the quantum algorithm for element distinctness by Ambainis [5]. Additionally, quantum walk algorithms can also be used to search and ind graph properties [33,43,48,75,76,86]. Quantum random walks can be seen as a quantum mechanical generalization of classical random walks.…”

Section: Quantum Random Walks 91 Problem Definition and Backgroundmentioning

confidence: 99%

Quantum Algorithm Implementations for Beginners

Abhijith

Adedoyin

Ambrosiano

et al. 2022

ACM Transactions on Quantum Computing

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As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims to explain the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.

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Discrete-time quantum walks and graph structures

Cited by 22 publications

References 23 publications

The average search probabilities of discrete-time quantum walks

The average search probabilities of discrete-time quantum walks

Brownian-Huygens Propagation: Modeling Wave Functions with Discrete Particle-Antiparticle Random Walks

Quantum Algorithm Implementations for Beginners

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