A systematic method to building Dirac quantum walks coupled to electromagnetic fields
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A systematic method to building Dirac quantum walks coupled to electromagnetic fields Gareth Jay1 · Fabrice Debbasch2 · Jingbo Wang1 Received: 13 January 2019 / Accepted: 6 November 2020 / Published online: 20 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac quantum walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a 3D walk on the cubic lattice. Recently introduced DQWs on the triangular lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method’s effectiveness, a new 3D walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed. Furthermore, we show the simulation of the established DQWs can be efficiently implemented on a quantum computer. Keywords Quantum walk · Dirac equation · Lattices
1 Introduction Quantum walks are unitary quantum automata first proposed by Feynman [25,47] that can be viewed as formal generalisations of classical random walks. First introduced systematically by Aharonov et al. [1] and Myers [44], they have found application in quantum information and algorithmic development [3,12,13,24,29,34,38,41,43]. They
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Gareth Jay [email protected] Fabrice Debbasch [email protected] Jingbo Wang [email protected]
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Physics Department, The University of Western Australia, Perth, WA 6009, Australia
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LERMA, CNRS, Observatoire de Paris, Université PSL, Sorbonne Université, 75005 Paris, France
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can also be used as quantum simulators [10,11,18,23,28,33,45,48,51], where the lattice represents a discretisation of continuous space. More ambitiously, quantum walks may represent a potentially realistic discrete space-time underlying the apparently continuous physical universe [15]. It has been shown that several discrete-time quantum walks (DTQWs) defined on regular square lattices simulate the Dirac dynamics in various space-time dimensions and that these Dirac quantum walks (DQWs) can be coupled to various discrete gauge fields [4–7,14,17,19,22,23,42]. It has also more recently been shown that 2D DQWs can be defined on regular non-square lattices like the triangle and honeycomb lattice [8,30]. The aim of the present article is to extend these results by developing a new systematic approach to construct DQWs coupled to EM fields. The new approach is based on discretising the Hamiltonian [11] in terms of directional derivatives [8]. It presupposes the lattice is symmetric in the sense that for every edge extending from a vertex there exi
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