Dirac equation as a quantum walk over the honeycomb and triangular lattices

A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the Dirac equation in (1+ 1) curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in (3 + 1) curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.

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Erratum to: Massless Dirac equation from Fibonacci discrete-time quantum walk

Quantum Studies Mathematics and Foundations ◽

10.1007/s40509-015-0042-x ◽

2015 ◽

Vol 2 (3) ◽

pp. 253-254

Author(s):

Giuseppe Di Molfetta ◽

Lauchlan Honter ◽

Ben B. Luo ◽

Tatsuaki Wada ◽

Yutaka Shikano

Keyword(s):

Dirac Equation ◽

Discrete Time ◽

Quantum Walk ◽

Time Quantum

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Quantum walk as a simulator of nonlinear dynamics: Nonlinear Dirac equation and solitons

Physical Review A ◽

10.1103/physreva.92.052336 ◽

2015 ◽

Vol 92 (5) ◽

Cited By ~ 14

Author(s):

Chang-Woo Lee ◽

Paweł Kurzyński ◽

Hyunchul Nha

Keyword(s):

Nonlinear Dynamics ◽

Dirac Equation ◽

Quantum Walk ◽

Nonlinear Dirac Equation

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Dynamical Triangulation Induced by Quantum Walk

Symmetry ◽

10.3390/sym12010128 ◽

2020 ◽

Vol 12 (1) ◽

pp. 128 ◽

Cited By ~ 3

Author(s):

Quentin Aristote ◽

Nathanaël Eon ◽

Giuseppe Di Molfetta

Keyword(s):

Dirac Equation ◽

Local Density ◽

Quantum Walk ◽

Triangular Grid ◽

Global Behavior ◽

Local Curvature ◽

Long Run ◽

Pachner Moves ◽

Dynamical Triangulation ◽

Random Fluctuations

We present the single-particle sector of a quantum cellular automaton, namely a quantum walk, on a simple dynamical triangulated 2 - manifold. The triangulation is changed through Pachner moves, induced by the walker density itself, allowing the surface to transform into any topologically equivalent one. This model extends the quantum walk over triangular grid, introduced in a previous work, by one of the authors, whose space-time limit recovers the Dirac equation in (2+1)-dimensions. Numerical simulations show that the number of triangles and the local curvature grow as t α e − β t 2 , where α and β parametrize the way geometry changes upon the local density of the walker, and that, in the long run, flatness emerges. Finally, we also prove that the global behavior of the walker, remains the same under spacetime random fluctuations.

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