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arXiv:1605.01605 · Physical Review A

Quantum walks and non-Abelian discrete gauge theory

Pablo Arnault, Giuseppe Di Molfetta, Marc Brächet, Fabrice Debbasch

Published Jul 19, 2016• 56 citations
arXivHTMLPDF

Abstract

A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete $\text{U}(N)$ gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual $\text{U}(N)$ gauge fields in two-dimensional spacetime. A discrete generalization of the usual $\text{U}(N)$ curvature is also constructed. An alternate interpretation of these results in terms of superimposed $\text{U}(1)$ Maxwell fields and $\text{SU}(N)$ gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e., nonquantum) motions in classical $\text{SU}(2)$ fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.

DOI
10.1103/physreva.94.012335

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A quantum computer, i.e. utilizing the resources of quantum physics, superposition of states and entanglement, could furnish an exponential gain in computing time. A simulation using such resources is called a quantum simulation. The advantage of quantum simulations over classical ones is well established at the theoretical, i.e. software level. Their practical benefit requires their implementation on a quantum hardware. The quantum computer, i.e. the universal one (see below), has not seen the light of day yet, but the efforts in this direction are both growing and diverse. Also, quantum simulation has already been illustrated by numerous experimental proofs of principle, thanks too small-size and specific-task quantum computers or simulators. Quantum walks are particularly-studied quantum-simulation schemes, being elementary bricks to conceive any quantum algorithm, i.e. to achieve so-called universal quantum computation. The present thesis is a step more towards a simulation of quantum field theories based on discrete-time quantum walks (DTQWs). Indeed, it is shown, in certain cases, how DTQWs can simulate, in the continuum, the action of Yang-Mills gauge fields on fermionic matter, and the retroaction of the latter on the gauge-field dynamics. The suggested schemes preserve gauge invariance on the spacetime lattice, i.e. not only in the continuum. In the (1+2)-dimensional Abelian case, consistent lattice equivalents to both Maxwell's equations and the current conservation are suggested. In the (1+1)-dimensional non-Abelian case, a lattice version of the non-Abelian field strength is suggested. Moreover, it is shown how this fermionic matter based on DTQWs can be coupled to relativistic gravitational fields of the continuum, i.e. to curved spacetimes, in 1+2 dimensions.
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