Quantum walking in curved spacetime: discrete metric

Pablo Arrighi1, Giuseppe Di Molfetta2, and Stefano Facchini3

1Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, and IXXI, Lyon, France
2Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France and Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, Dr. Moliner 50, 46100-Burjassot, Spain
3Aix-Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators-differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.

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