# Quantum Walking in a Discrete Geometry (master student internship)

Supervisors: Pablo Arrighi (Professor), Giuseppe Di Molfetta (Associate Professor).

Contact details: pablo.arrighi@univ-amu.fr or giuseppe.dimolfetta@lif.univ-mrs.fr

Place: Laboratoire d’Informatique Fondamentale (LIF), Natural Computing team (CaNa).

Scientific environment: The CaNa research group (Pablo Arrighi, Giuseppe Di Molfetta, Kevin Perrot, Sylvain Sené) seeks to capture at the formal level some of the fundamental paradigms of theoretical physics and biology, via the models and approaches of theoretical computer science and discrete mathematics. The group is located in Luminy, Marseille, France, and benefits from a rich scientific environment with the Cellular Automata experts of I2M (Pierre Guillon, Guillaume Theyssier) and the physicists from CPT (Alberto Vega, Thomas Krajewski).

Theme: Quantum Walks are the natural, quantum version of Random Walks. According to Feynman, quantum mechanics is a but theory of probabilities “with minus signs”. Indeed, one could say that probabilities get replaced by complex numbers. More specifically, whereas a random walker moves either left, or right, with probability a half, a quantum walker moves left with complex amplitude α, and right with amplitude β, where |α|^2 +|β|^2 =1. Why consider such a generalization? Simply because it turns out that this is how particles behave.

For instance, one can easily show [1] that the continuum limit of some quantum walk is the Dirac Equation, the equation which governs the propagation of the electron. Discrete Geometry approximates geometry via polygons. For instance a surface (e.g. a sphere) can be approximated by gluing triangles side be side that’s

an example of a 2D simplicial complex. The aim of this internship is to study the continuum limit of quantum walks over simplicial complexes. Why do this?

First of all one may wonder if there is a quantum walk over the 2D euclidean plane, but triangulated, which still has the Dirac Equation as its continuum limit. The steps to follow towards this aim seem relatively clear to us, yet answering this question would be the student’s main task, and achieving it would already mean a successful internship.

Still, once this is done, one may wonder about the continuum limits of quantum walks over simplicial complexes that are less regular. For instance when an electron propagates on a graphene sheet, the deformations of that sheet have an influence upon its propagation, which is then governed by the Dirac equation in “curved space”[2]. This is the case also whenever general relativity comes into play. Thus, one could hope that the continuum limit of quantum walk on a triangulated sphere, for instance, coincides with the Dirac equation in curved space that corresponds to the sphere.

It may be the case, moreover, that the geometry of the simplicial complex is able to encode other fields than the gravitational field such as the electromagnetic field. We have already seen this sort of effects happen with the mass, for instance: it turns out that a qu antum walk on a line with mass m, is equivalent to a quantum walk without mass, but on a cylinder [3]. What happens if the cylinder is irregular?

Quantum walks over discrete geometries seem to constitute a simple, discrete time, discrete space model à la Computer Science, but one that is able to grasp, with much elegance, a number of physical phenomena. They also constitute quantum algorithms (numerical schemes) for simulating physics on a Quantum Computer or other quantum simulation device. Do get in touch for more information!

References:

[1] http://arxiv.org/abs/1307.3524

[2] http://arxiv.org/abs/1212.5821 http://arxiv.org/abs/1505.07023

[3] http://arxiv.org/abs/1607.08191