From this Friday (20.10) up to the end of the month we have the pleasure to receive Giuseppe Di Molfeta at CBPF. Giuseppe has many contributions to the topic of quantum walks. More specifically he employs quantum walks to simulate all sort of systems: from neutrino oscillations and Dirac equation, all the way up to gravity! The latter is the subject of the talk he will deliver in the Theory Seminar. See the details below, and be sure to be there!
Title: Quantum walking in curved spacetime
Speaker: Giuseppe Di Molfetta (Université Aix-Marseille )
Coordinates: seminar room 6th floor, CBPF. 25.10, 14h30
Abstract:In the framework of Quantum Simulation, a crucial topic for the exploration of physical situations where experiments are currently hard or impossible to setup (e.g. quantum gravity), Quantum Walks (QW) are increasingly recognized as prominent models. A discrete-time QW is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). We introduce Grouped QWs, a generalization of the usual QWs where (i) the input is allowed a simple prior encoding and (ii) the local unitary coin is allowed to act on larger than usual neighborhoods. In  it was shown that the continuum limit of this class of QWs leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1 + 1) curved spacetime . Therefore a certain QW provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice.
Here we take a step further and discretize the coin operator itself, only allowing, as elementary local unitary operator, the identity (no propagation) or the Pauli X operator (full-speed propagation). This discretization has the practical advantage of allowing easier experimental implementation, as well as of being of interest for studying the quantization of the metric. We prove that we can obtain the Dirac equation in the case of constant background metric. We also thoroughly analyze the non-constant metric case showing how, due to a non-differentiability issue in the discrete model, a new term arises in the differential equation, deviating from the usual Dirac equation.
 P. Arrighi, S. Facchini, M. Forets, Quantum Inf. Process. (2016) 15: 3467
 G. Di Molfetta, F. Debbasch, M. E. Brachet, Phys. Rev. A 88.4 (2013): 042301