Chirality from quantum walks without a quantum coin

Abstract

Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom-called the coin and represented by a finite-dimensional Hilbert space-is associated to each node. Scalar quantum walks are QWs with a one-dimensional coin. We propose a general strategy allowing one to construct scalar QWs on a broad variety of graphs, which admit embedding in Eulidean spaces, thus having a direct geometric interpretation. After reviewing the technique that allows one to regroup cells of nodes into new nodes, transforming finite spatial blocks into internal degrees of freedom, we prove that no QW with a two-dimensional coin can be derived from an isotropic scalar QW in this way. Finally we show that the Weyl and Dirac QWs can be derived from scalar QWs in spaces of dimension up to three, via our construction.

Scattering and perturbation theory for discrete-time dynamics

“…[19][20][21][22][23][24][25] and the recent reviews [26,27]) which consist of a translational invariant network of local quantum gates implementing the discrete-time evolution of a discrete set of quantum systems, each one interacting only with a finite number of neighbours. QCAs and in particular Quantum Walks (QWs) [28], which can be thought of as the one particle sectors of QCAs, have been considered as a simulation tool for relativistic quantum fields [29][30][31][32][33][34] and as discrete approaches for studying the foundations of Quantum Field Theory [34][35][36][37][38].…”