They have been introduced in both discrete and continuous time, and both formulations have been connected, in the 2000’s, by F. Strauch and then A. Childs, more extensively. They have been used, in the 2000’s, to design various quantum algorithms, such as local versions of Grover’s algorithm, or element-distinctness algorithms, and have been suggested by A. Childs as models of universal computation in the late 2000’s. QWs have also been used to directly simulate various physical quantum states and dynamics, such as molecular binding, topologically-protected transport, and relativistic quantum dynamics. We will focus on this last application.
On the one hand, we will show that discrete-time QWs (DTQWs) can, in the continuous-spacetime limit, reproduce a broad spectrum of relativistic quantum dynamics of spin-1/2 fermions, coupled to both Abelian or non-Abelian Yang-Mills fields, in possibly-curved spacetimes. On the other hand, we will show that several of the aforementioned continuous-spacetime gauge theories can be extended at the level of the lattice. We will, e.g., suggest a lattice equivalent to Maxwell’s equations in (2+1)D spacetime, that is, a lattice dynamics for the Abelian gauge bosons, consistent with the DTQW-based fermionic dynamics. We will also suggest a gauge-covariant non-Abelian field strength, in (1+1)D spacetime. During the presentation, we may as well present various phenomenal regimes of DTQWs coupled to Yang-Mills gauge fields. Note that one-dimensional quantum walks have been experimentally implemented, with, e.g., spin-dependent optical lattices, in A. Alberti’s group, or integrated photonics, in F. Sciarrino’s group, with position-dependent phase catchings.