Quantum Walks and Lattice Gauge Theories
February 6th, 2019 PABLO ARNAULT Universitat de València and CSIC

I will show, in a single-particle framework, that discrete-time quantum walks (DTQWs) provide simple, natural spacetime discretizations of relativistic field theories (the walker typically coincides, in the continuum limit, with a Dirac field), that is: not only (i) are they unitary, but (ii) the evolution operator is also local, i.e., from one instant to the next one, the particle remains within a certain spacetime-lattice ‘lightcone’ (a spacetime-lattice counterpart to the standard, continuum one). In the discretizations provided by standard lattice gauge theories (LGTs), both properties (i) and (ii) are not straightforward at all, need to be evaluated, and may not be obtainable.

In the first part of the talk, I will present various such DTQWs, which are all connected to their continuum counterpart by a limit procedure in which the space step is kept proportional to the time step (ballistic scaling), the proportionality factor coinciding, in the continuum limit, with the speed of light. I will show that one can describe, in this framework, couplings of the walker to Abelian Yang-Mills fields in one, two, and three dimensions, and also to gravitational fields in one, two, and three dimensions. I will also show a model with non-Abelian Yang-Mills coupling in one dimension. We will see that the models with Yang-Mills couplings have exact lattice gauge invariance, and I will briefly mention the work in progress on the dynamics of the gauge fields, both for Yang-Mills and gravitational fields, the latter being associated to work in progress on the general-relativistic lattice covariance of DTQWs. In the second part of the talk, I will present a DTQW having a continuous-time limit while keeping space discrete (in contrast with the previous models), which coincides with the Hamiltonian formulation of LGTs, i.e., that of Kogut and Susskind, which is extensively used in works on the quantum simulation of LGTs. As any DTQW, this time discretization is unitary and has a local evolution operator, but then, by Meyer’s no-go lemmas, it must break translational invariance, that is, the staggered description of chiral symmetry.

I insist on the fact that all results are in a single-particle, i.e., classical-field framework. In other words, there are, in what I will present, neither multiple quantum particles in fixed number, nor quantized fields, i.e., an arbitrary and unfixed number of particles (or quasiparticles). These are obviously next steps to take, and there are already some works in the literature, (i) both theoretical and experimental for a fixed number of two particles, but also (ii) theoretical for an arbitrary and unfixed number of particles, both in the free and in the interacting case. Now, if we do not restrict ourselves to DTQW schemes, i.e., to local evolution operators in discrete time, but consider the continuous-time, Kogut-Susskind formulation of LGTs, there are of course, as mentioned above, an extensive number of works suggesting, in this framework, descriptions of quantized fermionic fields with qubits, through the Jordan-Wigner transformation, and there is even an experimental proof of principle.

Seminar, February 6, 2019, 15:00. ICFO’s Seminar Room

Hosted by Prof. Maciej Lewenstein