**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**