Towards quantum cellular automata meeting lattice gauge theory more closely

This is a pedagogical review of essentially to families of works, Refs. [1, 2] on the one hand, and Refs. [3, 4] on the other hand, and it contains a few novelties. We first review the new discretization of the Dirac equation by discrete-time quantum walk (DQW) which is introduced in Ref. [1]. This discretization has the following properties. It is unitary and strictly local, as any DQW discretization, but also, and this is the new part, it is both independent of the Clifford-algebra representation used to write down the Dirac equation and very similar to usual lattice-gauge-theory (LGT) discretizations, with an on-site mass term. Moreover, we remind that this new discretization avoids fermion doubling thanks to a natural Wilson term which moreover does not break unitarity. Reference [2] is a follow-up to that work, in which we define an action functional based on that new discretization of the Dirac equation; we do not do any review of that reference in particular. All this was done in the single-particle sector, that is, at the level of classical fields (relativistic quantum mechanics). Then, we give a brief introduction to the extensions to the multi-particle sector, i.e., to quantum fields, which have been done by Arrighi et al., and which use more standard, older DQWs; upgraded in a certain way to the multi-particle sector, DQWs become so-called quantum cellular automata (QCAs), and these QCAs can be used to discretize and further quantum simulate quantum field theories (QFTs), both in 1 + 1 dimensions [3] and in 2 + 1 and 3 + 1 dimensions [4]. This brief introduction to Refs. [3, 4] only covers the fermionic part, not the gauge-field part.

[1] P. Arnault, “Clifford algebra from quantum automata and unitary Wilson fermions,” Phys. Rev. A 106, 012201 (2022), arXiv:2105.12314.

[2] P. Arnault and C. Cedzich, “A single-particle framework for unitary lattice gauge theory in discrete time,” New J. Phys. 24, 123031 (2022).

[3] P. Arrighi, C. Bény, and T. Farrelly, “A quantum cellu- lar automaton for one-dimensional QED,” Quantum Inf. Process. 19 (2020).

[4] N. Eon, G. Di Molfetta, G. Magnifico, and P. Arrighi, “A relativistic discrete spacetime formulation of 3+1 QED,” Quantum 7, 1179 (2023).