Space-like dynamics in a reversible cellular automaton

Abstract

In this paper we study the space evolution in the Rule 54 reversible cellular automaton, which is a paradigmatic example of a deterministic interacting lattice gas. We show that the spatial translation of time configurations of the automaton is given in terms of local deterministic maps with the support that is small but bigger than that of the time evolution. The model is thus an example of space-time dual reversible cellular automaton, i.e. its dual is also (in general different) reversible cellular automaton. We provide two equivalent interpretations of the result; the first one relies on the dynamics of quasi-particles and follows from an exhaustive check of all the relevant time configurations, while the second one relies on purely algebraic considerations based on the circuit representation of the dynamics. Additionally, we use the properties of the local space evolution maps to provide an alternative derivation of the matrix product representation of multi-time correlation functions of local observables positioned at the same spatial coordinate.

Exact Thermalization Dynamics in the “Rule 54” Quantum Cellular Automaton

“…This result generalizes the findings of Ref. [47] to the quantum case. Specifically, we find that the leading left and right eigenvectors can be expressed in terms of the 3 × 3 local tensors A sr , B sr and the boundary vector jbi (the explicit expressions are reported in the Supplemental Material [62]) ð5Þ that fulfill the following set of local relations, which we term "zipping conditions" ð6aÞ ð6bÞ…”

“…In this Letter, we present an exact solution for the thermalization dynamics in a quantum circuit that provably exhibits typical features of interacting many-body systems: the quantum version of the "Rule 54" cellular automaton [40]. The latter can be regarded as the simplest interacting integrable system and over the last years has been shown to provide an ideal ground for studying interacting manybody dynamics [41][42][43][44][45][46][47][48][49][50][51][52][53][54].…”

Exact solution of the Floquet-PXP cellular automaton

“…[38]; (iv) the properties of the dual system to the RCA201 where propagation is in the space rather than time direction, and the consequences of this duality, cf. [27]. We hope to report on some of these in the near future.…”

“…Furthermore, when the sequence of applied gates is repeated periodically we refer to those as Floquet systems. The circuit platform is not only useful in unitary quantum many-body framework, but also in classical deterministic systems of continuous [26] or discrete variables (RCAs) [27]. Moreover, so-called duality symmetries under the swap of space and time axes allow for remarkable advancements in analytic tractability [20,26,27].…”

“…If deterministic, they can either be reversible or not, where the former (RCA [1], see also [30]) can be considered as a model of classical many-body Hamiltonian (or symplectic) dynamics. The RCA201/Floquet-PXP is a deterministic RCA, closely related to the now much studied RCA54/Floquet-FA [27,[31][32][33][34][35][36][37][38][39][40][41]. Just like the RCA54, the RCA201 (see detailed definitions below) is a one-dimensional lattice of binary variables with local three-site gates applied simultaneously to two halves (of even/odd indexed sites) of the lattice in two successive half time-steps.…”