Cylindrical Matter: A beyond-quantum many-body system for efficient classical simulation of quantum pure-Ising like systems

Even simplified models of quantum many-body systems can be difficult to analyse. However, taking inspiration from the foundations of physics, one may wonder whether there are practical advantages to constructing alternative beyond-quantum descriptions of many-body systems. We explore this question in the context of quantum interactions that are diagonal in the computational basis. We construct a hypothetical model of a continuous time dynamical many-body system that is based upon lattices of interacting particles called "cylindrical bits", a concept first introduced in [6]. In the language of [5] our toy model is {\it non-free}, as we need spatial constraints on how the particles interact to ensure valid probabilities. We investigate these constraints and explore the resulting `entangled' states that can exist. Certain pure {\it quantum} entangled systems can be faithfully mimicked by our cylindrical worlds. This allows us to simulate efficiently classically, in the sense of sampling measurement outcomes, a variety of previously unknown quantum systems. Examples include some states created by pure Ising interactions algebraically decaying faster than $\sim 1/r^{3D/2}$, with spatial dimension $D$, under measurements in the $Z$ eigenbasis or eigenbases of $aX+bY$ for $a,b \in \mathbb{R}$. We also explore whether another choice of non-quantum `particle' could expand the applicability of the classical simulation by defining and partially optimising a figure-of-merit that attempts to capture how useful various possibilities may be.