Spontaneously interacting qubits from Gauss-Bonnet

TL;DR

This paper constructs critical geometric metrics incorporating the Gauss-Bonnet term that spontaneously break symmetry to produce qubit-like structures, extending previous work on Ricci scalar-based models.

Contribution

It introduces a new class of KAQ critical metrics using a generalized loss functional with the Gauss-Bonnet term, expanding the framework for emergent qubit systems.

Findings

Existence of KAQ critical metrics with Gauss-Bonnet term

Natural classes of metrics include distributions like GUE, GOE, GSE

Tools developed for extension to other loss functionals

Abstract

Building on previous constructions examining how a collection of small, locally interacting quantum systems might emerge via spontaneous symmetry breaking from a single-particle system of high dimension, we consider a larger family of geometric loss functionals and explicitly construct several classes of critical metrics which "know about qubits" (KAQ). The loss functional consists of the Ricci scalar with the addition of the Gauss-Bonnet term, which introduces an order parameter that allows for spontaneous symmetry breaking. The appeal of this method is two-fold: (i) the Ricci scalar has already been shown to have KAQ critical metrics and (ii) exact equations of motions are known for loss functionals with generic curvature terms up to two derivatives. We show that KAQ critical metrics, which are solutions to the equations of motion in the space of left-invariant metrics with fixed determinant, exist for loss functionals that include the Gauss-Bonnet term. We find that exploiting the subalgebra structure leads us to natural classes of KAQ metrics which contain the familiar distributions (GUE, GOE, GSE) for random Hamiltonians. We introduce tools for this analysis that will allow for straightfoward, although numerically intensive, extension to other loss functionals and higher-dimension systems.