Cayley graphs and complexity geometry

TL;DR

This paper establishes a rigorous connection between quantum complexity geometry and Cayley graphs, enabling the application of geometric group theory to analyze complexity, including properties like negative curvature and average complexity growth.

Contribution

It rigorously links quantum complexity geometry to Cayley graphs, allowing geometric group theory tools to analyze complexity properties and dynamics.

Findings

Complexity geometry can be modeled as Cayley graphs of finite groups.

The notion of δ-hyperbolicity describes negative curvature in complexity geometry.

Exact average complexity as a function of time computed in a large N random circuit model.

Abstract

The basic idea of quantum complexity geometry is to endow the space of unitary matrices with a metric, engineered to make complex operators far from the origin, and simple operators near. By restricting our attention to a finite subgroup of the unitary group, we observe that this idea can be made rigorous: the complexity geometry becomes what is known as a Cayley graph. This connection allows us to translate results from the geometrical group theory literature into statements about complexity. For example, the notion of $\delta$-hyperbolicity makes precise the idea that complexity geometry is negatively curved. We report an exact (in the large N limit) computation of the average complexity as a function of time in a random circuit model.