Cayley graphs and complexity geometry

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.

arXiv comment16 pages, 3 figures

References in corpus (3)

• Complexity Equals Action
• Operator growth in the SYK model
• The hyperbolic geometry of random transpositions

Cited by in corpus (14)

• Quantum Complexity of Time Evolution with Chaotic Hamiltonians
• The Complexity Geometry of a Single Qubit
• Complexity Growth in Integrable and Chaotic Models
• WdW-patches in AdS${}_3$ and complexity change under conformal transformations II
• BPS Chaos
• Resource theory of quantum uncomplexity
• Circuit Complexity in Topological Quantum Field Theory
• Complexity Geometry and Schwarzian Dynamics
• Universality in long-distance geometry and quantum complexity
• Circuit Complexity of Knot States in Chern-Simons theory
• Stringy Effects and the Role of the Singularity in Holographic Complexity
• Polynomial Equivalence of Complexity Geometries
• Late-Time Saturation of Black Hole Complexity
• How smooth is quantum complexity?