Cayley graphs and complexity geometry

TL;DR

The paper investigates quantum complexity geometry by restricting to a large finite subgroup of the unitary group and equipping it with a Cayley graph metric, enabling rigorous analysis of complexity growth. It shows that the geometry exhibits negative curvature features via δ-hyperbolicity and computes the average complexity in a random-circuit model using the permutation subgroup, revealing linear growth up to a critical time t ~ n/2 and a subsequent slowdown. For the general Cayley-graph setting, it discusses expander properties and cutoff phenomena, illustrating how random walks approach equilibrium with sharp transitions, and suggests these features may have holographic interpretations. Overall, the work provides a bridge between geometric group theory and quantum complexity, offering exact large-$K$ results and guiding principles for understanding complexity geometry beyond continuous Lie-group models.

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 $δ$-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.

Figures (3)

• Figure 1: Cayley graph of $\mathbb{Z}_3 \star \mathbb{Z}_3$. The elements of the group live on the vertices of the triangles; the edges of the Cayley graph live on the edges of the triangles. Going around a blue (yellow) triangle clockwise corresponds to multiplication by $R_x$ ($R_y$). Going around a blue (yellow) triangle counterclockwise corresponds to multiplication by $R_x^{-1}$ ($R_y^{-1}$). The triangles have three sides since $R_x^3 = R_y^3 = 1$. This Cayley graph happens to give a uniform tiling of the hyperbolic plane, which shows that $\mathbb{Z}_3 \star \mathbb{Z}_3$ is $\delta$-hyperbolic.
• Figure 2: A large triangle on the Poincare disk. Even though the distance between the three points grows without bound, any point on one side of the triangle is close to some other point on another side of the triangle. The triangle is $\delta$-thin, where $\delta$ is of order the curvature scale.
• Figure 3: The average complexity as a function of time in the random circuit model (solid black) defined in the text. In the large $K$ limit, there is a phase transition at $t=2^{K-1}$ (red), where the second derivative is discontinuous. The dashed curve is what was conjectured in, e.g., figure 1 of brown17.