Complexity is Simple

TL;DR

This paper explores the applicability of Lloyd's computational bound to holographic complexity, analyzing the roles of simple and orthogonalizing gates in black hole models and their implications for complexity growth.

Contribution

It translates Lloyd's assumptions into bulk language, distinguishes between gate types, and examines their relevance to black hole complexity and the Weak Gravity Conjecture.

Findings

Large black holes use simple gates and violate Lloyd's assumptions.

Orthogonalization is feasible near phase transitions for small black holes.

Supports a connection between complexity bounds and the Weak Gravity Conjecture.

Abstract

In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in Brown et al.