Reflections on Virasoro circuit complexity and Berry phase

TL;DR

This work connects circuit complexity defined on Virasoro symmetry groups to group-theoretic Berry phases in two-dimensional CFTs. In the large central charge limit, the computational cost matches the Berry connection, enabling the Berry phase to be expressed in terms of Virasoro circuit complexity, up to a boundary term dependent only on endpoints. The authors derive a simple proportionality between Virasoro circuit complexity and the logarithm of the reference-target state overlap, linking these ideas to holographic complexity and path-integral optimization via the Liouville action. Extending to full CFT, they discuss how left-right sectors combine and how these results illuminate the relationship between geometric actions, Berry phases, and holographic proposals for complexity.

Abstract

Recently, the notion of circuit complexity defined in symmetry group manifolds has been related to geometric actions which generally arise in the coadjoint orbit method in representation theory and play an important role in geometric quantization. On the other hand, it is known that there exists a precise relation between geometric actions and Berry phases defined in group representations. Motivated by these connections, we elaborate on a relation between circuit complexity and the group theoretic Berry phase. As the simplest setup relevant for holography, we discuss the case of two dimensional conformal field theories. In the large central charge limit, we identify the computational cost function with the Berry connection in the unitary representation of the Virasoro group. We then use the latter identification to express the Berry phase in terms of the Virasoro circuit complexity. The former can be seen as the holonomy of the Berry connection along the path in the group manifold which defines the protocol. In addition, we derive a proportionality relation between Virasoro circuit complexity and the logarithm of the inner product between a particularly chosen reference state and the prepared target state. In this sense, the logarithmic formula turns out to be approximating the complexity up to some additive constant if the building blocks of the circuit are taken to be the underlying symmetry gates. Predictions based on this formula have recently been shown to coincide with the holographic complexity proposals and the path integral optimization procedure. The found connections may therefore help to better understand such coincidences. We also discuss that our findings, put together with earlier observations, may suggest a connection between the Virasoro Berry phase and the complexity measure in the path integral optimization proposal.