  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.
