Reflections on Virasoro circuit complexity and Berry phase

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