Computational complexity measures for CFTs from geometric actions on Virasoro and Kac-Moody orbits

Computational complexity is a concept from quantum information theory measuring the number of operations required to reach a target state from a reference state. In the last years, following the work of Susskind et al, a number of proposals were established for realising this concept in asymptotically AdS geometry for gaining new insight into the quantum structure of black holes and wormholes. In view of establishing a rigorous holographic duality for these gravity complexity proposals, a definition of complexity in quantum field theory is needed, i.e. for infinite dimensional Hilbert spaces. While significant progress has been achieved for free QFT, a generally accepted definition of complexity in (strongly) interacting QFT is still absent. Here I report on recent work on advancing approaches for 2d CFTs that are based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups. We determine the optimal circuits for these complexity measures and calculate the corresponding costs for examples of optimal transformations. In the Virasoro case, we find that for all choices of reference state except for the vacuum state, the complexity only measures the cost associated to phase changes, while assigning zero cost to the non-phase changing part of the transformation. For Kac-Moody groups in contrast, there do exist non-trivial optimal transformations beyond phase changes that contribute to the complexity, yielding a finite gauge invariant result. Furthermore, we also show that the alternative complexity proposal of path integral optimization is equivalent to the Virasoro proposal studied here. Finally, we sketch a new proposal for a complexity definition for the Virasoro group that measures the cost associated to non-trivial transformations beyond phase changes. This proposal is based on a cost function given by a metric on the Lie group of conformal transformations. The minimization of the corresponding complexity functional is achieved using the Euler-Arnold method yielding the Korteweg-de Vries equation as equation of motion.