Quantum Computation as Gravity

TL;DR

The complexity functional in this setup can be written as the Polyakov action of two-dimensional gravity or as the geometric action on the coadjoint orbits of the Virasoro group, and it is argued that gravity sets the rules for optimal quantum computation in conformal field theories.

Abstract

We formulate Nielsen's geometric approach to complexity in the context of two dimensional conformal field theories, where series of conformal transformations are interpreted as unitary circuits. We show that the complexity functional can be written as the Polyakov action of two dimensional gravity or, equivalently, as the geometric action on the coadjoint orbits of the Virasoro group. This way, we argue that gravity sets the rules for optimal quantum computation in conformal field theories.