Quantum Computation as Gravity

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.

v2 includes major text revision and clarifications. Appendices added with a summary of our setup, discussion on different complexity metrics and Euler-Arnold approach to Virasoro circuits