# Quantum Computation as Gravity

Caputa, Paweł (Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan) ; Magan, Javier M. (Instituto Balseiro, Centro Atomico Bariloche S. C. de Bariloche, Rio Negro, R8402AGP, Argentina)

12 June 2019

Abstract: We formulate Nielsen’s geometric approach to circuit complexity in the context of two-dimensional conformal field theories, where series of conformal transformations are interpreted as “unitary circuits” built from energy-momentum tensor gates. We show that the complexity functional in this setup 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.

Published in: Physical Review Letters 122 (2019)