Quantum Computation as Gravity
Abstract
We formulate Nielsen's geometric approach to circuit complexity in the context of twodimensional conformal field theories, where series of conformal transformations are interpreted as "unitary circuits" built from energymomentum tensor gates. We show that the complexity functional in this setup can be written as the Polyakov action of twodimensional 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.
 Publication:

Physical Review Letters
 Pub Date:
 June 2019
 DOI:
 10.1103/PhysRevLett.122.231302
 arXiv:
 arXiv:1807.04422
 Bibcode:
 2019PhRvL.122w1302C
 Keywords:

 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 v2 includes major text revision and clarifications. Appendices added with a summary of our setup, discussion on different complexity metrics and EulerArnold approach to Virasoro circuits