Quantum Computation as Gravity

TL;DR

This paper connects quantum complexity in 2D conformal field theories to 2D gravity, showing that the complexity functional aligns with the Polyakov action and gravity influences optimal quantum computation.

Contribution

It formulates Nielsen's complexity approach within 2D CFTs, linking it to 2D gravity and the Virasoro group, providing a novel geometric perspective.

Findings

Complexity functional equals the Polyakov action.

Gravity governs optimal quantum circuits in 2D CFTs.

Complexity can be viewed as geometric action on Virasoro coadjoint orbits.

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.