Time Evolution of Complexity in Abelian Gauge Theories - And Playing Quantum Othello Game -

TL;DR

This paper investigates the time evolution of quantum complexity in Abelian gauge theories, discretizing the gauge group and spacetime to explicitly evaluate complexity growth, revealing conditions for dual black hole correspondence.

Contribution

It introduces a universal gate set for Abelian gauge theories and analyzes the conditions for maximal complexity growth related to black hole duality.

Findings

Complexity grows exponentially with entropy for certain Hamiltonians.

Maximal nonlocality in Abelian gauge theories is necessary for large complexity.

Explicit evaluation of complexity evolution in discretized gauge theories.

Abstract

Quantum complexity is conjectured to probe inside of black hole horizons (or wormhole) via gauge gravity correspondence. In order to have a better understanding of this correspondence, we study time evolutions of complexities for generic Abelian pure gauge theories. For this purpose, we discretize $U(1)$ gauge group as $\mathbf{Z}_N$ and also continuum spacetime as lattice spacetime, and this enables us to define a universal gate set for these gauge theories, and evaluate time evolutions of the complexities explicitly. We find that for a generic class of diagonal Hamiltonians to achieve a large complexity $\sim \exp(\mbox{entropy})$, which is one of the conjectured criteria necessary to have a dual black hole, the Abelian gauge theory needs to be maximally nonlocal.