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

TL;DR

This work probes how quantum complexity evolves in Abelian gauge theories by discretizing U(1) to Z_N on a lattice and defining a universal gate set. It shows that achieving exponential-in-entropy complexity requires maximally nonlocal Hamiltonians, linking nonlocality to potential gravity duals, and demonstrates how locality constraints suppress the maximum attainable complexity. Through both classical random-flux models and fully quantum analyses, the paper characterizes how complexity grows and saturates, and how nonlocal interactions (embodied by Othello/CNOT-like gates) accelerate growth. The results illuminate when gauge theories may replicate the fast information processing associated with black holes and provide a framework for comparing locality, entanglement, and complexity in gauge/gravity duality contexts.

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.

Figures (13)

• Figure 1: Eternal black hole with extremal surface anchored at the boundary time $t$ where $t$ runs upward in both CFTs (Left Figure). As time evolves upward, the wormhole inside black holes keeps growing linearly with respect to time $t$ (Right Figure).
• Figure 2: Two-dimensional spacial lattice. Dynamical variables live on links.
• Figure 3: Magnetic flux operator with unit strength on plaquette $p$. A flux loop has positive strength $+1$ if we see it along the direction $i\to j \to k \to l \to i$.
• Figure 4: Othello game representation of a basis of the physical Hilbert space in $\mathbf{Z}_2$ gauge theory. General physical states are given by superpositions of these Othello configurations.
• Figure 5: A time evolution of a state in the random flux model. The state at a step $t$ is obtained from the state at $(t-1)$ by acting a single qudit gate randomly. The white disk represents that the magnetic flux in the plaquette is zero, and other color disks do that the magnetic fluxes are nonzero.