Disentangling strategies and entanglement transitions in unitary circuit games with matchgates

TL;DR

This paper investigates entanglement phase transitions in unitary circuit games involving matchgate dynamics, introducing a minimal representation of fermionic Gaussian states and analyzing different disentangling strategies.

Contribution

It develops a minimal matchgate circuit representation for fermionic Gaussian states and studies entanglement transitions under various disentangling strategies in matchgate-based circuits.

Findings

Different entanglement transitions observed with braiding gates versus generic matchgates.

A generalized Yang-Baxter relation is derived for updating fermionic Gaussian states.

Disentangling procedures reduce circuit complexity and entanglement in the system.

Abstract

In unitary circuit games, two competing parties, an "entangler" and a "disentangler", can induce an entanglement phase transition in a quantum many-body system. The transition occurs at a certain rate at which the disentangler acts. We analyze such games within the context of matchgate dynamics, which equivalently corresponds to evolutions of non-interacting fermions. We first investigate general entanglement properties of fermionic Gaussian states (FGS). We introduce a representation of FGS using a minimal matchgate circuit capable of preparing the state and derive an algorithm based on a generalized Yang-Baxter relation for updating this representation as unitary operations are applied. This representation enables us to define a natural disentangling procedure that reduces the number of gates in the circuit, thereby decreasing the entanglement contained in the system. We then explore different strategies to disentangle the systems and study the unitary circuit game in two different scenarios: with braiding gates, i.e., the intersection of Clifford gates and matchgates, and with generic matchgates. For each model, we observe qualitatively different entanglement transitions, which we characterize both numerically and analytically.