Bipartite unitary gates and billiard dynamics in the Weyl chamber

TL;DR

This paper links the long-term behavior of bipartite quantum gates to billiard dynamics in the Weyl chamber, showing ergodic properties lead to universal averages in large dimensions.

Contribution

It introduces a novel connection between quantum gate dynamics and billiard systems, providing a new framework for understanding entanglement evolution.

Findings

Average gate behavior converges to Haar measure ensemble for large t.

Entanglement entropy averages match those of random unitaries in large N.

Billiard dynamics explain ergodic properties of quantum gate sequences.

Abstract

Long time behavior of a unitary quantum gate $U$, acting sequentially on two subsystems of dimension $N$ each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a $3D$ billiard. Due to ergodicity of such a dynamics an average along a trajectory $V^t$ stemming from a generic two-qubit gate $V$ in the canonical form tends for a large $t$ to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber - the minimal $3D$ set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension $N$ the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on $U(N^2)$.