Bipartite unitary gates and billiard dynamics in the Weyl chamber

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 $U(N^2)$.

References in corpus (9)

• Quantum brachistochrone curves as geodesics: obtaining accurate control protocols for time-optimal quantum gates
• Entangling power of time-evolution operators in integrable and nonintegrable many-body systems
• Optimizing for an arbitrary perfect entangler: I. Functionals
• Solution to the quantum Zermelo navigation problem
• Entanglement, quantum randomness, and complexity beyond scrambling
• Diagonal unitary entangling gates and contradiagonal quantum states
• Universal extensions of restricted classes of quantum operations
• Discording power of quantum evolutions
• Statistical bounds on the dynamical production of entanglement

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• Computing Graph Edit Distance with Algorithms on Quantum Devices
• Bulk-boundary correspondence from hyper-invariant tensor networks
• On the Fidelity Robustness of CHSH--Bell Inequality via Filtered Random States
• Asymptotic $\ast$--distribution of permuted Haar unitary matrices