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 (27)

• Geometric quantum computation with NMR
• Time Optimal Control in Spin Systems
• Optimal Creation of Entanglement Using a Two--Qubit Gate
• On the Entangling Power of Quantum Evolutions
• A geometric theory of non-local two-qubit operations
• Optimal control theory for unitary transformations
• Quantum Brachistochrone
• Entanglement of Quantum Evolutions
• Quantum dynamics as a physical resource
• Experimental demonstration of a non-destructive controlled-NOT quantum gate for two independent photon-qubits
• Composed ensembles of random unitary matrices
• Quantum brachistochrone curves as geodesics: obtaining accurate control protocols for time-optimal quantum gates
• Characterization of non-local gates
• Entangling power and operator entanglement in qudit systems
• Entangling power of time-evolution operators in integrable and nonintegrable many-body systems
• Optimizing for an arbitrary perfect entangler: I. Functionals
• Geometry of Quantum Computation with Qutrits
• Unitary quantum gates, perfect entanglers and unistochastic maps
• Solution to the quantum Zermelo navigation problem
• Entanglement, quantum randomness, and complexity beyond scrambling
• Symmetry criteria for quantum simulability of effective interactions
• Can local dynamics enhance entangling power?
• Diagonal unitary entangling gates and contradiagonal quantum states
• Universal extensions of restricted classes of quantum operations
• Discording power of quantum evolutions
• Measures of operator entanglement of two-qubit gates
• Statistical bounds on the dynamical production of entanglement

Cited by in corpus (10)

• Creating ensembles of dual unitary and maximally entangling quantum evolutions
• Generating random quantum channels
• Entanglement, quantum randomness, and complexity beyond scrambling
• Construction and local equivalence of dual-unitary operators: from dynamical maps to quantum combinatorial designs
• Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength
• Entangling power of multipartite unitary gates
• 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