2017/10/31 by Antonio Mandarino, Tomasz Linowski, Karol Życzkowski
Computer Science · Mathematics · Physics and Astronomy · #Bipartite graph #Combinatorics #Dimension (graph theory) #Discrete mathematics #Dynamical billiards #Ergodicity #Geometry #Haar measure #Mathematics #Measure (data warehouse) #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum entanglement #Quantum gate #Quantum many-body systems #Quantum mechanics #Qubit #Unitary state #quant-ph
paper · pdf · doi:10.1103/physreva.98.012335
published as Phys. Rev. A 98, 012335 (2018)
arxiv created 2018/07/19 · openalex publication_date 2018/07/30 · arxiv updated 2018/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 three-dimensional (3D) billiard. Due to ergodicity of such a dynamics an average along a trajectory Vt 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(N2).