Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits

We investigate a class of local quantum circuits on chains of d-level systems (qudits) that share the so-called ‘dual unitarity’ property. In essence, the latter property implies that these systems generate unitary dynamics not only when propagating in time, but also when propagating in space. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over SU(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SU}(d)$$\end{document}, e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time t in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of t sites. For general dual unitary circuits of qubits (d=2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d=2)$$\end{document} and a family of their extensions to higher d>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>2$$\end{document}, we provide an explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all t, if only the local 2-qubit gates are different from a SWAP (non-interacting gate).