Hopf algebras and solvable unitary circuits

Exactly solvable models in quantum many-body dynamics provide valuable insights into many interesting physical phenomena and serve as platforms for rigorously investigating fundamental theoretical questions. Nevertheless, they are extremely rare and existing solvable models and solution techniques have serious limitations. In this paper, we present a family of exactly solvable unitary circuits that model quantum many-body dynamics in discrete space and time. Unlike many previous solvable models, one can exactly compute the full quantum dynamics initialized from any matrix product state in this family of models. The time evolution of local observables and correlations, the linear growth of Rényi entanglement entropy, spatiotemporal correlations, and out-of-time-order correlations are all exactly computable. A key property of these models enabling the exact solution is that any time-evolved local operator is an exact matrix product operator with a finite bond dimension, even at arbitrarily long times, which we prove using the underlying <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msup><a:mi mathvariant="double-struck">C</a:mi><a:mo>*</a:mo></a:msup></a:math>-(weak) Hopf algebra structure along with tensor network techniques. We lay down the general framework for construction and solution of this family of models, and give several explicit examples. In particular, we study in detail a model constructed out of a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msup><c:mi mathvariant="double-struck">C</c:mi><c:mo>*</c:mo></c:msup></c:math>-weak Hopf algebra that is very close to a Floquet version of the PXP model, and the exact results we obtain may shed light on the phenomenon of quantum many-body scars, and more generally, Floquet quantum dynamics in constrained systems.