Integrability of open boundary driven quantum circuits

TL;DR

This paper explores the integrability of open boundary quantum circuits with specific boundary conditions, identifying solutions for free and interacting models, and connecting discrete and continuous time dynamics with boundary interactions.

Contribution

It provides the most general solutions to the boundary Yang-Baxter equation for open quantum circuits with free and interacting bulk models, and links discrete Floquet dynamics to Lindblad-type continuous evolution.

Findings

Integrable boundary conditions are characterized for free and interacting models.

Non-factorizable boundary matrices lead to non-trivial boundary interactions.

Certain boundary solutions correspond to open quantum systems with particle injection/removal.

Abstract

In this paper, we address the problem of Yang-Baxter integrability of doubled quantum circuit of qubits (spins 1/2) with open boundary conditions where the two circuit replicas are only coupled at the left or right boundary. We investigate the cases where the bulk is given by elementary six vertex unitary gates of either the free fermionic XX type or interacting XXZ type. By using the Sklyanin's construction of reflection algebra, we obtain the most general solutions of the boundary Yang-Baxter equation for such a setup. We use this solution to build, from the transfer matrix formalism, integrable circuits with two step discrete time Floquet (aka brickwork) dynamics. We prove that, only if the bulk is a free-model, the boundary matrices are in general non-factorizable, and for particular choice of free parameters yield non-trivial unitary dynamics with boundary interaction between the two chains. Then, we consider the limit of continuous time evolution and we give the interpretation of a restricted set of the boundary terms in the Lindbladian setting. Specifically, for a particular choice of free parameters, the solutions correspond to an open quantum system dynamics with the source terms representing injecting or removing particles from the boundary of the spin chain.