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Integrability of open boundary driven quantum circuits

Chiara Paletta, Tomaž Prosen

TL;DR

We study Yang–Baxter integrability of doubled open quantum circuits built from two replicas coupled at a single boundary, examining bulk gates of XX and XXZ/XXX types. Using Sklyanin’s reflection algebra, we derive boundary K-matrices and construct a two-step Floquet circuit whose transfer matrix commutes for open boundaries, revealing that non-factorizable boundary terms arise only for the XX (free fermion) bulk. In the continuous-time limit, certain boundary terms can be cast as Lindbladian dissipators, but only for a restricted set of parameters; this links integrability to open-system dynamics with boundary particle injection/removal. Overall, the XX bulk uniquely admits nontrivial boundary interactions compatible with Yang–Baxter integrability, while interacting bulk cases preserve factorized boundaries, highlighting a tight interplay between bulk statistics and boundary integrability with potential implications for exact NESS constructions and boundary-driven transport.

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.

Integrability of open boundary driven quantum circuits

TL;DR

We study Yang–Baxter integrability of doubled open quantum circuits built from two replicas coupled at a single boundary, examining bulk gates of XX and XXZ/XXX types. Using Sklyanin’s reflection algebra, we derive boundary K-matrices and construct a two-step Floquet circuit whose transfer matrix commutes for open boundaries, revealing that non-factorizable boundary terms arise only for the XX (free fermion) bulk. In the continuous-time limit, certain boundary terms can be cast as Lindbladian dissipators, but only for a restricted set of parameters; this links integrability to open-system dynamics with boundary particle injection/removal. Overall, the XX bulk uniquely admits nontrivial boundary interactions compatible with Yang–Baxter integrability, while interacting bulk cases preserve factorized boundaries, highlighting a tight interplay between bulk statistics and boundary integrability with potential implications for exact NESS constructions and boundary-driven transport.

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.
Paper Structure (50 sections, 124 equations, 11 figures)

This paper contains 50 sections, 124 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Quantum integrable models
  3. Quantum circuits
  4. Open quantum circuit
  5. Integrability in boundary driven diffusive Lindbladian systems
  6. Motivation of the paper
  7. Short summary of the paper
  8. Set up: building an integrable quantum circuit
  9. Quantum integrability in a nutshell
  10. Periodic Boundary Condition
  11. Open Boundary Condition
  12. Two-step Floquet dynamics
  13. Periodic boundary condition
  14. Open boundary conditions
  15. Our construction
  16. ...and 35 more sections

Figures (11)

  • Figure 1: Graphical representation of the right reflection algebra \ref{['KRreflectionalgebra']}.
  • Figure 2: Double row transfer matrix \ref{['doublerowtransfermat']}.
  • Figure 3: Quantum circuit with periodic boundary conditions.
  • Figure 4: Quantum circuit with open boundary conditions.
  • Figure 5: Elementary quantum gate given as the tensor product of two qubit.
  • ...and 6 more figures