Quantum Zeno Effect in Noisy Integrable Quantum Circuits for Impurity Models

We theoretically study the open quantum system dynamics (in the Trotterized limit) of integrable quantum circuits in the presence of onsite dephasing noise with a spin-$\frac{1}{2}$ impurity interacting at the edge. Using a combination of Bethe Ansatz (BA) and exact diagonalization (ED), we study the dynamics of both the bulk and the impurity for the XXX (Heisenberg) and the XX qubit chains in the presence and absence of bulk noise. In the absence of noise, we show that the impurity exhibits two distinct phases, the bound mode phase where the impurity keeps oscillating in time, and the Kondo phase where it decays with Kondo time $t_K$. Turning on the bulk dephasing noise, we find for the two models that in the long time limit in both regimes the quantum Zeno effect takes place where the dynamics of the impurity magnetization slows down as the noise strength $\gamma$ increases. The impurity magnetization in the bound mode regime shows the opposite effect, decaying faster as the noise strength increases for short times ($t \ll 1/\gamma$). We show that the bulk KPZ dynamics of the XXX model is converted to diffusive dynamics as in the XX case studied before by V. Alba, driving both systems to the Zeno effect for the impurity in the long time limit.