Generalized Gibbs ensembles in weakly interacting dissipative systems and digital quantum computers

TL;DR

This work proposes using a digital quantum computer to showcase the activation of integrable effects in weakly dissipative integrable systems, which can be analyzed using scattering between the Bogoliubov quasiparticles caused by the dissipation.

Abstract

Identifying use cases with superconducting circuits not critically affected by the inherent noise is a pertinent challenge. Here, we propose using a digital quantum computer to showcase the activation of integrable effects in weakly dissipative integrable systems. Dissipation is realized by coupling the system's qubits to ancillary ones that are periodically reset. We compare the digital reset protocol to the usual Lindblad continuous-time evolution by considering non-interacting integrable systems dynamics, which can be analyzed using scattering between the Bogoliubov quasiparticles caused by the dissipation. If not dominant, the inherent noise would cause extra scattering but would not critically change the physics. A corresponding quantum computer implementation would illuminate the possibilities of stabilizing exotic states in nearly integrable quantum materials.

Figures (7)

• Figure 1: (a) Time evolution from an initial thermal mode occupation with $\beta=0.323$ to a highly non-thermal steady state distribution, stabilized by our choice of Lindblad operators, Eq. \ref{['eq::Li']}. (b) Relative error $\sum_q |(\langle n_q\rangle (t) - \langle n_q\rangle_0 (t)) / \langle n_q\rangle_0(t) |/L$ of the occupations $\langle n_q\rangle (t)$ obtained with Euler method with time steps $\epsilon\delta t=0.01,...,0.6$ and the reference $\langle n_q\rangle_0 (t)$ evaluated with smallest $\epsilon\delta t= 0.005$. At late times differences are tiny. (c) Steady state expectation values of local conserved quantities \ref{['eq::isingC']}. With increasing support, the importance of even conserved quantities decays exponentially. Expectation values of odd observables are zero due to symmetry. Parameters: $J = 1, h = 0.6, L=10^5$.
• Figure 2: Comparison of exact and GGE Ansatz dynamics at scaled time $\epsilon t$ for (a) $\langle \sigma^x_{L/2} \sigma^x_{L/2+1} \rangle$ and (b) $\langle \sigma^y_{L/2} \sigma^y_{L/2+1} \rangle$. Exact evolution is obtained with tensor network representation for the full Liouvillian at four different coupling strengths of the dissipator $\epsilon = 1, 0.5, 0.1, 0.05$. Parameters: $J = 1, h = 0.6, L=80, \chi=100$.
• Figure 3: Scheme of our dissipative transverse field Ising realization, similar to Refs. mi23matthies22 and realistic to implement with a digital quantum computer. In this setup, the system's qubits are coupled to ancillary ones. After every $T$ system-ancilla-coupling propagations, ancilla qubits are reset to the spin-down state.
• Figure 4: (a) Time evolution of the mode occupation from an initial infinite temperature state. A highly non-thermal steady state distribution is reached, which could be stabilized by the system-ancilla coupling in a digital quantum computer. Parameters: $J = 0.8, h=0.45, h_A = 0.8, T = 6$, $L=500, \lambda_{\tau} = \sqrt{\epsilon} = 0.1$. (b) Decay of correlations $|\langle S^{yy}_{i,i+\ell} \rangle|$, Eq. \ref{['eq::isingC']}, as a function of $\ell$ in the steady-state GGE and the ground state for the same parameters as in panel (a). As a signature of the stabilized non-thermal GGE, operators that overlap with local conserved quantities of transverse field Ising models show a slower decay of spatial correlations compared to the ground state. (c) Different choices of system-ancilla coupling parameters (field $h_A$ and cycle duration $T$) yield different correlation lengths $\xi$. Quite generically, longer cycles lead to slower decay of spatial correlations and thus more non-thermal states. Other paramters are the same as in panel (a) and (b): $J = 0.8, h=0.45, L=500$.
• Figure 5: (a) Convergence to the steady state mode occupation at different iterative steps $k$. In the $k=0$ step, the steady state is approximated by a thermal state. In the following iterative steps, additional leading conserved operators are added to a truncated GGE. A decent convergence is obtained in finite number of steps. (b) After the initial improvement of results with increasing number of iterative steps, for chosen parameters, $k>18$ iterative steps fail to improve the results further. However, this happens in the regime where results are converged for all practical purposes. Parameters: $J=1, h=0.6, L=10^5$. (c) Ratio of computing times $t_t/t_i$, where $t_t$ corresponds to time evolution with $\epsilon\delta t =0.6$ and $t_i$ to calculation with the iterative scheme, as a function of $(1/L)\sum_q |\langle \dot{n}_q \rangle|$, characterizing the accuracy of steady state calculation. Points are labeled by the number of iterative step taken for $t_i$ calculation. The two methods are comparable. Which one is more efficient in absolute terms depends on parameters. Parameters: $J=1, L=10^5$.