Tensor-Network-Based Unraveling of Non-Markovian Dynamics in Large Spin Chains via the Influence Martingale Approach

Abstract

Classical simulation of open quantum system dynamics remains challenging due to the exponential growth of the Hilbert space, the need to accurately capture dissipation and decoherence, and the added complexity of memory effects in the non-Markovian regime. We develop an efficient algorithm for simulating both Markovian and non-Markovian dynamics in large one-dimensional quantum systems. Extending the Tensor Jump Method, which combines TDVP-based tensor-network evolution with a Suzuki-Trotter decomposition of stochastic trajectories, our approach incorporates time-dependent decay rates-treating positive rates as time-inhomogeneous Markovian processes and negative rates via the Influence Martingale formalism to unravel time-local non-Markovian dynamics. This resource-efficient framework enables scalable simulations of open-system dynamics in the non-Markovian regime, as demonstrated for a one-dimensional transverse-field Ising chain comprising up to 100 spin qubits.

Figures (8)

• Figure 1: Plot of decay strength $\gamma_k(t)$ and shifted Decay strength $r_k(t)$ vs time
• Figure 2: Left panel: Time dependence of $N_{\mathrm{jump}}(t)/N_{\mathrm{traj}}$ in Non-Markovian scenario with dephasing noise. Right panel: Corresponding plots for the Markovian case.
• Figure 3: Time evolution of the martingale factor $\mu_t$ for all stochastic trajectories under non-Markovian dephasing noise. The horizontal line indicates the ensemble-averaged value of $\mu_t$ across all trajectories. The decay rate $\gamma(t)$, scaled by a factor of 0.5 to match the plot scale, is overlaid to illustrate the martingale behavior during intervals where $\gamma(t)$ attains negative values.
• Figure 4: Evolution of the local expectation $\langle X^{[i]} \rangle$ at each site of a 5-spin transverse Ising chain under Markovian dephasing noise with constant rate $\gamma(t) = \gamma_\infty$.
• Figure 5: Time evolution of the local expectation $\langle X^{[i]} \rangle$ at each site of a 5-spin transverse-field Ising chain subject to non-Markovian dephasing with rate $\gamma(t)$ [Eq. (\ref{['eq:chosengamma']})]. The shifted rates $r(t)$ (scaled by a factor of 0.1 to match the plot scale) are overlaid to highlight the behavior of the local expectation during intervals where $\gamma(t)$ becomes negative and is shifted to positive values $r(t)$.