Direct probing of the simulation complexity of open quantum many-body dynamics

TL;DR

This work probes the intrinsic complexity of open quantum many-body dynamics by linking simulation cost to physically meaningful measures like correlation length and mixing time, across both classical tensor-network and quantum-jump-based approaches. It develops a dilation-free stochastic framework for simulating Markovian and non-Markovian dynamics, and introduces average correlation-length scaling to bound quantum and classical resources. The key finding is a dissipation-enabled separation: stronger dissipation can reduce quantum simulation complexity at long times, while classical TN complexity does not necessarily decrease, revealing regimes where quantum methods may outperform classical ones. The results span boundary-driven spin chains and chaotic regimes, and point to strategies such as randomised dissipation to accelerate convergence and guide ground-state preparation.

Abstract

Simulating open quantum systems is key to understanding non-equilibrium processes, as persistent influence from the environment induces dissipation and can give rise to steady-state phase transitions. A common strategy is to embed the system-environment into a larger unitary framework, but this obscures the intrinsic complexity of the reduced system dynamics. Here, we investigate the computational complexity of simulating open quantum systems, focusing on two physically relevant parameters -- correlation length and mixing time -- and explore whether it can be comparable (or even lower) to that of simulating their closed counterparts. In particular, we study the role of dissipation in simulating open-system dynamics using both quantum and classical methods, where the classical complexity is characterised by the bond dimension and operator entanglement entropy. Our results show that dissipation affects correlation length and mixing time in distinct ways at intermediate and long timescales. Moreover, we observe numerically that in classical tensor network simulations, classical complexity does not decrease with stronger dissipation, revealing a separation between quantum and classical resource scaling.

Figures (9)

• Figure 1: Illustration of the dissipative spin system under study. The boundary-driven quantum spin chain $S$ is coupled with strength $\gamma$ to two baths at its edges, $i=0$ and $j=N-1$, and the set of jump operators $\left\{L_{k}\right\}$ acts locally at the boundary sites. Spatial separation of $d=1$ means two sites are nearest neighbours with coupling $J$.
• Figure 2: Time evolution of spin observables and trace distance for a dissipative boundary-driven spin chain system $S$ of size $N = 25$ under increasing dissipation rates $\gamma$. (a–c) Expectation values of the local spin operators $\langle \sigma^x_r(t) \rangle$, $\langle \sigma^y_r(t) \rangle$, and $\langle \sigma^z_r(t) \rangle$, respectively, under Lindbladian dynamics for different values of $\gamma$. (d) Trace distance between consecutive timesteps, $D = ||\rho(t) - \rho(t - \delta t)||$, indicating convergence toward a steady state. In all panels, $\gamma \in [0, 3]$, with step size of $0.05$, where increasing values are shown in progressively lighter shades of blue. Simulations were performed using MPO methods with $\chi = 250$.
• Figure 3: Evolution of spin-$Z$ correlations in a dissipative boundary-driven spin chain $S$ with $N=25$ sites as a function of $\gamma$, $t$, and $d$. (a) Two‐point correlation $C^z_d(t=T)$ at the final time $T=10$, plotted for spatial separations $d=1,\dots,N-1$ (curves range from deep blue at $d=1$ to deep red at $d=24$); labels indicate the shortest and longest distances. Inset showing the distance‐averaged peak correlation $\frac{1}{N-1}\sum_{d=1}^{N-1}\max_t\bigl|C^z_d(t)\bigr|$ versus $\gamma$. (b) Maximum correlation across distances as a function of $t$. The data ranges for $\gamma \in [0, 0.3]$ with step size of $0.05$.
• Figure 4: Tensor network simulations for estimating the complexity of classical simulations. In all panels $\gamma=0.05$. We compute the correlation error, indicative of the classical simulability of $S$: (a) as a function of $t$ for $N=25$ and $\chi\in[50,225]$, (b) as a function of $\chi$ for $N \in [10,25]$ and $t=3$; inset: growth of $\chi$ over $t$, and (c) as a function of $N\in[10,25]$ for $\chi\in [50,225]$ and $t=3$. (d) Minimum $\chi_{\min}$ to reach $\epsilon_{\chi_{\min}}\le 0.05$ as a function of $N \in [10,25]$ for $t \in \{3,4,5\}$.
• Figure 5: Operator entanglement entropy growth under time evolution. Operator entanglement entropy as a function of $t$ for $N=25$, $\chi = 250$, and $\gamma \in [0, 0.3]$ with step size of $0.05$. Each data point is obtained by calculating the operator space entanglement entropy of the bipartition between the two halves of the MPS.