Digital Quantum Simulation of the Lindblad Master Equation and Its Nonlinear Extensions via Quantum Trajectory Averaging

TL;DR

The paper tackles the challenge of simulating open quantum systems governed by the Lindblad master equation over long times and with many dissipation channels by introducing a trajectory-averaged digital quantum simulation framework. It maps quantum trajectories to quantum circuits via a 1-dilation scheme (with a 2-dilation extension for nonlinear Lindblad master equations), enabling postselection-free, deterministic simulation of the standard LME and controlled exploration of NLME dynamics through a tunable postselection strength $\eta$. The authors provide rigorous error scaling analyses, show that the NLME reduces to LME or ENHH in appropriate limits, and validate the approach on dissipative XXZ chains, as well as open-system many-body localization and postselected skin-effect scenarios. This work enables efficient, scalable digital quantum simulations of open-system dynamics and the interplay between non-Hermitian evolution and dissipation, with potential applications to cutting-edge theories in non-Hermitian physics and monitored quantum dynamics.

Abstract

Since precisely controlling dissipation in realistic environments is challenging, digital simulation of the Lindblad master equation (LME) is of great significance for understanding nonequilibrium dynamics in open quantum systems. However, achieving long-time simulations for complex systems with multiple dissipation channels remains a major challenge, both theoretically and experimentally. Here, we propose a 1-dilation digital scheme for simulating the LME based on quantum trajectory averaging without postselection. By rigorously matching the stochasticity inherent in quantum trajectories with the probabilistic outcomes of quantum measurements, our method effectively translates the classically established quantum jump algorithm into executable quantum circuits. A key advantage of our method is that it overcomes the exponential suppression of success probability seen in some existing postselection-dependent schemes, especially for long-time evolution or systems with numerous jump operators. Moreover, the scheme can be extended to a 2-dilation framework for the nonlinear LME with postselection, bridging the full LME and non-Hermitian Hamiltonian dynamics. This extended scheme provides a digital approach for exploring the interplay between non-Hermitian Hamiltonians and dissipative terms within a monitored quantum dynamics framework.

Figures (6)

• Figure 1: Realizing stochastic quantum trajectories via quantum circuit measurements. (a) In the classical algorithm, the system undergoes Hamiltonian evolution (black) followed by a stochastic choice between non-unitary evolution (red double lines) and quantum jumps $L_\mu$ (blue). The branching probabilities are determined by Eq. (\ref{['tadt']}). (b) In the circuit implementation, the unitary $U_0$ corresponds to the Hamiltonian part, while the operators $A$ and $B$ (Eq. (\ref{['ABC']})) represent the non-unitary and jump components. Ancilla measurements (black triangles) facilitate the stochasticity for LME, while NLME implementation further requires postselection on the measurement outcomes.
• Figure 2: Scheme of the digital quantum trajectory simulation: (a) the 2-dilation method for the NLME as Eq. (\ref{['NLME']}), (b) the 1-dilation method for the LME as Eq. (\ref{['LME']}), and (c) the 1-dilation method for the evolution of ENHH as Eq. (\ref{['ENHH']}). The system is initialized in state $|\phi(0)\rangle$. The auxiliary qubits are initialized in state $|00\rangle_a$ in (a) and in state $|0\rangle_a$ in both (b) and (c). At each time step $\delta t$, the operations inside the brackets are executed. The $U_0$ represents the Hamiltonian simulation of the time evolution $e^{-iH\delta t}$. The gate $U_{\mu \ge 1}$, corresponding to the $\mu$-th dissipation source, is given by Eq. (\ref{['U']}) in (a), Eq. (\ref{['U0']}) in (b), and Eq. (\ref{['U1']}) in (c). After each measurement of auxiliary qubit, the postselection, denoted by 'P', is required only in (a) and (c), making their realization probabilistic. In contrast, the method in (b) is deterministic as it does not require postselection. In all three cases, the auxiliary qubit must be repeatedly reset to its initial state, denoted by 'R'.
• Figure 3: Numerical simulation of dissipative XXZ spin chain. We study the evolution of a 5-site dissipative XXZ chain from $t=1$ to $10$. The system is initialized in the all-spin-up state, with parameters $(J, \Delta , \gamma)=(1, 2, 0.5)$. (a) Exact results of the spin-up probability at the first site, $P_1= {\rm Tr} (\sigma_{1}^+ \sigma_{1}^- \rho)$, and the averaged nearest-neighbor correlations, $C_{zz}=\frac{1}{L-1}\sum_{j=1}^{L-1} \langle \sigma_{l+1}^z \sigma_l^z \rangle$, are shown as the blue dashed line and the solid magenta line, respectively. Circular and rhombus markers with error bars denote the results of our digital simulation method with time step $\delta t =0.1$, using 100 and 1000 trajectories, respectively. (b) Scaling of the error of the density matrix $\rho$ at $t=10$ with the simulation time step $\delta t$, obtained by averaging over 100,000 trajectories. Data ponts correspond to $\delta t = (1.0,0.5,0.2,0.1,0.05,0.02,0.01)$, and the solid line shows a fit to the data.
• Figure S1: Simulation of two-level atom under monitoring spontaneous emission. We study the NLME evolution from $t=0$ to $10$. The atom is initialized in excited state,with parameters $(J,\gamma)=(1,0.5)$. (a) Time evolution of the excited state probability $P_e$ for different values of $\eta$. Digital simulation results (data points) from 1000 experimental rounds with time step of $\delta t =0.1$ are compared with exact solutions (lines) computed by the vectorization method. (b)-(d) The number of valid trajectories $K_{eff}$, standard deviation of $P_e$ (SD), and standard error of $P_e$ (SE) at time $t=10$ as a function of the time step $\delta t$ (=0.01,0.05,0.1,0.25,0.5) for different $\eta$ (=0.0,0.25,0.5,0.75,0.95,1.0). Each point comes from 10,000 experimental rounds.
• Figure S2: Simulation of localization and thermalization in an open system. The probability distribution of the spin-up state in a 10-site chain evolves from the initial state $|\uparrow \downarrow \uparrow \cdots \downarrow \rangle$. The dynamics is govern by the LME in Eq. (\ref{['Local']}) with $J=1$, $V=2$, $\gamma=1$ and $\eta=0$. The jump operators take the form as Eq. (\ref{['Ll']}) with $\alpha=0$, $\beta=\pi$ in (a) and $\alpha=0$, $\beta=0$ in (b). We simulate the evolution by the 1-dilation method with $\delta t=0.01$ and an average over 100 trajectories. The results show that the system is localized in (a) and thermalized in (b). The evolution curves of dIPR in both (a) and (b) are shown in (c).