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arXiv:2504.00121 · Physical Review A

Digital quantum simulation of the Lindblad master equation and its nonlinear extensions via quantum trajectory averaging

Yu-Guo Liu, Heng Fan, Shu Chen

Published Jun 26, 2026• 2 citations
arXivHTMLPDF

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.

DOI
10.1103/q33r-j5tl

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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.
The Lindblad master equation (LME) describing the Markovian dynamics of the quantum open system can be understood as the evolution of the effective non-Hermitian Hamiltonian balanced with random quantum jumps. Here we investigate the balance-breaking dynamics by partly eliminating jumps from postselection experiments. To describe this dynamics, a nonlinear Lindblad master equation (NLME) is derived from the quantum trajectory method. However, the NLME shows significant advantages in analytical analysis over the quantum trajectory method. Using the NLME, we classify the dynamics into two classes. In the trivial class, the process of reducing jumps is completely equivalent to weakening the coupling from the environment. In contrast, the nontrivial class presents more complex dynamics. We study a prototypical model within this class and demonstrate the existence of the postselected skin effect whose steady state is characterized by the accumulation of particles on one side. The steady-state distribution can be fitted by a scale-invariant tanh function which is different from the uniform distribution of LME. Furthermore, the NLME can give a reasonable framework for studying the interplay and competition between the non-Hermitian Hamiltonians and dissipative terms. We show this by capturing the characteristics of the trajectory-averaged entanglement entropy influenced by non-Hermitian skin effect and Zeno effect in the model of postselected skin effect.
For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.
We present a quantum algorithm for simulating a family of Markovian master equations that can be realized through a probabilistic application of unitary channels and state preparation. Our approach employs a second-order product formula for the Lindblad master equation, achieved by decomposing the dynamics into dissipative and Hamiltonian components and replacing the dissipative segments with randomly compiled, easily implementable elements. The sampling approach eliminates the need for ancillary qubits to simulate the dissipation process and reduces the gate complexity in terms of the number of jump operators. We provide a rigorous performance analysis of the algorithm. We also extend the algorithm to time-dependent Lindblad equations, generalize the family of Markovian master equations it can be applied to, and explore applications beyond the Markovian noise model. A new error bound, in terms of the diamond norm, for second-order product formulas for time-dependent Liouvillians is provided that might be of independent interest.
Every open-system dynamics can be associated to infinitely many stochastic pictures, called unravelings, which have proved to be extremely useful in several contexts, both from the conceptual and the practical point of view. Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.
We present a universal framework for simulating NNN-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs dXt=A(t)Xt dt+∑j=1JBj(t)Xt dWtj dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j dXt​=A(t)Xt​dt+∑j=1J​Bj​(t)Xt​dWtj​ to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.
The Lindblad master equation is a fundamental tool for describing the evolution of open quantum systems, but its computational complexity poses a significant challenge, especially for large systems. This article introduces a stochastic representation of the Lindblad dissipator that addresses this challenge by bundling the Lindblad operators. The stochastic dissipator maintains the Lindblad form, ensuring completely positive and trace-preserving dynamics. We demonstrate the effectiveness of this method by considering a Morse oscillator coupled to a spin bath. Our numerical experiments show that a small number of stochastically bundled operators can accurately capture the system's dynamics, even when the Hilbert space dimension is large. This method offers a new perspective on open quantum systems and provides a computationally efficient way to simulate their dynamics.