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arXiv:2509.19887 · Journal of Physics A Mathematical and Theoretical

Dynamically optimal unraveling schemes for simulating Lindblad equations

Yu Hua Cao, Mingfeng He, Xiantao Li

Published Apr 22, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract Stochastic unraveling schemes are powerful computational tools for simulating Lindblad equations, offering significant reductions in memory requirements. However, this advantage is accompanied by increased stochastic uncertainty, and the question of optimal unraveling remains open. In this work, we investigate unraveling schemes driven by Brownian motion or Poisson processes and present a comprehensive parametric characterization of these approaches. For the case of a single Lindblad operator and one noise term, this parametric family provides a complete description for unraveling scheme with pathwise norm-preservation. We further analytically derive dynamically optimal quantum state diffusion (DO-QSD) and dynamically optimal quantum jump process that minimize the growth rate of the variance of an observable locally in time. Compared to jump process ansatz, DO-QSD offers two notable advantages: firstly, the variance for DO-QSD can be rigorously shown not to exceed that of any jump-process ansatz locally in time; secondly, it has very simple expressions. Numerical results demonstrate that the proposed DO-QSD scheme may achieve substantial reductions in the variance of observables and the resulting simulation error.

DOI
10.1088/1751-8121/ae5afd

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Abstract Stochastic unraveling schemes are powerful computational tools for simulating Lindblad equations, offering significant reductions in memory requirements. However, this advantage is accompanied by increased stochastic uncertainty, and the question of optimal unraveling remains open. In this work, we investigate unraveling schemes driven by Brownian motion or Poisson processes and present a comprehensive parametric characterization of these approaches. For the case of a single Lindblad operator and one noise term, this parametric family provides a complete description for unraveling scheme with pathwise norm-preservation. We further analytically derive dynamically optimal quantum state diffusion (DO-QSD) and dynamically optimal quantum jump process that minimize the growth rate of the variance of an observable locally in time. Compared to jump process ansatz, DO-QSD offers two notable advantages: firstly, the variance for DO-QSD can be rigorously shown not to exceed that of any jump-process ansatz locally in time; secondly, it has very simple expressions. Numerical results demonstrate that the proposed DO-QSD scheme may achieve substantial reductions in the variance of observables and the resulting simulation error.
Stochastic unravelings represent a useful tool to describe the dynamics of open quantum systems, and standard methods, such as quantum state diffusion (QSD), call for the complete positivity of the open-system dynamics. Here, we present a generalization of QSD, which also applies to positive, but not completely positive evolutions. The rate and the action of the diffusive processes involved in the unraveling are obtained by applying a proper transformation to the operators which define the master equation. The unraveling is first defined for semigroup dynamics and then extended to a definite class of time-dependent generators. We test our approach on a prototypical model for the description of exciton transfer, keeping track of relevant phenomena, which are instead disregarded within the standard, completely positive framework.
The jump unraveling of a quantum master equation decomposes the dynamics of an open quantum system into abrupt jumps, interspersed by periods of coherent dynamics when no jumps occur. Such open quantum systems are ubiquitous in quantum optics and mesoscopic physics, hence the need for efficient techniques for their stochastic simulation. Numerical simulation techniques fall into two main categories. The first splits the evolution into small time steps and determines stochastically for each step if a jump occurs or not. The second, known as Monte Carlo wave-function simulation, is based on the reduction of the norm of an initially pure state in the conditional no-jump evolution. It exploits the fact that the purity of the state is preserved by the finest unraveling of the master equation. In this paper, we present an alternative method for the simulation of the quantum jump unraveling, inspired by the classical Gillespie algorithm. The method is particularly well suited for situations in which a large number of trajectories is required for relatively small systems. It allows for non-purity-preserving dynamics, such as the ones generated by partial monitoring and channel merging. We describe the algorithm in detail and discuss relevant limiting cases. To illustrate it, we include four example applications of increasing physical complexity and discuss the performance of the algorithm across regimes of interest for open quantum systems simulation. Publicly available implementations of our code are provided in julia and mathematica.
Stochastic unravelings allow to efficiently simulate open system dynamics, yet their application has traditionally been restricted to master equations that preserve both Hermiticity and trace. In this work, we introduce a general framework that extends piecewise-deterministic unravelings to arbitrary trace-nonpreserving master equations, requiring only positivity and Hermiticity of the dynamics. Our approach includes, as special cases, unravelings of arbitrary dynamics in the Heisenberg picture, evolutions interpolating between fully Lindblad and non-Hermitian Hamiltonian generators, and equations employed in the derivation of full counting statistics, for which we show it can be used to obtain the moments of the associated probability distribution. The framework is suitable for both trace-decreasing and trace-increasing processes through stochastic disappearance and replication of the stochastic realizations, and it is compatible with different unraveling schemes and with reverse jumps in the non-Markovian regime. Thereby, our approach provides a powerful and versatile simulation method that significantly broadens the applicability of stochastic techniques for open system dynamics.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.
A method for stochastic unraveling of general time-local quantum master equations (QME) which involve the reduced density operator at time t only is proposed. The present kind of jump algorithm enables a numerically efficient treatment of QMEs that are not of Lindblad form. So it opens large fields of application for stochastic methods. The unraveling can be achieved by allowing for trajectories with negative weight. We present results for the quantum Brownian motion and the Redfield QMEs as test examples. The algorithm can also unravel non-Markovian QMEs when they are in a time-local form like in the time-convolutionless formalism.