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Stochastic unraveling of Redfield master equations and its application to electron transfer problems

Kondov, I, Kleinekathöfer, U, Schreiber, M

Published Jul 9, 2003• 15 citations
OpenAlex

Abstract

A method for stochastic unraveling of general time-local quantum master equations (QMEs) is proposed. The present kind of jump algorithm allows a numerically efficient treatment of QMEs which are not in Lindblad form, i.e. are not positive semidefinite by definition. The unraveling can be achieved by allowing for trajectories with negative weights. Such a property is necessary, e.g. to unravel the Redfield QME and to treat various related problems with high numerical efficiency. The method is successfully tested on the damped harmonic oscillator and on electron transfer models including one and two reaction coordinates. The obtained results are compared to those from a direct propagation of the reduced density matrix (RDM) as well as from the standard quantum jump method. Comparison of the numerical efficiency is performed considering both the population dynamics and the RDM in the Wigner phase space representation.

Related papers

Stochastic unraveling of Redfield master equations and its application to electron transfer problems

Ivan Kondov, Ulrich Kleinekathöfer, Michael Schreiber

A method for stochastic unraveling of general time-local quantum master equations (QMEs) is proposed. The present kind of jump algorithm allows a numerically efficient treatment of QMEs which are not in Lindblad form, i.e., are not positive semidefinite by definition. The unraveling can be achieved by allowing for trajectories with negative weights. Such a property is necessary, e.g., to unravel the Redfield QME and to treat various related problems with high numerical efficiency. The method is successfully tested on the damped harmonic oscillator and on electron transfer models including one and two reaction coordinates. The obtained results are compared to those from a direct propagation of the reduced density matrix (RDM) as well as from the standard quantum jump method. Comparison of the numerical efficiency is performed considering both the population dynamics and the RDM in the Wigner phase space representation.

arXiv:physics/0307050•The Journal of Chemical Physics•Sep 18, 2003•25 citations

Stochastic unraveling of time-local quantum master equations beyond the Lindblad class

Ulrich Kleinekathöfer, Ivan Kondov, Michael Schreiber

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.

arXiv:quant-ph/0208084•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 3, 2002•20 citations

Stochastic Bloch-Redfield theory: Quantum jumps in a solid-state environment

Nicolas Vogt, Jan Jeske, Jared H. Cole

We discuss mapping the Bloch-Redfield master equation to Lindblad form and then unraveling the resulting evolution into a stochastic Schr\"odinger equation according to the quantum-jump method. We give two approximations under which this mapping is valid. This approach enables us to study solid-state systems of much larger sizes than is possible with the standard Bloch-Redfield master equation, while still providing a systematic method for obtaining the jump operators and corresponding rates. We also show how the stochastic unraveling of the Bloch-Redfield equations becomes the kinetic Monte Carlo algorithm in the secular approximation when the system-bath-coupling operators are given by tunneling operators between system eigenstates. The stochastic unraveling is compared to the conventional Bloch-Redfield approach with the superconducting single-electron transistor (SSET) as an example.

arXiv:1310.0559•Physical Review B•Nov 19, 2013•24 citations

Quantum Trajectories for Time-Local Non-Lindblad Master Equations

Tobias Becker, Ché Netzer, André Eckardt

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.

arXiv:2306.14876•Physical Review Letters•Oct 17, 2023•11 citations

Quantum jump unravelings for non-Markovian open system dynamics: a review

Federico Settimo, Jyrki Piilo

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.

quant-pharXiv:2605.07797•May 11, 2026•0 citations

Variational quantum simulation of time-local quantum master equations via quantum jump

Zhihao Lan, Jie Liu, Zhenyu Li +1 more

Strong coupling and environmental memory render many open quantum systems intractable to classical computation. To overcome this barrier, we present a variational quantum algorithm capable of solving generalized form time-local quantum master equations directly on noisy intermediate-scale quantum (NISQ) processors. Our approach combines four key innovations: (1) a pair-vector stochastic Schrödinger equation to unravel time-local quantum master equations, (2) deterministic evolution projected onto parameterized circuits via McLachlan's principle and stochastic jump implementation via singular-value decomposition, (3) using Hadamard tests circuits to measure the jump rates and the element of reduced density matrices, and (4) using no-evolution sampling to sample trajectories. The resulting protocol remains resilient under realistic hardware noise. We validate it on both classical simulators and a 66-qubit superconducting processor for solving the Redfield master equation and fourth-order time-local non-Markovian master equation and find that it can successfully capture key features of complex open quantum dynamics-including non-Markovian oscillations and strong-coupling effects. Our approach establishes a practical pathway toward scalable simulations of open quantum systems in the NISQ era.

The Journal of Chemical Physics•Oct 9, 2025•2 citations
A method for stochastic unraveling of general time-local quantum master equations (QMEs) is proposed. The present kind of jump algorithm allows a numerically efficient treatment of QMEs which are not in Lindblad form, i.e. are not positive semidefinite by definition. The unraveling can be achieved by allowing for trajectories with negative weights. Such a property is necessary, e.g. to unravel the Redfield QME and to treat various related problems with high numerical efficiency. The method is successfully tested on the damped harmonic oscillator and on electron transfer models including one and two reaction coordinates. The obtained results are compared to those from a direct propagation of the reduced density matrix (RDM) as well as from the standard quantum jump method. Comparison of the numerical efficiency is performed considering both the population dynamics and the RDM in the Wigner phase space representation.
A method for stochastic unraveling of general time-local quantum master equations (QMEs) is proposed. The present kind of jump algorithm allows a numerically efficient treatment of QMEs which are not in Lindblad form, i.e., are not positive semidefinite by definition. The unraveling can be achieved by allowing for trajectories with negative weights. Such a property is necessary, e.g., to unravel the Redfield QME and to treat various related problems with high numerical efficiency. The method is successfully tested on the damped harmonic oscillator and on electron transfer models including one and two reaction coordinates. The obtained results are compared to those from a direct propagation of the reduced density matrix (RDM) as well as from the standard quantum jump method. Comparison of the numerical efficiency is performed considering both the population dynamics and the RDM in the Wigner phase space representation.
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.
We discuss mapping the Bloch-Redfield master equation to Lindblad form and then unraveling the resulting evolution into a stochastic Schr\"odinger equation according to the quantum-jump method. We give two approximations under which this mapping is valid. This approach enables us to study solid-state systems of much larger sizes than is possible with the standard Bloch-Redfield master equation, while still providing a systematic method for obtaining the jump operators and corresponding rates. We also show how the stochastic unraveling of the Bloch-Redfield equations becomes the kinetic Monte Carlo algorithm in the secular approximation when the system-bath-coupling operators are given by tunneling operators between system eigenstates. The stochastic unraveling is compared to the conventional Bloch-Redfield approach with the superconducting single-electron transistor (SSET) as an example.
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.
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.
Strong coupling and environmental memory render many open quantum systems intractable to classical computation. To overcome this barrier, we present a variational quantum algorithm capable of solving generalized form time-local quantum master equations directly on noisy intermediate-scale quantum (NISQ) processors. Our approach combines four key innovations: (1) a pair-vector stochastic Schrödinger equation to unravel time-local quantum master equations, (2) deterministic evolution projected onto parameterized circuits via McLachlan's principle and stochastic jump implementation via singular-value decomposition, (3) using Hadamard tests circuits to measure the jump rates and the element of reduced density matrices, and (4) using no-evolution sampling to sample trajectories. The resulting protocol remains resilient under realistic hardware noise. We validate it on both classical simulators and a 66-qubit superconducting processor for solving the Redfield master equation and fourth-order time-local non-Markovian master equation and find that it can successfully capture key features of complex open quantum dynamics-including non-Markovian oscillations and strong-coupling effects. Our approach establishes a practical pathway toward scalable simulations of open quantum systems in the NISQ era.