Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1710.03431 · Physical Review A

Quantum trajectories for time-dependent adiabatic master equations

Ka-Wa Yip, Tameem Albash, Daniel A. Lidar

Published Feb 28, 2018• 23 citations
arXivHTMLPDF

Abstract

We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension $N$ instead of a complex density matrix of dimension ${N}^{2}$, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor $N$ advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor ${N}^{2}$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.

DOI
10.1103/physreva.97.022116

Related papers

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

Accelerated Simulation of Non-Markovian Quantum Systems Using Quantum Trajectory

Yanfang Wang, Bo Yue, Shibei Xue

Quantum trajectory can be viewed as a numerical simulation method for wavefunctions which consume smaller memories compared to density matrices in master equations. In this paper, we adopt quantum trajectory to simulate the evolution of a non-Markovian quantum system modeled by an augmented system and further use parallel computing to improve efficiency. Compared to the Lindblad master equation solver in QuTiP, it can run more efficiently using less memory. In the example of a 8-spin-chain non-Markovian system, it only takes about 0.57% of the memory footprint and 4.12% of the runtime of the solver for the Lindblad master equation.

Chinese Control Conference (CCC)•Jul 24, 2023•0 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

Unraveling Open Quantum Dynamics with Time-Dependent Variational Monte Carlo

Christian Apostoli, Jacopo D'Alberto, Marco G. Genoni +2 more

We introduce a method to simulate open quantum many-body dynamics by combining time-dependent variational Monte Carlo (tVMC) with quantum trajectory techniques. Our approach unravels the Lindblad master equation into an ensemble of stochastic Schrödinger equations for a variational ansatz, avoiding the exponential cost of density matrix evolution. The method is compatible with generic ansatzë, including expressive neural-network wavefunctions. We derive the nonlinear stochastic equations of motion for the variational parameters and employ suitable Stratonovich numerical solvers. To validate our approach, we simulate quenches in the locally dissipative long-range Ising model in a transverse field, accurately capturing non-equilibrium magnetization and spin squeezing dynamics relevant to trapped-ion and Rydberg atom experiments. The framework is computationally efficient, scalable on high-performance computing platforms, and can be readily integrated into existing tVMC implementations. This work enables large-scale simulations of complex, dissipative quantum systems in higher dimensions, with broad implications for quantum technology and fundamental science.

arXiv:2506.23928•Jun 30, 2025•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

Quantum trajectory entanglement in various unravelings of Markovian dynamics

Tatiana Vovk, Hannes Pichler

The cost of classical simulations of quantum many-body dynamics is often determined by the amount of entanglement in the system. In this paper, we study entanglement in stochastic quantum trajectory approaches that solve master equations describing open quantum system dynamics. First, we introduce and compare adaptive trajectory unravelings of master equations. Specifically, building on [Phys. Rev. Lett. 128, 243601 (2022)], we study several greedy algorithms that generate trajectories with a low average entanglement entropy. Second, we consider various conventional unravelings of a one-dimensional open random Brownian circuit and locate the transition points from area- to volume-law-entangled trajectories. Third, we compare various trajectory unravelings using matrix product states with a direct integration of the master equation using matrix product operators. We provide concrete examples of dynamics, for which the simulation cost of stochastic trajectories is exponentially smaller than the one of matrix product operators.

Physical Review A•Jul 8, 2024•14 citations
We describe a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension NNN instead of a complex density matrix of dimension N2{N}^{2}N2, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor NNN advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor N2{N}^{2}N2 is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for 8-qubit quantum annealing examples. We also apply the quantum trajectories method to a 16-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.
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.
Quantum trajectory can be viewed as a numerical simulation method for wavefunctions which consume smaller memories compared to density matrices in master equations. In this paper, we adopt quantum trajectory to simulate the evolution of a non-Markovian quantum system modeled by an augmented system and further use parallel computing to improve efficiency. Compared to the Lindblad master equation solver in QuTiP, it can run more efficiently using less memory. In the example of a 8-spin-chain non-Markovian system, it only takes about 0.57% of the memory footprint and 4.12% of the runtime of the solver for the Lindblad master equation.
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 introduce a method to simulate open quantum many-body dynamics by combining time-dependent variational Monte Carlo (tVMC) with quantum trajectory techniques. Our approach unravels the Lindblad master equation into an ensemble of stochastic Schrödinger equations for a variational ansatz, avoiding the exponential cost of density matrix evolution. The method is compatible with generic ansatzë, including expressive neural-network wavefunctions. We derive the nonlinear stochastic equations of motion for the variational parameters and employ suitable Stratonovich numerical solvers. To validate our approach, we simulate quenches in the locally dissipative long-range Ising model in a transverse field, accurately capturing non-equilibrium magnetization and spin squeezing dynamics relevant to trapped-ion and Rydberg atom experiments. The framework is computationally efficient, scalable on high-performance computing platforms, and can be readily integrated into existing tVMC implementations. This work enables large-scale simulations of complex, dissipative quantum systems in higher dimensions, with broad implications for quantum technology and fundamental science.
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 cost of classical simulations of quantum many-body dynamics is often determined by the amount of entanglement in the system. In this paper, we study entanglement in stochastic quantum trajectory approaches that solve master equations describing open quantum system dynamics. First, we introduce and compare adaptive trajectory unravelings of master equations. Specifically, building on [Phys. Rev. Lett. 128, 243601 (2022)], we study several greedy algorithms that generate trajectories with a low average entanglement entropy. Second, we consider various conventional unravelings of a one-dimensional open random Brownian circuit and locate the transition points from area- to volume-law-entangled trajectories. Third, we compare various trajectory unravelings using matrix product states with a direct integration of the master equation using matrix product operators. We provide concrete examples of dynamics, for which the simulation cost of stochastic trajectories is exponentially smaller than the one of matrix product operators.