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:2010.08492 · Physical Review A

Quantum jump Monte Carlo approach simplified: Abelian symmetries

Katarzyna Macieszczak, Dominic C. Rose

Published Apr 5, 2021• 11 citations
arXivHTMLPDF

Abstract

We consider Markovian dynamics of a finitely dimensional open quantum system featuring a weak unitary symmetry, i.e., when the action of a unitary symmetry on the space of density matrices commutes with the master operator governing the dynamics. We show how to encode the weak symmetry in quantum stochastic dynamics of the system by constructing a weakly symmetric representation of the master operator: a symmetric Hamiltonian, and jump operators connecting only the symmetry eigenspaces with a fixed eigenvalue ratio. In turn, this representation simplifies both the construction of the master operator as well as quantum jump Monte Carlo simulations, where, for a symmetric initial state, stochastic trajectories of the system state are supported within a single symmetry eigenspace at a time, which is changed only by the action of an asymmetric jump operator. Our results generalize directly to the case of multiple Abelian weak symmetries.

DOI
10.1103/physreva.103.042204

Related papers

Stochastic dynamics of quantum jumps

Heinz‐Peter Breuer, Francesco Petruccione

The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form ${\mathit{tsum}}_{\mathit{i}}$${\mathit{A}}_{\mathit{i}}$\ensuremath{\bigotimes}${\mathit{B}}_{\mathit{i}}$, where ${\mathit{A}}_{\mathit{i}}$ and ${\mathit{B}}_{\mathit{i}}$ are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 1, 1995•56 citations

Weak unitary symmetries of open quantum dynamics: beyond quantum master equations

Calum A. Brown, Robert L. Jack, Katarzyna Macieszczak

Abstract We consider Markovian open quantum dynamics with weak unitary symmetries. Starting from the quantum master equation (QME) for the system alone, it is known that the joint dynamics of the system and its environment can be obtained by dilation, leading to a closed dynamics for a continuous matrix product state. Performing counting measurements on the environment gives rise to stochastic dynamics of quantum trajectories for the system, which when averaged yield back the QME. In this work, we identify necessary and sufficient conditions under which the dynamics of these different descriptions retain the weak symmetry of the QME and we characterise the resulting symmetries of the different descriptions in terms of their generators. We find that the joint dynamics always features a separable symmetry directly related to that of the QME, but for quantum trajectories the corresponding symmetry is present only if the counting measurement satisfies certain conditions.

arXiv:2506.19814•New Journal of Physics•Jan 7, 2026•0 citations

Stochastic dynamics of individual quantum systems: Stationary rate equations

W. G. Teich, G. Mahler

An open quantum system is usually characterized by a reduced ensemble density matrix, the dynamics of which is governed by a generalized Master equation. Transforming this equation of motion into the instantaneous diagonal basis of the corresponding reduced density matrix, we can separate the coherent and incoherent part of the dynamics: The coherent dynamics is incorporated in the time development of the diagonal basis states, while the coupling to the reservoirs leads to simple rate equations. Interpreting these rate equations as a stochastic point process allows one to simulate the stochastic time evolution (random telegraph signals, ``quantum jumps'') of single-quantum systems. The diagonal representation can be considered as a generalization of the dressed-state picture of open quantum systems. Numerical simulations (``quantum Monte Carlo'') allow one to derive various dynamical properties (including correlation functions) of single-quantum systems. This concept is applied to different two- and three-level scenarios (\ensuremath{\lambda} and \ensuremath{\nu} configuration), and its limitations are discussed.

Physical Review A•Mar 1, 1992•55 citations

Gillespie algorithm for quantum jump trajectories

Marco Radaelli, Gabriel Teixeira Landi, Felix C. Binder

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.

arXiv:2303.15405•Physical Review A•Dec 13, 2024•6 citations

Open system dynamics with non-Markovian quantum jumps

Jyrki Piilo, Kari Härkönen, Sabrina Maniscalco +1 more

We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.

arXiv:0902.3609•Physical Review A•Jun 24, 2009•138 citations

Deterministic Quantum Jump (DQJ) Method for Weakly Dissipative Systems

Marcus Meschede, Ludwig Mathey

Physical quantum systems are generically coupled to an environment, resulting in open system dynamics. A typical approach to simulating this dynamics is to propagate the density matrix of the system via the Lindblad master equation. This approach is numerically challenging due to the size of the density matrix, which has led to the development of quantum jump methods, which unravel the density matrix into an ensemble of state vectors. These methods utilize a stochastic sampling of the quantum jump times, which becomes inefficent for weakly dissipative dynamics, in which jumps are rare events. Here, we propose the deterministic quantum jump (DQJ) method, which we show to outperform standard quantum jump methods in the weakly dissipative regime, by removing the error of stochastic sampling. We describe the methodology at the single-jump and two-jump level, reconstructing the density matrix at the corresponding level. We demonstrate the performance of the method for two examples, the dissipative transverse-field Ising model, and the dissipative Kerr oscillator. Given that quantum technologies such as quantum computing have weakly dissipative quantum dynamics as their central focus, we propose this method to be utilized in that context, for exploring and understanding quantum technology platforms.

quant-pharXiv:2603.04066•Mar 5, 2026•0 citations
We consider Markovian dynamics of a finitely dimensional open quantum system featuring a weak unitary symmetry, i.e., when the action of a unitary symmetry on the space of density matrices commutes with the master operator governing the dynamics. We show how to encode the weak symmetry in quantum stochastic dynamics of the system by constructing a weakly symmetric representation of the master operator: a symmetric Hamiltonian, and jump operators connecting only the symmetry eigenspaces with a fixed eigenvalue ratio. In turn, this representation simplifies both the construction of the master operator as well as quantum jump Monte Carlo simulations, where, for a symmetric initial state, stochastic trajectories of the system state are supported within a single symmetry eigenspace at a time, which is changed only by the action of an asymmetric jump operator. Our results generalize directly to the case of multiple Abelian weak symmetries.
The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Ai{\mathit{A}}_{\mathit{i}}Ai​\ensuremath{\bigotimes}Bi{\mathit{B}}_{\mathit{i}}Bi​, where Ai{\mathit{A}}_{\mathit{i}}Ai​ and Bi{\mathit{B}}_{\mathit{i}}Bi​ are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.
Abstract We consider Markovian open quantum dynamics with weak unitary symmetries. Starting from the quantum master equation (QME) for the system alone, it is known that the joint dynamics of the system and its environment can be obtained by dilation, leading to a closed dynamics for a continuous matrix product state. Performing counting measurements on the environment gives rise to stochastic dynamics of quantum trajectories for the system, which when averaged yield back the QME. In this work, we identify necessary and sufficient conditions under which the dynamics of these different descriptions retain the weak symmetry of the QME and we characterise the resulting symmetries of the different descriptions in terms of their generators. We find that the joint dynamics always features a separable symmetry directly related to that of the QME, but for quantum trajectories the corresponding symmetry is present only if the counting measurement satisfies certain conditions.
An open quantum system is usually characterized by a reduced ensemble density matrix, the dynamics of which is governed by a generalized Master equation. Transforming this equation of motion into the instantaneous diagonal basis of the corresponding reduced density matrix, we can separate the coherent and incoherent part of the dynamics: The coherent dynamics is incorporated in the time development of the diagonal basis states, while the coupling to the reservoirs leads to simple rate equations. Interpreting these rate equations as a stochastic point process allows one to simulate the stochastic time evolution (random telegraph signals, ``quantum jumps'') of single-quantum systems. The diagonal representation can be considered as a generalization of the dressed-state picture of open quantum systems. Numerical simulations (``quantum Monte Carlo'') allow one to derive various dynamical properties (including correlation functions) of single-quantum systems. This concept is applied to different two- and three-level scenarios (\ensuremath{\lambda} and \ensuremath{\nu} configuration), and its limitations are discussed.
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.
We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.
Physical quantum systems are generically coupled to an environment, resulting in open system dynamics. A typical approach to simulating this dynamics is to propagate the density matrix of the system via the Lindblad master equation. This approach is numerically challenging due to the size of the density matrix, which has led to the development of quantum jump methods, which unravel the density matrix into an ensemble of state vectors. These methods utilize a stochastic sampling of the quantum jump times, which becomes inefficent for weakly dissipative dynamics, in which jumps are rare events. Here, we propose the deterministic quantum jump (DQJ) method, which we show to outperform standard quantum jump methods in the weakly dissipative regime, by removing the error of stochastic sampling. We describe the methodology at the single-jump and two-jump level, reconstructing the density matrix at the corresponding level. We demonstrate the performance of the method for two examples, the dissipative transverse-field Ising model, and the dissipative Kerr oscillator. Given that quantum technologies such as quantum computing have weakly dissipative quantum dynamics as their central focus, we propose this method to be utilized in that context, for exploring and understanding quantum technology platforms.