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

Journal of Chemical Theory and Computation

Integrating Self-Initialized Local Thermalizing Lindblad Operators for Variational Quantum Algorithm with Quantum Jump: Implementation and Performance

Zhihao Lan, WanZhen Liang

Published Nov 26, 2024• 5 citations
Publisher

Abstract

Quantum computing holds great potential for simulating quantum systems, such as molecular systems, due to its inherent ability to represent and manipulate quantum states. However, simulating nonunitary dynamics of open quantum systems on a quantum computer is still challenging. Here, we present a scheme denoted as VQS-QJ-LTLME, which adopts the trajectory average of quantum jump Monte Carlo wave function variational evolution to evolve the system's density matrix and local thermalizing Lindblad operators to describe the system-environment interactions. This combination allows the quantum circuit to be initialized after a period of evolution, thereby eliminating the accumulation of errors. The VQS-QJ-LTLME algorithm requires only log_2 qubits for system size $n$ and holds a time complexity of O(Tn^3), making it particularly suitable for operation on noisy intermediate-scale quantum devices. To accelerate the Monte Carlo sampling process in VQS-QJ-LTLME, we introduce an efficient sampling method, named No-evolution sampling (NES). The VQS-QJ-LTLME aided by NES requires simulating only (n^3+1) trajectories on quantum computers, and then makes 10^6 ~ 10^7 samplings on classical computers in a few seconds. To demonstrate the performance of the VQS-QJ-LTLME algorithm, we simulate the dynamics of a two-level spin-boson model and a four-level reduced Fenna-Matthews-Olson system with real superconducting quantum computers and classical simulators. It is shown that the simulations produced by VQS-QJ-LTLME algorithm align closely with those obtained from the purely classical methods.

DOI
10.1021/acs.jctc.4c00490

Related papers

Variational Quantum Simulation of Lindblad Dynamics via Quantum State Diffusion

Jianming Luo, Kaihan Lin, Xing Bao Gao

Quantum simulation of dynamics in open quantum systems is crucial but poses a significant challenge due to the non-Hermitian nature leading to nonunitary evolution and the limited quantum resources on current quantum computers. Here we introduce a variational hybrid quantum-classical algorithm designed for simulating the time evolution governed by the Lindblad master equation. Our approach involves on a stochastic unveiling of the density matrix, transforming the Lindblad equation into a wave function-based quantum state diffusion (QSD) method with the aim of reducing qubit requirements. We then apply variational quantum simulation (VQS) to efficiently capture the nonunitary evolution in QSD. We demonstrate our QSD-VQS algorithm by investigating the quantum dynamics in a two-level system subjected to an amplitude damping channel and a four-level transverse field Ising model within a dissipative environment including time-independent and periodic Hamiltonian cases. The results reveal its promising utility with upcoming hardware in the near future.

The Journal of Physical Chemistry Letters•Mar 22, 2024•21 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

QFlux: An Open-Source Toolkit for Quantum Dynamics Simulations on Quantum Computers. Part V - Adaptive Variational Quantum Algorithms for Open Quantum Systems

Saurabh Shivpuje, Alexander V. Soudackov, Xiaohan Dan +8 more

Simulating open quantum systems remains one of the most demanding problems in quantum dynamics, as environmental interactions lead to non-unitary evolution that challenges standard quantum simulation techniques. Variational quantum algorithms (VQAs) offer a practical way forward by combining classical optimization with quantum hardware, making them well suited for near-term devices. Here, in Part V of the QFlux implementation series, we introduce adaptive variational-ansatz methods for solving the Lindblad master equation. This tutorial presents the Python-based qmad module of the QFlux toolkit, which is designed to help students and researchers easily build and explore adaptive VQA strategies. The tutorial walks through each step of the workflow and applies the methods to two illustrative examples. The first is the amplitude-damping channel, which models spontaneous emission in a two-level system. The second is the Fenna-Matthews-Olson (FMO) complex, a prominent model for excitonic energy transfer in photosynthesis. The two workflows utilizing adaptive variational approaches implemented here are the unrestricted adaptive variational quantum dynamics (UAVQD) scheme and the stochastic Schrödinger equation (SSE) based trajectory method. By working through these examples, readers gain both intuition and hands-on experience with variational techniques for simulating dissipative quantum dynamics.

•Jan 30, 2026•0 citations

Noisy quantum circuit simulation with the tensor jump method

Maximilian Fröhlich, Aaron Sander, Martin Eigel +2 more

Classical simulation of noisy quantum circuits is essential for validating algorithms, benchmarking hardware, and assessing error-mitigation strategies, but remains limited by the exponential cost of density-matrix methods and the high variance of standard trajectory sampling. We introduce a variance-aware tensor network framework that combines the tensor jump method with local TDVP gate evolution on matrix product states and sparse Pauli-Lindblad hardware noise models. Gates are applied as short variational evolutions on the MPS manifold, while noise is sampled per circuit window from Pauli-Lindblad jump sets with state-independent hazards and dissipative contractions that reduce to irrelevant global factors after renormalization. The method supports correlated multi-qubit Lindblad noise consistent with hardware connectivity, including long-range operators on non-adjacent qubits, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. We develop two unbiased variance-aware unravelings. An analog unitary-mixture unraveling matches the Lindblad generator exactly under symmetric Gaussian or two-point angle laws, while a projector-jump unraveling yields state-independent hazards and closed-form variance laws. Both retain the standard 1/sqrt(N) Monte Carlo convergence but with reduced prefactors. Empirically, projector sampling strongly reduces trajectory variance and bond-dimension growth across many circuit architectures, whereas analog sampling is most effective at weak noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM's 127-qubit kicked-Ising benchmark with long-range depolarizing noise, achieving reduced Monte Carlo variance and favorable MPS bond-dimension growth compared with standard Kraus-insertion baselines.

quant-pharXiv:2607.01323•Jul 3, 2026•0 citations

QFlux: An Open-Source Toolkit for Quantum Dynamics Simulations on Quantum Computers. Part V - Adaptive Variational Quantum Algorithms for Open Quantum Systems

Saurabh Shivpuje, Alexander V. Soudackov, Xiaohan Dan +8 more

Simulating open quantum systems remains one of the most demanding problems in quantum dynamics, as environmental interactions lead to non-unitary evolution that challenges standard quantum simulation techniques. Variational quantum algorithms (VQAs) offer a practical way forward by combining classical optimization with quantum hardware, making them well suited for near-term devices. Here, in Part V of the QFlux implementation series, we introduce adaptive variational-ansatz methods for solving the Lindblad master equation. This tutorial presents the Python-based qmad module of the QFlux toolkit, which is designed to help students and researchers easily build and explore adaptive VQA strategies. The tutorial walks through each step of the workflow and applies the methods to two illustrative examples. The first is the amplitude-damping channel, which models spontaneous emission in a two-level system. The second is the Fenna-Matthews-Olson (FMO) complex, a prominent model for excitonic energy transfer in photosynthesis. The two workflows utilizing adaptive variational approaches implemented here are the unrestricted adaptive variational quantum dynamics (UAVQD) scheme and the stochastic Schrödinger equation (SSE) based trajectory method. By working through these examples, readers gain both intuition and hands-on experience with variational techniques for simulating dissipative quantum dynamics.

•Apr 5, 2026•0 citations

Towards Robust Variational Quantum Simulation of Lindblad Dynamics via Stochastic Magnus Expansion

Jia-Cheng Huang, Hao-En Li, Yi-Cheng Wang +3 more

In this paper, we introduce a novel and general framework for the variational quantum simulation of Lindblad equations. Building on the close relationship between the unraveled Lindblad dynamics, stochastic Magnus integrators, and variational quantum simulation, we propose a high-order scheme for solving the quantum state diffusion equation using exponential integrators. This formulation facilitates the simulation of wavefunction trajectories within the established framework of variational quantum algorithms for time evolution. Our algorithm significantly enhances robustness in two key aspects: the stability of the simulation with large time steps, and the reduction in the number of quantum trajectories required to accurately simulate the Lindblad dynamics in terms of the ensemble average. We demonstrate the effectiveness of our algorithm through numerical examples in both classical and quantum implementations, including the transverse-field Ising model (TFIM) with damping, the Fenna-Matthews-Olson (FMO) complex, and the radical pair model (RPM). The simulation accuracy can be systematically improved, and the algorithm remains reliable even in highly oscillatory regimes. These methods are expected to be applicable to a broader class of open quantum systems beyond the specific models considered in this study.

arXiv:2503.22099•PRX Quantum•Sep 16, 2025•3 citations
Quantum computing holds great potential for simulating quantum systems, such as molecular systems, due to its inherent ability to represent and manipulate quantum states. However, simulating nonunitary dynamics of open quantum systems on a quantum computer is still challenging. Here, we present a scheme denoted as VQS-QJ-LTLME, which adopts the trajectory average of quantum jump Monte Carlo wave function variational evolution to evolve the system's density matrix and local thermalizing Lindblad operators to describe the system-environment interactions. This combination allows the quantum circuit to be initialized after a period of evolution, thereby eliminating the accumulation of errors. The VQS-QJ-LTLME algorithm requires only log_2 qubits for system size nnn and holds a time complexity of O(Tn^3), making it particularly suitable for operation on noisy intermediate-scale quantum devices. To accelerate the Monte Carlo sampling process in VQS-QJ-LTLME, we introduce an efficient sampling method, named No-evolution sampling (NES). The VQS-QJ-LTLME aided by NES requires simulating only (n^3+1) trajectories on quantum computers, and then makes 10^6 ~ 10^7 samplings on classical computers in a few seconds. To demonstrate the performance of the VQS-QJ-LTLME algorithm, we simulate the dynamics of a two-level spin-boson model and a four-level reduced Fenna-Matthews-Olson system with real superconducting quantum computers and classical simulators. It is shown that the simulations produced by VQS-QJ-LTLME algorithm align closely with those obtained from the purely classical methods.
Quantum simulation of dynamics in open quantum systems is crucial but poses a significant challenge due to the non-Hermitian nature leading to nonunitary evolution and the limited quantum resources on current quantum computers. Here we introduce a variational hybrid quantum-classical algorithm designed for simulating the time evolution governed by the Lindblad master equation. Our approach involves on a stochastic unveiling of the density matrix, transforming the Lindblad equation into a wave function-based quantum state diffusion (QSD) method with the aim of reducing qubit requirements. We then apply variational quantum simulation (VQS) to efficiently capture the nonunitary evolution in QSD. We demonstrate our QSD-VQS algorithm by investigating the quantum dynamics in a two-level system subjected to an amplitude damping channel and a four-level transverse field Ising model within a dissipative environment including time-independent and periodic Hamiltonian cases. The results reveal its promising utility with upcoming hardware in the near future.
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.
Simulating open quantum systems remains one of the most demanding problems in quantum dynamics, as environmental interactions lead to non-unitary evolution that challenges standard quantum simulation techniques. Variational quantum algorithms (VQAs) offer a practical way forward by combining classical optimization with quantum hardware, making them well suited for near-term devices. Here, in Part V of the QFlux implementation series, we introduce adaptive variational-ansatz methods for solving the Lindblad master equation. This tutorial presents the Python-based qmad module of the QFlux toolkit, which is designed to help students and researchers easily build and explore adaptive VQA strategies. The tutorial walks through each step of the workflow and applies the methods to two illustrative examples. The first is the amplitude-damping channel, which models spontaneous emission in a two-level system. The second is the Fenna-Matthews-Olson (FMO) complex, a prominent model for excitonic energy transfer in photosynthesis. The two workflows utilizing adaptive variational approaches implemented here are the unrestricted adaptive variational quantum dynamics (UAVQD) scheme and the stochastic Schrödinger equation (SSE) based trajectory method. By working through these examples, readers gain both intuition and hands-on experience with variational techniques for simulating dissipative quantum dynamics.
Classical simulation of noisy quantum circuits is essential for validating algorithms, benchmarking hardware, and assessing error-mitigation strategies, but remains limited by the exponential cost of density-matrix methods and the high variance of standard trajectory sampling. We introduce a variance-aware tensor network framework that combines the tensor jump method with local TDVP gate evolution on matrix product states and sparse Pauli-Lindblad hardware noise models. Gates are applied as short variational evolutions on the MPS manifold, while noise is sampled per circuit window from Pauli-Lindblad jump sets with state-independent hazards and dissipative contractions that reduce to irrelevant global factors after renormalization. The method supports correlated multi-qubit Lindblad noise consistent with hardware connectivity, including long-range operators on non-adjacent qubits, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. We develop two unbiased variance-aware unravelings. An analog unitary-mixture unraveling matches the Lindblad generator exactly under symmetric Gaussian or two-point angle laws, while a projector-jump unraveling yields state-independent hazards and closed-form variance laws. Both retain the standard 1/sqrt(N) Monte Carlo convergence but with reduced prefactors. Empirically, projector sampling strongly reduces trajectory variance and bond-dimension growth across many circuit architectures, whereas analog sampling is most effective at weak noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM's 127-qubit kicked-Ising benchmark with long-range depolarizing noise, achieving reduced Monte Carlo variance and favorable MPS bond-dimension growth compared with standard Kraus-insertion baselines.
Simulating open quantum systems remains one of the most demanding problems in quantum dynamics, as environmental interactions lead to non-unitary evolution that challenges standard quantum simulation techniques. Variational quantum algorithms (VQAs) offer a practical way forward by combining classical optimization with quantum hardware, making them well suited for near-term devices. Here, in Part V of the QFlux implementation series, we introduce adaptive variational-ansatz methods for solving the Lindblad master equation. This tutorial presents the Python-based qmad module of the QFlux toolkit, which is designed to help students and researchers easily build and explore adaptive VQA strategies. The tutorial walks through each step of the workflow and applies the methods to two illustrative examples. The first is the amplitude-damping channel, which models spontaneous emission in a two-level system. The second is the Fenna-Matthews-Olson (FMO) complex, a prominent model for excitonic energy transfer in photosynthesis. The two workflows utilizing adaptive variational approaches implemented here are the unrestricted adaptive variational quantum dynamics (UAVQD) scheme and the stochastic Schrödinger equation (SSE) based trajectory method. By working through these examples, readers gain both intuition and hands-on experience with variational techniques for simulating dissipative quantum dynamics.
In this paper, we introduce a novel and general framework for the variational quantum simulation of Lindblad equations. Building on the close relationship between the unraveled Lindblad dynamics, stochastic Magnus integrators, and variational quantum simulation, we propose a high-order scheme for solving the quantum state diffusion equation using exponential integrators. This formulation facilitates the simulation of wavefunction trajectories within the established framework of variational quantum algorithms for time evolution. Our algorithm significantly enhances robustness in two key aspects: the stability of the simulation with large time steps, and the reduction in the number of quantum trajectories required to accurately simulate the Lindblad dynamics in terms of the ensemble average. We demonstrate the effectiveness of our algorithm through numerical examples in both classical and quantum implementations, including the transverse-field Ising model (TFIM) with damping, the Fenna-Matthews-Olson (FMO) complex, and the radical pair model (RPM). The simulation accuracy can be systematically improved, and the algorithm remains reliable even in highly oscillatory regimes. These methods are expected to be applicable to a broader class of open quantum systems beyond the specific models considered in this study.