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Journal of Chemical Theory and Computation

Variational Quantum Simulation of Open Quantum Dynamics via Non-Markovian Stochastic Schrödinger Equation with Complex Frequency Modes

Yukai Guo, Xing Bao Gao

Published Sep 8, 2025• 1 citation
Publisher

Abstract

Simulating non-Markovian open quantum dynamics is crucial for understanding complex quantum systems, yet it poses significant challenges for standard quantum hardware. These challenges stem from the non-Hermitian nature of such dynamics, which results in nonunitary evolution, as well as constraints imposed by limited quantum resources. To address this, we propose a hybrid quantum-classical algorithm designed for simulating dissipative dynamics in systems coupled to non-Markovian environments. Our approach includes formulating a non-Markovian Stochastic Schrödinger equation with complex frequency modes (cNMSSE) where the non-Markovianity is characterized by the mode excitation. We then employ variational quantum simulation to capture the nonunitary evolution within the cNMSSE framework, leading to a substantial reduction in qubit requirements. To demonstrate our approach, we investigated dissiaptive dynamics in the spin-boson model (SBM) and excitation energy transfer processes in both a prototype dimer system and the biologically relevant Fenna-Matthews-Olson (FMO) complex.

DOI
10.1021/acs.jctc.5c00517

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Simulating non-Markovian open quantum dynamics is crucial for understanding complex quantum systems, yet it poses significant challenges for standard quantum hardware. These challenges stem from the non-Hermitian nature of such dynamics, which results in nonunitary evolution, as well as constraints imposed by limited quantum resources. To address this, we propose a hybrid quantum-classical algorithm designed for simulating dissipative dynamics in systems coupled to non-Markovian environments. Our approach includes formulating a non-Markovian Stochastic Schrödinger equation with complex frequency modes (cNMSSE) where the non-Markovianity is characterized by the mode excitation. We then employ variational quantum simulation to capture the nonunitary evolution within the cNMSSE framework, leading to a substantial reduction in qubit requirements. To demonstrate our approach, we investigated dissiaptive dynamics in the spin-boson model (SBM) and excitation energy transfer processes in both a prototype dimer system and the biologically relevant Fenna-Matthews-Olson (FMO) complex.
Quantum simulation of non-Markovian open quantum dynamics is essential but challenging for standard quantum computers due to their non-Hermitian nature, leading to non-unitary evolution, and the limitations of available quantum resources. Here we introduce a hybrid quantum-classical algorithm designed for simulating dissipative dynamics in system with non-Markovian environment. Our approach includes formulating a non-Markovian Stochastic Schrödinger equation with complex frequency modes (cNMSSE) where the non-Markovianity is characterized by the mode excitation. Following this, we utilize variational quantum simulation to capture the non-unitary evolution within the cNMSSE framework, leading to a substantial reduction in qubit requirements. To demonstrate our approach, we investigated the spin-boson model and dynamic quantum phase transitions (DQPT) within transverse field Ising model (TFIM). Significantly, our findings reveal the enhanced DQPT in TFIM due to non-Markovian behavior.
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.
We introduce an extended formulation of the non-Markovian stochastic Schrödinger equation with complex frequency modes (extended cNMSSE), designed for simulating open quantum system dynamics under arbitrary spectral densities. This extension employs non-exponential basis sets to expand the bath correlation functions, overcoming the reliance of the original cNMSSE on exponential decompositions of the spectral density. Consequently, the extended cNMSSE is applicable to environments beyond those characterized by Debye-type spectral densities. The flexibility to employ general basis functions is particularly advantageous for handling spectral densities with higher-order poles, for which exponential decompositions are often inaccurate or unavailable. The extended cNMSSE is implemented in a pseudo-Fock space using conventional ladder operators and solved efficiently via matrix product state techniques, preserving the favorable linear-scaling and wavefunction-based nature of the original method. Benchmark simulations across four representative cases, including discrete spectral density, Ohmic spectral density with exponential and algebraic cutoffs, and critically damped Brownian spectral density, demonstrate excellent agreement with results of hierarchy of forward-backward stochastic Schrödinger equations and extended hierarchical equation of motion.
The Non-Markovian Stochastic Schrodinger Equation (NMSSE) offers a promising approach for open quantum simulations, especially in large systems, owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end,we propose a scheme that combines neural network techniques with simulations of the non-Markovian stochastic Schrodinger equation. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs),along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thereby substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.
The theory of open quantum systems has many applications ranging from simulating quantum dynamics in condensed phases to better understanding quantum-enabled technologies. At the center of theoretical chemistry are the developments of methodologies and computational tools for simulating charge and excitation energy transfer in solutions, biomolecules, and molecular aggregates. As a variety of these processes display non-Markovian behavior, classical computer simulation can be challenging due to exponential scaling with existing methods. With quantum computers holding the promise of efficient quantum simulations, in this paper, we present a new quantum algorithm based on Kraus operators that capture the exact non-Markovian effect at a finite temperature. The implementation of the Kraus operators on the quantum machine uses a combination of singular value decomposition (SVD) and optimal Walsh operators that result in shallow circuits. We demonstrate the feasibility of the algorithm by simulating the spin-boson dynamics and the exciton transfer in the Fenna-Matthews-Olson (FMO) complex. The NISQ results show very good agreement with the exact ones.
The non-Markovian stochastic Schrödinger equation (NMSSE) offers a promising approach for open quantum simulations owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end, we propose a scheme that utilizes neural networks to extract effective information from long-time simulations of the non-Markovian stochastic Schrödinger equation, fine-tunes the networks using converged short-time data, and predicts long-term system evolution, eliminating the influence of stochastic oscillations. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs), along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thus substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.