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Neural quantum states for non-Markovian open quantum dynamics in bosonic environments

Qingdong Zhu, Long Cao, Liwei Ge, Zhiqian Huang, Igor Ying Zhang, Xin Xu, Xiao Zheng

Published Oct 28, 2026• 0 citations
PDF (CC BY)Publisher

Abstract

Condensed-phase quantum dynamics are strongly shaped by vibrational, solvent, and protein environments, yet their nonperturbative, non-Markovian description requires a rapidly growing high-dimensional representation as environmental complexity increases. Here, we extend neural quantum states (NQS) to the dissipaton-embedded quantum master equation (DQME) for bosonic environments using a complex-valued restricted Boltzmann machine adapted to nonbinary bosonic occupations. An explicit filtering scheme further concentrates the neural representation on dynamically relevant dissipaton configurations. The resulting NQS-DQME approach reproduces representative quantum dissipative dynamics while effectively compressing the reduced density tensor (RDT). For an anharmonic model of condensed-phase hydrogen transfer, the method also reproduces weak-coupling tunneling dynamics. At stronger system-bath coupling, however, errors become localized in a small subset of RDT elements, showing that a small global variational residual does not necessarily ensure accurate chemically relevant observables. This limitation identifies observable-level error control as a key challenge for reliable long-time NQS propagation.

DOI
10.26434/chemrxiv.15011173/v1

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This work integrates the physics-informed neural network (PINN) approach into the neural quantum state framework to simulate open quantum system dynamics, to circumvent the computationally expensive time-dependent variational principle required in conventional variational methods. The proposed PINN-DQME method employs time-encoded neural networks within a time-domain decomposition strategy to represent the evolution governed by the dissipaton-embedded quantum master equation (DQME). We implement and validate this approach in the single-impurity Anderson model, benchmarking the PINN-DQME results against the numerically exact hierarchical equations of motion. The PINN-DQME method demonstrates high accuracy in simulating quantum dissipative dynamics at high temperatures, where non-Markovian effects are weak. However, for strongly non-Markovian dynamics at low temperatures, it encounters challenges with error accumulation during time propagation, highlighting an area for future refinement in applying PINNs to complex quantum dynamical settings.
Non-Markovianity, the intricate dependence of an open quantum system on its temporal evolution history, holds tremendous implications across various scientific disciplines. However, accurately characterizing the complex non-Markovian effects has posed a formidable challenge for numerical simulations. While quantum computing technologies show promise, a universal theory enabling practical quantum algorithm implementation has been elusive. We address this gap by introducing the dissipaton-embedded quantum master equation in second quantization (DQME-SQ). This exact and compact theory offers two key advantages: representability by quantum circuits and universal applicability to any Gaussian environment. We demonstrate these capabilities through digital quantum simulations of non-Markovian dissipative dynamics in both bosonic and fermionic environments. The DQME-SQ framework opens a new horizon for the efficient exploration of complex open quantum systems by leveraging the rapidly advancing quantum computing technologies.
Numerous variational methods have been proposed for solving quantum many-body systems, but they often face exponentially increasing computational complexity as the Hilbert space dimension grows. To address this, we introduce a novel approach using quantum neural networks to simulate the dissipative dynamics of many-body open quantum systems. This method combines neural-network quantum state representation with the time-dependent variational principle, both implemented via quantum algorithms. This results in accurate open quantum dynamics described by the Lindblad quantum master equation, exemplified by the spin-boson and transverse field Ising models. Our approach avoids the computational expense of classical algorithms and demonstrates the potential advantages of quantum computing for many-body simulations. To reduce measurement errors, we introduce a projection reset procedure, which could benefit other quantum simulations. In addition, our approach can be extended to simulate non-Markovian quantum dynamics.
In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.