# Algorithms and software for open quantum system dynamics

Alex W. Chin, Jonathan Keeling, Dvira Segal, Haobin Wang

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Alex W. Chin, Jonathan Keeling, Dvira Segal, Haobin Wang. Algorithms and software for open quantum system dynamics. The Journal of Chemical Physics, 2025, 163 (5), 10.1063/5.0289390. hal-05406536

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# Editorial: Algorithms and Software for Open Quantum System Dynamics

Alex Chin,¹ Jonathan Keeling,² Dvira Segal,³,⁴ and Haobin Wang⁵

¹) Sorbonne Université, CNRS, Institut des NanoSciences de Paris, 4 place Jussieu, 75005 Paris, France

²) SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, United Kingdom

³) Department of Chemistry, University of Toronto, 80 Saint George St., Toronto, Ontario, M5S 3H6, Canada

⁴) Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, 60 Saint George St., Toronto, Ontario, M5S 1A7, Canada

⁵) Department of Chemistry, University of Colorado Denver, Science Building 1151 Arapahoe St, Denver, Colorado 80204, United States

(Dated: 9 December 2025)

## I. INTRODUCTION AND CONTEXT

This special issue of the Journal of Chemical Physics is dedicated to work developing and applying algorithms to allow accurate and efficient simulations of realistic open quantum systems, and progress towards simulating many-body or high-dimensional open quantum systems. The papers included cover a range of aspects of this topic, with papers describing conceptual approaches to tackling such problems, papers introducing computational algorithms and describing software packages, and papers applying such methods to practically relevant questions involving open quantum system dynamics. As discussed in the following, the range of applications presented here includes the photophysics of organic molecules and light harvesting systems, charge carrier dynamics, the physics of colloidal quantum dots, transition metal dichalcogenides, plasmonic systems, and effects of strong light-matter coupling on organic molecules.

A wide range of methods are described in this special issue, illustrative of the wide variety of approaches currently being explored across this field. Several of these methods make use of concepts related to tensor networks¹,². The key concept here is that high-order tensors, i.e. those with many indices, can often be efficiently compressed in terms of networks of lower-order tensors, summing over the intermediate bonds. There are several distinct ways in which these ideas can be applied in modeling open quantum systems. Some involve explicit representation of the environment, particularly using “chain mappings” to map the environment to a linear chain of sites³,⁴. One can then use matrix product states (a subclass of tensor networks, where the network is linear) to efficiently represent the state of the chain-mapped environment. Work using this approach presented in this special issue includes Refs. 5–7. A second, similar, context is the tensor network reformulation⁸,⁹ of the multiconfigurational time-dependent Hartree formalism¹⁰; this formalism is designed to efficiently tackle large Hilbert spaces, and so is again based on explicit representations of the environment. Examples of work related to this approach in this special issue include Refs. 11–13. A third, distinct, context is methods that use tensor

networks to represent the process tensor. Here, instead of explicitly representing the environment, its properties are described by the process tensor, a multilinear map from the sequence of operations to the final state of the system¹⁴⁻¹⁹. References 20–24 in this issue involve these kinds of approaches, and Refs. 25 and 26 also study closely related approaches. Other papers in this special issue²⁷,²⁸ also apply matrix product state methods in other contexts.

Among the papers in this special issue, a significant number fall within the Journal’s section of “Chemical Physics Software”²⁹. These papers describe some of the many software packages that are available for general use by researchers in this field. The benefits of making scientific software available have long been recognized³⁰⁻³⁴, both in terms of reproducibility, the ability to compare and benchmark different methods, lowering the barriers to entry for the field, and maximizing the benefits that arise from the work of those who develop scientific software. It is inspiring to see the number of open-source software packages described in the special issue, and we hope that collecting in one place a summary of the current status of many of these packages will serve as a useful resource for the field.

## II. SUMMARY OF AREAS COVERED

In the following, we provide an overview of the contributions to this special issue, divided into three broad categories. In Section II A we cover papers that discuss concepts for new algorithms or theoretical approaches. In Section II B we include papers that talk about software packages, computational algorithms, or particular developments related to computational methods. Finally, in Section II C we summarize the papers that discuss specific applications.

### A. Concepts

A rich palette of theoretical and computational advances are covered by this special issue, ranging from
problems related to abstract modeling of open quantum systems to methods that could be combined with *ab initio* descriptions of complex chemical systems. Many contributions illustrate promising new physical and/or computational ideas that could support the development of new software—in addition to papers providing formal descriptions of existing open source packages which are discussed below in Section II B. In surveying these strikingly diverse contributions, two broad conceptual categories emerge: (1) time-independent problems and (2) the real-time simulation of dissipative quantum dynamics.

In the first category, novel use of Recurrent Neural Networks for describing the ground state of dipolar rotors is given by Serwatka and Roy³⁵, while Maitra et al.³⁶ extend the DeePMD neural network architecture to obtain electronic structure properties (excited state energies, forces and non-adiabatic couplings) with machine learning methods. A promising development of the inch-worm quantum Monte Carlo algorithms (covering non-equilibrium, as well as ground state many body properties) is also presented by Erpenbeck et al.³⁷.

The second category covers a wider range of techniques, many of which advance the simulation of dynamics of dissipative dynamics in specific contexts. For example, several articles present developments within the framework of mixed quantum-classical and semiclassical approaches for chemically complex systems, such as a formalism for including optical cavity modes in treating polaritonic chemistry³⁸⁻⁴⁰, mixed QM-classical approaches for non-Markovian solid-state systems⁴¹, and extensions of semi-classical methods using Generalized Quantum Master Equations⁴²,⁴³. Weight et al. ⁴⁴ present a fully benchmarked demonstration of the spin-mapping approach to non-adiabatic dynamics on *ab initio* models of four test molecules. Hogg et al. complete this grouping with a tutorial overview of how quasiclassical tools can be used to treat open system problems⁴⁵.

The intrinsic complexity of simulating non-Markovian dynamics also points to the virtues of finding alternative or simpler approximate models that can capture the key phenomenology of non-Markovian systems. In this spirit, Chen and Franco introduce a quasiparticle-based approach to bosonic thermal environments⁴⁶, while Attal et al. develop a coarse-graining approach for intermediate-sized environments using effective energy states⁴⁷. Ferracin and Smirne extend recent ideas about so-called ‘Markovian Embeddings’ to reduce the complexity of problems involving fermionic baths⁴⁸, and Xie et al. ⁴⁹ demonstrate the advantages of Markovian closures in a bosonic context via a new pure-state unraveled hybrid-bath method. Takahashi and Borrelli⁵⁰ present an efficient low-rank approximation of structured system-environment couplings that leads to numerical advantages in TEDOPA-like simulations of open dynamics.

Comparing and understanding the conceptual underpinnings and validity of the many different types of master equations that have been proposed is a central topic of the field, and this comes across in the large num-

ber of articles we received for this special issue. In particular, the reader will find significant new developments in the theory of Hierarchical Equations of Motion (HEOM)⁶,⁵¹, the Nakajima-Zwanzig equation⁵², polaron master equations⁵³, influence functionals for path integrals (for bosons and fermions, using neural networks and tensor networks)²⁴,²⁶,⁵⁴, the use of tensor trains and machine learning for obtaining reduced density matrices²⁵, and the dissipative equations of motion (DEOM) applied to spin baths and spin-lattice coupling⁵⁵,⁵⁶. Yang et al. ⁵⁷ also demonstrate how secular Born-Markov master equations can be used to look at non-equilibirum transport in dissipative spin-chains.

While master equations have been the main focus of open system theory from its inception, the difficulties of going from the dynamics of reduced density matrices to more complex but experimentally relevant objects, such as multi-time correlation functions, have recently lead to a more generalized ‘reduced’ object known as the process tensor. This exciting new development, and its related software development, is described in several articles in this topical issue, and the concept of using it for quantum control is developed in Ref. 23.

The other major strand in open system theory involves approaches that treat the entire system-environment wave function as a many body problem, an approach that is particularly suited for the full-rank problems presented by realistic vibronic models in non-adiabatic chemical dynamics. Again, the reader will find a number of striking advances for wave-function-based approaches in this special issue, ranging from the application of neural networks and machine learning methods for wave function dynamics on classical and quantum computers⁵⁸, to simulation strategies for stochastic Schrödinger equations⁵⁹⁻⁶¹. Tensor Networks are also playing a prominent role in full wave function methods⁷, and Le Dé et al. present an interesting comparison of how accurately temperature effects can be captured in the Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm (T-TEDOPA) and hierarchical equations of motion (HEOM) master equation approaches⁶. Strachan, Purkayastha, and Clark⁶² also extend chain mapping approaches to extract dynamical maps for reduced density matrices coupled to multiple fermionic baths, which is critical for studying many-body transport. Our conceptual category of wave function-based approaches is finally complemented by new results involving other types of wave function ansätze, such as Multiconfigurational Ehrenfest with Full Cloning, and trajectory approaches based on non-Gaussian states with the truncated Wigner approximation¹¹,⁶³.

## B. Algorithms and software packages

The special issue includes several papers introducing algorithms and open source software packages for accurate and efficient simulations of open quantum sys-
tems with areas of applications covering nonadiabatic
charge transfer processes, excitation energy transfer in
light-harvesting complexes and disordered media, such as
organic photovoltaics, and quantum nanoelectronic de-
vices. Broadly, these software packages can be divided
into two groups: (1) model system simulations and (2)
chemical and large scale simulations, which we discuss
in turn below. Several of the codes presented here were
developed for multicore processors and for a graphics pro-
cessing unit (GPU).

# 1. Minimal model systems

Model system simulations address the fundamental
challenge of accurately describing quantum systems that
are *strongly* coupled to baths, which are often *struc-
tured* resulting in *non-Markovian* dynamics. Additional
challenging aspects that are addressed by modern algo-
rithms and software packages are handling many body
interactions within the system, and possibly involving
long-range interactions, treating time-dependent Hamil-
tonians and optimizing quantum control protocols under
dissipation, calculating multitime correlation functions as
relevant for spectroscopy, treating multiple environments
and possibly of different types, bosonic, fermionic, and
anharmonic, and simulating observables not only of the
system but also jointly with the environments, address-
ing quantum thermodynamic setups. Canonical models
commonly treated by these software packages include the
spin-boson model, the single-impurity Anderson model,
the Frenkel exciton model, the Holstein model, and their
extensions.

Several algorithms featured in this special issue lever-
age tensor network methods to simulate non-Makrovian
and strongly coupled open quantum systems. These ap-
proaches represent quantum states and operators using
matrix product states (MPS) and matrix product opera-
tors (MPO), respectively, and employ tensor networks to
represent the process tensor (PT) or the closely-related
influence matrix functional. A notable example is the
ACE computational toolkit²⁰, which efficiently encodes
and compresses the influence of different environments on
quantum systems via the PT-MPO formalism. Similarly,
the Python package OQuPy leverages the PT formal-
ism, representing it in a tensor network form, thus en-
abling efficient-compact simulations of strongly coupled
system bath and non-Markovian dynamics²¹. MPS and
MPO are also used in MPSQD, a software package that
enables simulations of closed and open system quantum
dynamics²⁷, with the latter performed using the HEOM
method. The multi-layer multi-configuration time de-
pendent Hartree (ML-MCTDH) method is also shown to
greatly benefit from tensor network states, with Ref. 12
releasing an algorithm that automatically constructs an
optimal tree tensor network representation for the wave-
function. Related to that, the MPSDYNAMICS.JL soft-
ware package⁵ describes a tensor network representation

of the T-TEDOPA, which is based on environment-chain
mapping⁶. To simulate quantum thermodynamic ma-
chines, such as quantum engines, Koyanagi and Tanimura
developed two codes to simulate different thermodynamic
processes of relevance for quantum heat engines. T-
QFPE⁶⁴ performs simulations via the quantum Fokker-
Planck equations, while HEOM-MB⁶⁵ uses the HEOM
approach extended to describe multiple baths.

While the above mentioned software packages mainly
target systems coupled to bosonic-harmonic environ-
ments, other packages focus on fermionic problems at
strong system-electrode, modeled by the single-impurity
Anderson model and its extensions, and realized, e.g.,
in quantum dot junctions. Nestmann et al. introduce
REALTIMETRANSPORT to compute the memory kernel
in the Nakajima-Zwanzig formulation using a renormal-
ized perturbation theory⁶⁶. The KELDYSHQFT code⁶⁷
by Ritz et al. allows the calculation of the real frequency
dynamical correlation functions using functional renor-
malization group methods.

## 2. Chemical and large scale systems

Contrasting minimal model-system simulations, the
second category of packages collected in this special is-
sue aims to capture carrier quantum dynamics in realistic
settings of chemical environments and disordered mate-
rials. These problems often require large-scale and de-
tailed molecular simulations. To achieve this goal, one
needs to integrate electronic structure calculations, and,
in some cases, molecular dynamics simulations, into the
open quantum system computational workflow.

The package PYCTRAMER described by Liu et al.⁶⁸
offers an automatic workflow to simulate charge trans-
port processes in disordered media, such as organic pho-
tovoltaic materials. Beginning with quantum chemical
and molecular dynamics simulations, minimal models are
constructed to allow rate constant calculations at differ-
ent levels of Fermi’s Golden rule. Associated with this,
the KYLIN-V package by Xu et al.⁶⁹ enables the calcu-
lation of dynamics and spectroscopic properties of large
systems, supporting a range of methods such as the time-
dependent density matrix renormalization group (TD-
DMRG). This package is accelerated via MPS and MPO
implementations. MESOHOPS is a wavefunction ap-
proach for open quantum systems, which builds on the
non-Markovian quantum state diffusion formalism. The
package developed by Citty et al.⁷⁰ allows simulations of
mesoscale excited state processes in molecular materials
with 10⁶ molecules. The PYSCES code by Myers et al.
interfaces nonadiabatic semicalssical dynamics with elec-
tronic structure calculations on-the-fly⁷¹. Accelerated by
GPUs, it enables studies of solvation and charge transfer
dynamics in condensed phases.
### C. Applications

The third broad area of articles in this special issue are those focused on using open quantum systems to answer physical questions—both fundamental questions about the generic properties of open quantum systems, as well as questions more directly tied to specific applications.

The first category of fundamental questions includes several papers. Takahashi et al.⁷² present work on the ever-present spin-boson problem, specifically mapping out the behavior of the finite-temperature model with sub-Ohmic spectra. Two papers explore questions of bath-mediated interactions, Brenes et al.⁷³ discuss synchronization via such interactions, and Xu, Stockburger, and Ankerhold⁷⁴ discuss bath-mediated interactions in a many-body spin system. Kim and Franco⁷⁵ explore how one may determine which bath modes are responsible for a particular energy transfer process (using methods introduced by these authors in Ref.⁵²). Turning toward more specific applications, contributions include work by Boettcher et al.⁷⁶ on quantum Otto engines, a form of heat engine, and exploring how bath observables may provide useful information on the processes involved.

Turning to papers based on particular physical systems, key physical contexts include organic molecular systems²⁸,⁷⁷⁻⁷⁹, colloidal quantum dots⁸⁰, transition metal dichalcogenides (TMD)⁴¹,⁸¹, and plasmonic system⁸².

A recurring challenge for open quantum systems methods is the calculation of optical spectra. Lambertson et al.⁷⁸ discuss calculating the optical spectra of organic molecules with complex vibrational environments, such as effects beyond the Frank-Condon approximation. Krotz and Tempelaar⁴¹ discuss exciton optical line-shapes in TMD materials, accounting for exciton-phonon scattering. A number of papers explore the behavior of systems with strong light-matter coupling, leading to polaritons—hybrid light-matter particles. Sokolovskii and Groenhof⁷⁹ discuss relaxation processes in molecular polaritons, with particular consideration of the challenges of dealing with multiple molecules. Peng and Rabani⁸⁰ examine polariton assisted energy transfer between colloidal quantum dots, accounting for phonon effects.

A final area to mention is that of charge carrier dynamics. Zhou, Gao, and Shuai²⁸ discuss charge carrier dynamics in organic semiconductors, while Liu et al.⁸¹ consider TMD materials such as MoSe₂, and take into account interband recombination and intraband relaxation processes. Mäck, Thoss, and Rudge⁷⁷ discuss molecules interacting with metal surfaces and the resulting electronic friction, while Dall'Osto et al.⁸² focus on hot carrier dynamics in plasmonic systems.

### III. CONCLUSIONS

This Editorial highlights the growing number of algorithms, software packages, concepts, and applications in

the area of open quantum system dynamics, as showcased in the contributions to this special issue. Collectively, this special issue demonstrates a strong activity in the field of open quantum systems. It features efforts aimed at both advancing conceptual understanding and enabling realistic simulations of quantum systems with newly designed theoretical and computational tools.

Applications of open quantum systems span a wide range of topics, including quantum dynamics, quantum thermodynamics, and quantum transport. The physical platforms studied are equally diverse, ranging from organic molecules and colloidal quantum dots to layered materials and plasmonic systems, with players including interacting electrons, nuclei, spins, and photons. Methods range from those targeting fundamental-abstract models to techniques aiming to perform simulations in great detail.

A substantial portion of the featured works falls under the “Algorithms and software packages” section, reflecting the community’s growing commitment to open-source developments, thus accessibility and reproducibility. The availability of scientific software will enable systematic benchmarking of different methods. Furthermore, it will lower the entry barrier for researchers new to the field and enhance the impact of methodological advances. The packages presented here showcase current capabilities and directions in the field. Our hope is that they serve both as a resource and as an inspiration for continuing developments.

### ACKNOWLEDGMENTS

The guest editors would like to thank all authors who contributed articles to this special issue, and the journal editors and staff for their assistance.

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