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The Journal of Chemical Physics

Algorithms and software for open quantum system dynamics

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

Published Aug 1, 2025• 3 citations
Publisher
DOI
10.1063/5.0289390

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Abstract We report the implementation of a hierarchical equations of motion (HEOM) module within the open‐source Libra software. It includes the standard and scaled HEOM algorithms for computing the dynamics of open quantum systems interacting with a harmonic bath. The module allows the computing of the evolution of the reduced density matrix, as well as spectral lineshapes. The truncation, filtering, and “update list” schemes, as well as OpenMP parallelization, allow for further computational saving. The package is written in a mix of C++ and Python languages, delivering the best compromise between user friendliness and efficiency. The Python layer of the package takes advantage of standard Python libraries, such as h5py, which allows efficient storage and retrieval of the generated results. The package can be seamlessly used within Jupyter notebooks; its careful design shall provide the maximal convenience and intuitiveness to its users.

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Abstract Many‐body open quantum systems (OQSs) have a profound impact on various subdisciplines of physics, chemistry, and biology. Thus, the development of a computer program capable of accurately, efficiently, and versatilely simulating many‐body OQSs is highly desirable. In recent years, we have focused on the advancement of numerical algorithms based on the fermionic hierarchical equations of motion (HEOM) theory. Being in‐principle exact, this approach allows for the precise characterization of many‐body correlations, non‐Markovian memory, and non‐equilibrium thermodynamic conditions. These efforts now lead to the establishment of a new computer program, HEOM for QUantum Impurity with a Correlated Kernel, version 2 (HEOM‐QUICK2), which, to the best of our knowledge, is currently the only general‐purpose simulator for fermionic many‐body OQSs. Compared with version 1, the HEOM‐QUICK2 program features more efficient solvers for stationary states, more accurate treatment of non‐Markovian memory, and improved numerical stability for long‐time dissipative dynamics. Integrated with quantum chemistry software, HEOM‐QUICK2 has become a valuable theoretical tool for the precise simulation of realistic many‐body OQSs, particularly the single atomic or molecular junctions. Furthermore, the unprecedented precision achieved by HEOM‐QUICK2 enables accurate simulation of low‐energy spin excitations and coherent spin relaxation. The unique usefulness of HEOM‐QUICK2 is demonstrated through several examples of strongly correlated quantum impurity systems under non‐equilibrium conditions. Thus, the new HEOM‐QUICK2 program offers a powerful and comprehensive tool for studying many‐body OQSs with exotic quantum phenomena and exploring applications in various disciplines. This article is categorized under: Data Science > Computer Algorithms and Programming Software > Simulation Methods Theoretical and Physical Chemistry > Statistical Mechanics

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In this thesis, we develop an automated framework for open-loop quantum control design, called the Generalized Control of Noiseless Subspaces (GCNS) framework, that targets a broad class of open quantum systems subject to the Born-Markov approximation. The framework is automated, highly generalized, computationally efficient, and extensible. It rests on an eleven-point Formal Framework Scope that specifies the model classes, control standards, information preservation standards, and control input classes it supports. It additionally rests on a nine-point set of Software Implementation Guidelines (SIG) that specifies how any software implementation of the framework should ideally perform with respect to efficiency, automaticity, and scalability. In constructing the framework, we develop novel theorems for controllability in Quantum Noiseless Subsystems, novel algorithms for the efficient, automated detection and parameterization of subsystem codes, and novel algorithms for the systematic selection and synthesis of control Hamiltonians from large candidate sets of feasible controls. Additionally, we develop the theory of Transition-Effort-based Control Input Synthesis: a versatile approach to generating control input signals, subject to a wide potential range of constraints, for the purpose of noiselessly effecting quantum operations on encoded quantum states. The thesis provides a comprehensive examination of a MATLAB- and Java-based implementation of the framework called the QBox library. The examination covers supporting algorithms critical for meeting the efficiency and extensibility guidelines in the SIG. It additionally covers all major aspects of the software library's anatomy, its application programming interface, and its suite of supplementary tools assisting control design. To validate the GCNS framework, its principal workflow is applied to two open quantum system models selected from the quantum information processing literature, based on trapped ions and superconducting transmon qutrits. The models and workflow parameters are selected to demonstrate a broad cross-section of the algorithms, subroutines, and software tools developed in the thesis. The examples further establish the exceptional scope and configurability of the framework, highlighting its importance to rapid design iteration in the noisy intermediate-scale quantum computing era.

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Abstract We report the implementation of a hierarchical equations of motion (HEOM) module within the open‐source Libra software. It includes the standard and scaled HEOM algorithms for computing the dynamics of open quantum systems interacting with a harmonic bath. The module allows the computing of the evolution of the reduced density matrix, as well as spectral lineshapes. The truncation, filtering, and “update list” schemes, as well as OpenMP parallelization, allow for further computational saving. The package is written in a mix of C++ and Python languages, delivering the best compromise between user friendliness and efficiency. The Python layer of the package takes advantage of standard Python libraries, such as h5py, which allows efficient storage and retrieval of the generated results. The package can be seamlessly used within Jupyter notebooks; its careful design shall provide the maximal convenience and intuitiveness to its users.
Abstract Many‐body open quantum systems (OQSs) have a profound impact on various subdisciplines of physics, chemistry, and biology. Thus, the development of a computer program capable of accurately, efficiently, and versatilely simulating many‐body OQSs is highly desirable. In recent years, we have focused on the advancement of numerical algorithms based on the fermionic hierarchical equations of motion (HEOM) theory. Being in‐principle exact, this approach allows for the precise characterization of many‐body correlations, non‐Markovian memory, and non‐equilibrium thermodynamic conditions. These efforts now lead to the establishment of a new computer program, HEOM for QUantum Impurity with a Correlated Kernel, version 2 (HEOM‐QUICK2), which, to the best of our knowledge, is currently the only general‐purpose simulator for fermionic many‐body OQSs. Compared with version 1, the HEOM‐QUICK2 program features more efficient solvers for stationary states, more accurate treatment of non‐Markovian memory, and improved numerical stability for long‐time dissipative dynamics. Integrated with quantum chemistry software, HEOM‐QUICK2 has become a valuable theoretical tool for the precise simulation of realistic many‐body OQSs, particularly the single atomic or molecular junctions. Furthermore, the unprecedented precision achieved by HEOM‐QUICK2 enables accurate simulation of low‐energy spin excitations and coherent spin relaxation. The unique usefulness of HEOM‐QUICK2 is demonstrated through several examples of strongly correlated quantum impurity systems under non‐equilibrium conditions. Thus, the new HEOM‐QUICK2 program offers a powerful and comprehensive tool for studying many‐body OQSs with exotic quantum phenomena and exploring applications in various disciplines. This article is categorized under: Data Science > Computer Algorithms and Programming Software > Simulation Methods Theoretical and Physical Chemistry > Statistical Mechanics
In this thesis, we develop an automated framework for open-loop quantum control design, called the Generalized Control of Noiseless Subspaces (GCNS) framework, that targets a broad class of open quantum systems subject to the Born-Markov approximation. The framework is automated, highly generalized, computationally efficient, and extensible. It rests on an eleven-point Formal Framework Scope that specifies the model classes, control standards, information preservation standards, and control input classes it supports. It additionally rests on a nine-point set of Software Implementation Guidelines (SIG) that specifies how any software implementation of the framework should ideally perform with respect to efficiency, automaticity, and scalability. In constructing the framework, we develop novel theorems for controllability in Quantum Noiseless Subsystems, novel algorithms for the efficient, automated detection and parameterization of subsystem codes, and novel algorithms for the systematic selection and synthesis of control Hamiltonians from large candidate sets of feasible controls. Additionally, we develop the theory of Transition-Effort-based Control Input Synthesis: a versatile approach to generating control input signals, subject to a wide potential range of constraints, for the purpose of noiselessly effecting quantum operations on encoded quantum states. The thesis provides a comprehensive examination of a MATLAB- and Java-based implementation of the framework called the QBox library. The examination covers supporting algorithms critical for meeting the efficiency and extensibility guidelines in the SIG. It additionally covers all major aspects of the software library's anatomy, its application programming interface, and its suite of supplementary tools assisting control design. To validate the GCNS framework, its principal workflow is applied to two open quantum system models selected from the quantum information processing literature, based on trapped ions and superconducting transmon qutrits. The models and workflow parameters are selected to demonstrate a broad cross-section of the algorithms, subroutines, and software tools developed in the thesis. The examples further establish the exceptional scope and configurability of the framework, highlighting its importance to rapid design iteration in the noisy intermediate-scale quantum computing era.