Simulating quantum dynamics at finite temperature in complex chemical systems remains a central challenge in quantum chemistry and materials science. In many cases, it is advantageous to focus on the dynamics of a subsystem of interest, represented by its reduced density matrix, whose interaction with the surrounding environment gives rise to open-system behavior. The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a formally exact framework for capturing such non-Markovian dynamics, where memory effects play a defining role. Here, in Part VI of the QFlux tutorial series, we explore methods for simulating non-Markovian open quantum systems described by the GQME on quantum computers. The approach leverages the Sz.-Nagy dilation theorem to embed memory-dependent, non-unitary evolution into an enlarged unitary space suitable for quantum simulation. Using the spin-boson model as a prototypical example, we demonstrate how to formulate, implement, and analyze GQME dynamics within the QFlux open-source platform. This tutorial provides readers with both the theoretical foundation and practical tools required to study non-Markovian quantum dynamics, bridging fundamental concepts with executable workflows for quantum simulation.
Simulating quantum dynamics at finite temperature in complex chemical systems remains a central challenge in quantum chemistry and materials science. In many cases, it is advantageous to focus on the dynamics of a subsystem of interest, represented by its reduced density matrix, whose interaction with the surrounding environment gives rise to open-system behavior. The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a formally exact framework for capturing such non-Markovian dynamics, where memory effects play a defining role. Here, in Part VI of the QFlux tutorial series, we explore methods for simulating non-Markovian open quantum systems described by the GQME on quantum computers. The approach leverages the Sz.-Nagy dilation theorem to embed memory-dependent, non-unitary evolution into an enlarged unitary space suitable for quantum simulation. Using the spin-boson model as a prototypical example, we demonstrate how to formulate, implement, and analyze GQME dynamics within the QFlux open-source platform. This tutorial provides readers with both the theoretical foundation and practical tools required to study non-Markovian quantum dynamics, bridging fundamental concepts with executable workflows for quantum simulation.
Energy and charge transfer processes, which are central to photosynthesis, catalysis, and molecular electronics, are governed by quantum dynamics in contact with complex environments. Interactions with surrounding degrees of freedom give rise to dissipation, decoherence, and energy exchange, shaping the mechanisms that determine chemical reactivity and transport phenomena. In Part IV of the QFlux tutorial series, we extend our study of quantum dynamics to open quantum systems, where time evolution is intrinsically non-unitary. Using the Lindblad master equation as a unifying and practical framework, we show how environmental effects can be modeled, simulated, and visualized with QFlux , an open-source platform that bridges classical and quantum computation. The tutorial presents the theoretical foundations of open-system dynamics, constructs the Lindblad propagator, and implements representative simulations ranging from spin models to chemically relevant double-well potentials. Special attention is given to dilation methods, which embed non-unitary dynamics within an enlarged unitary space through ancillary qubits, enabling open-system simulations on quantum hardware. This installment provides both conceptual insight and computational methodology for exploring dissipative quantum dynamics, laying the groundwork for subsequent parts addressing variational, non-Markovian approaches.
We introduce QFlux, an open-source Python package for simulating quantum dynamics in chemical systems across multiple levels of theory using a unified classical and quantum computational framework. QFlux integrates deterministic wavefunction propagation, Lindblad master equations, generalized quantum master equations (GQMEs), operator-splitting techniques, and variational quantum algorithms within a modular architecture designed for systematic benchmarking and hybrid classical-quantum execution. The framework supports both closed-and open-system dynamics, including Markovian and non-Markovian regimes, and provides explicit mappings between classical propagators and quantum circuit constructions executable on NISQ hardware. Built atop established scientific and quantum software ecosystems, QFlux emphasizes reproducibility, interoperability, and controlled cross-validation between numerical approximations and hardware-oriented algorithms. Part I in this series of papers presents the theoretical foundations, software design principles, and methodological scope of QFlux, and positions it within the broader landscape of classical, tensor-network, and quantum-circuit-based simulation platforms. The series is designed to serve as a teaching reference for graduate students, a practical guide for researchers implementing custom quantum simulations, and a foundational reference for the broader QFlux ecosystem.
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.
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.
We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.
We present a modified approach for simulating electronically nonadiabatic dynamics based on the Nakajima-Zwanzig generalized quantum master equation (GQME). The modified approach utilizes the fact that the Nakajima-Zwanzig formalism does not require casting the overall Hamiltonian in system-bath form, which is arguably neither natural nor convenient in the case of the Hamiltonian that governs nonadiabatic dynamics. Within the modified approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density operator is fully captured by a memory kernel super-operator. A methodology for calculating the memory kernel from projection-free inputs is developed. Simulating the electronic dynamics via the modified approach, with a memory kernel obtained using exact or approximate methods, can be more cost effective and/or lead to more accurate results than direct application of those methods. The modified approach is compared to previously proposed GQME-based approaches, and its robustness and accuracy are demonstrated on a benchmark spin-boson model with a memory kernel which is calculated within the Ehrenfest method.