Several derivative and integral approximations are explored for discretizing the Nakajima-Zwanzig generalized quantum master equation (NZ-QME or GQME) to obtain discrete quantum master equation (DQME) hierarchies and relations between discrete memory kernel and reduced density matrix (RDM) elements. It is shown that the simplest forward-difference approximation does not allow the reliable determination of the discrete kernel elements, even in the infinitesimal time-step limit, and that discrete kernels obtained in earlier work are flawed, although the procedure can be remedied. The various approximations give rise to DQMEs that differ in structure and in the RDM-kernel relationships. It is shown that the use of a more accurate discretization based on the midpoint derivative and midpoint integral approximations leads to a DQME that exhibits endpoint effects, which reflect the weaker impact of the bath on the RDM during the first time step and which parallel those encountered in the small matrix decomposition of the path integral (SMatPI) with a symmetric factorization of the short-time propagator. The features of the DQME hierarchies and RDM-kernel relations are illustrated through analytical examples involving a simple integrodifferential equation and a scalar GQME model, as well as numerical results for a two-level system (TLS) coupled to a harmonic bath.
Several derivative and integral approximations are explored for discretizing the Nakajima-Zwanzig generalized quantum master equation (NZ-QME or GQME) to obtain discrete quantum master equation (DQME) hierarchies and relations between discrete memory kernel and reduced density matrix (RDM) elements. It is shown that the simplest forward-difference approximation does not allow the reliable determination of the discrete kernel elements, even in the infinitesimal time-step limit, and that discrete kernels obtained in earlier work are flawed, although the procedure can be remedied. The various approximations give rise to DQMEs that differ in structure and in the RDM-kernel relationships. It is shown that the use of a more accurate discretization based on the midpoint derivative and midpoint integral approximations leads to a DQME that exhibits endpoint effects, which reflect the weaker impact of the bath on the RDM during the first time step and which parallel those encountered in the small matrix decomposition of the path integral (SMatPI) with a symmetric factorization of the short-time propagator. The features of the DQME hierarchies and RDM-kernel relations are illustrated through analytical examples involving a simple integrodifferential equation and a scalar GQME model, as well as numerical results for a two-level system (TLS) coupled to a harmonic bath.
The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 112, 110401 (2014)] can be considered a discrete-time formulation of the Nakajima-Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 21, 5037 (2025)] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time t = 0. This work presents a detailed analysis of the discretization structure of the TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel KN and the continuous-time NZ-QME kernel K(NΔt). This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as Δt → 0. While the TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri's work. The relative performance of various schemes for either computing accurate K(NΔt) from exact dynamics or obtaining accurate dynamics from exact K(NΔt) warrants further investigation.
A recent article by Ivander, Lindoy and Lee [Nature Communications 15, 8087 (2024)] claims to discover the relationship between the generalized quantum master equation (GQME) and the path integral for a system coupled to a harmonic bath. However, this relationship was already established in 2020 by Makri in the context of the small matrix decomposition of the path integral (SMatPI) [J. Chem. Theory and Comput. 16, 4038 (2020)]. The procedure that this article uses in its Supplementary Information (SI) to obtain the various matrices follows the SMatPI decomposition steps for the alternative Trotter ordering. The absence of endpoint effects in the kernel matrices of the discretized GQME expression for the reduced density matrix (RDM) is the consequence of a crude GQME discretization and is not consistent with the SMatPI decomposition of an auxiliary matrix presented in the SI. This form is identical to the transfer tensor method (TTM) of Cerrillo and Cao [Phys. Rev. Lett. 112, 110401 (2014)]. Further, the Dyck path section of this article follows precisely the diagrammatic analysis developed by Wang and Cai in a recent paper [Communications in Computational Physics 36, 389 (2024)]. We elaborate on these three critiques in this Comment.
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.
The generalized quantum master equation (GQME) approach provides a rigorous framework for deriving the exact equation of motion for any subset of electronic reduced density matrix elements (e.g., the diagonal elements). In the context of electronic dynamics, the memory kernel and inhomogeneous term of the GQME introduce the implicit coupling to nuclear motion and dynamics of electronic density matrix elements that are projected out (e.g., the off-diagonal elements), allowing for efficient quantum dynamics simulations. Here, we focus on benchmark quantum simulations of electronic dynamics in a spin-boson model system described by various types of GQMEs. Exact memory kernels and inhomogeneous terms are obtained from short-time quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) simulations and are compared with those obtained from an approximate linearized semiclassical method, allowing for assessment of the accuracy of these approximate memory kernels and inhomogeneous terms. Moreover, we have analyzed the computational cost of the full and reduced-dimensionality GQMEs. The scaling of the computational cost is dependent on several factors, sometimes with opposite scaling trends. The TT-TFD memory kernels can provide insights on the main sources of inaccuracies of GQME approaches when combined with approximate input methods and pave the road for the development of quantum circuits that implement GQMEs on digital quantum computers.
This thesis describes the development of practical and efficient computational approaches to the quantum dynamics of complex systems. Most of the work presented here relies on the generalized quantum master equation (GQME) formalism, which provides a simple equation of motion of reduced dimensionality for a set of dynamical quantities, e.g., nonequilibrium averages and equilibrium time correlation functions. The reduced dimensionality of the GQME comes at a cost: the introduction of the memory kernel, which accounts for the influence of all ``excluded'' degrees of freedom. Focusing first on the second-order perturbative treatment of the memory kernel known as Redfield theory, I present a collaborative effort to extend its applicability into highly non-Markovian regions via a mode freezing approach. In this method, a portion of bath modes characterized by low frequencies are treated as sources of static disorder and used to calculate modified Redfield dynamics. Application of the method to the spin-boson and FMO complex models indicates that the Redfield+frozen modes scheme consistently produces dynamics that are as good or better than bare Redfield dynamics. Next, we explore GQME approach coupled to the self-consistent solution of the memory kernel, which requires the calculation of auxiliary kernels. Previous implementations of the method had shown impressive boosts in efficiency and, when approximate methods were used to calculate the auxiliary kernels, accuracy over direct calculation of nonequilibrium averages. We show that this method, when formulated from the Mori perspective, is equally applicable to nonequilibrium averages and equilibrium correlation functions. In addition, we examine the dependence of the improvements afforded by the GQME framework on the choice projection operator and kernel closure. In particular, we demonstrate that improvements in efficiency, which rely on short memory lifetimes, are sensitively dependent on the choice of projection operator, and that the choice of kernel closure directly dictates the improvements in accuracy. In addition, we present evidence that indicates that the success of the GQME formalism when the auxiliary kernels are calculated via semi- and quasi-classical methods is largely due to the exact sampling of bath operators at t=0 required by the calculation of specific kernel closures. Next, we provide analytical arguments that delineate when the GQME framework coupled to the self-consistent solution of the memory kernel is likely to provide improvements in efficiency and accuracy. Finally, we present a path integral framework that can efficiently render the partially Wigner-transformed canonical density operator for systems coupled linearly to harmonic baths. This approach permits the direct calculation of any thermodynamic quantity and can be integrated into dynamical schemes like the Ehrenfest, surface hopping, or linearized semi-classical initial value representation methods to calculate equilibrium correlation functions.
We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of the corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) a full density matrix GQME, (2) a single-population scalar GQME, (3) a populations-only GQME, and (4) a subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations is comparable to that obtained via the full density matrix GQMEs.