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 our response to the commentary piece by Makri {\it et al.} [arXiv:2410.08239], which raises critiques of our work [Nat. Commun. 15, 8087 (2024)]. In our paper, we considered various settings of open-quantum system dynamics, including non-commuting, non-diagonalizable system-bath coupling, and bosonic/spin/fermionic baths. For these, we showed a direct and explicit relationship between the discrete-time memory kernel (
K) of the generalized quantum master equation (GQME) and the discrete-time influence functions (
I) of the path integrals. As an application of this, we showed one can construct
K without projection-free dynamics inputs that conventional methods require, and we also presented a quantum sensing protocol that characterizes the bath spectral density from reduced system dynamics. As the Comment focused on the relationship between (
K) and
I in one specific setup (i.e., commuting, diagonalizable system-bath coupling with a bosonic bath), we focus on that aspect in this response. In summary, we could not find a set of equations that explicitly connect
I and
K from Makri's 2020 paper [J. Chem. Theory Comput. 16, 4038 (2020)]. Furthermore, while our analysis is specific to the choice of discretization of path-integral and GQME, we have not found issues with the GQME discretization employed. As per critiques on citations, in our paper, we note that we had acknowledged Makri's driven SMatPI work and Wang and Cai's tree-based SMatPI work for the number of Dyck paths needed for the computation of the memory kernel.
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.
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.
A small matrix decomposition of the path integral expression (SMatPI) that yields the reduced density matrix of a system interacting with a dissipative harmonic bath is obtained by recursively spreading the entangled influence functional terms over longer time intervals while simultaneously decreasing their magnitude until these terms become negligible. This allows summation over the path integral variables one by one through multiplication of small matrices with dimension equal to that of the bare system. The theoretical framework of the decomposition is described using a diagrammatic approach. Analytical and numerical calculations show that the necessary time length for the temporal entanglement to become negligible is practically the same as the bath-induced memory. The properties and structure of the propagator matrices are discussed, and applications to multistate systems are presented.