Efficient construction of generalized master equation memory kernels for multi-state systems from nonadiabatic quantum-classical dynamics

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Predicting rate kernels via dynamic mode decomposition

Simulating dynamics of open quantum systems is sometimes a significant challenge, despite the availability of various exact or approximate methods. Particularly when dealing with complex systems, the huge computational cost will largely limit the applicability of these methods. In this work, we investigate the usage of dynamic mode decomposition (DMD) to evaluate the rate kernels in quantum rate processes. DMD is a data-driven model reduction technique that characterizes the rate kernels using snapshots collected from a small time window, allowing us to predict the long-term behaviors with only a limited number of samples. Our investigations show that whether the external field is involved or not, the DMD can give accurate prediction of the result compared with the traditional propagations, and simultaneously reduce the required computational cost.

A deep learning approach to the measurement of long-lived memory kernels from generalized Langevin dynamics

Memory effects are ubiquitous in a wide variety of complex physical phenomena, ranging from glassy dynamics and metamaterials to climate models. The Generalized Langevin Equation (GLE) provides a rigorous way to describe memory effects via the so-called memory kernel in an integro-differential equation. However, the memory kernel is often unknown, and accurately predicting or measuring it via, e.g., a numerical inverse Laplace transform remains a herculean task. Here, we describe a novel method using deep neural networks (DNNs) to measure memory kernels from dynamical data. As a proof-of-principle, we focus on the notoriously long-lived memory effects of glass-forming systems, which have proved a major challenge to existing methods. In particular, we learn the operator mapping dynamics to memory kernels from a training set generated with the Mode-Coupling Theory (MCT) of hard spheres. Our DNNs are remarkably robust against noise, in contrast to conventional techniques. Furthermore, we demonstrate that a network trained on data generated from analytic theory (hard-sphere MCT) generalizes well to data from simulations of a different system (Brownian Weeks-Chandler-Andersen particles). Finally, we train a network on a set of phenomenological kernels and demonstrate its effectiveness in generalizing to both unseen phenomenological examples and supercooled hard-sphere MCT data. We provide a general pipeline, KernelLearner, for training networks to extract memory kernels from any non-Markovian system described by a GLE. The success of our DNN method applied to noisy glassy systems suggests that deep learning can play an important role in the study of dynamical systems with memory.

A derivation of the conditions under which bosonic operators exactly capture fermionic structure and dynamics

The dynamics of many-body fermionic systems are important in problems ranging from catalytic reactions at electrochemical surfaces to transport through nanojunctions and offer a prime target for quantum computing applications. Here, we derive the set of conditions under which fermionic operators can be exactly replaced by bosonic operators that render the problem amenable to a large toolbox of dynamical methods while still capturing the correct dynamics of n-body operators. Importantly, our analysis offers a simple guide on how one can exploit these simple maps to calculate nonequilibrium and equilibrium single- and multi-time correlation functions essential in describing transport and spectroscopy. We use this to rigorously analyze and delineate the applicability of simple yet effective Cartesian maps that have been shown to correctly capture the correct fermionic dynamics in select models of nanoscopic transport. We illustrate our analytical results with exact simulations of the resonant level model. Our work provides new insights as to when one can leverage the simplicity of bosonic maps to simulate the dynamics of many-electron systems, especially those where an atomistic representation of nuclear interactions becomes essential.

Tensor-Train Thermo-Field Memory Kernels for Generalized Quantum Master Equations

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.

An accurate and efficient Ehrenfest dynamics approach for calculating linear and nonlinear electronic spectra

Linear and nonlinear electronic spectra provide an important tool to probe the absorption and transfer of electronic energy. Here, we introduce a pure state Ehrenfest approach to obtain accurate linear and nonlinear spectra that is applicable to systems with large numbers of excited states and complex chemical environments. We achieve this by representing the initial conditions as sums of pure states and unfolding multi-time correlation functions into the Schrödinger picture. By doing this, we show that one can obtain significant improvements in accuracy over the previously used projected Ehrenfest approach and that these benefits are particularly pronounced in cases where the initial condition is a coherence between excited states. While such initial conditions do not arise when calculating linear electronic spectra, they play a vital role in capturing multidimensional spectroscopies. We demonstrate the performance of our method by showing that it is able to quantitatively capture the exact linear, 2D electronic spectroscopy, and pump-probe spectra for a Frenkel exciton model in slow bath regimes and is even able to reproduce the main spectral features in fast bath regimes.

Compact and complete description of non-Markovian dynamics

Generalized master equations provide a theoretically rigorous framework to capture the dynamics of processes ranging from energy harvesting in plants and photovoltaic devices to qubit decoherence in quantum technologies and even protein folding. At their center is the concept of memory. The explicit time-nonlocal description of memory is both protracted and elaborate. When physical intuition is at a premium, one would desire a more compact, yet complete, description. Here, we demonstrate how and when the time-convolutionless formalism constitutes such a description. In particular, by focusing on the dissipative dynamics of the spin-boson and Frenkel exciton models, we show how to: easily construct the time-local generator from reference reduced dynamics, elucidate the dependence of its existence on the system parameters and the choice of reduced observables, identify the physical origin of its apparent divergences, and offer analysis tools to diagnose their severity and circumvent their deleterious effects. We demonstrate that, when applicable, the time-local approach requires as little information as the more commonly used time-nonlocal scheme, with the important advantages of providing a more compact description, greater algorithmic simplicity, and physical interpretability. We conclude by introducing the discrete-time analog and a straightforward protocol to employ it in cases where the reference dynamics have limited resolution. The insights we present here offer the potential for extending the reach of dynamical methods, reducing both their cost and conceptual complexity.

Quasiclassical approaches to the generalized quantum master equation

The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels that are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper, we study the approximate solution of the GQME, obtained by calculating the kernels with two methods: Ehrenfest mean-field theory and spin-mapping. We test the approaches on a range of spin-boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin-mapping outperforms Ehrenfest for all the systems studied. The problem of unphysical negative electronic populations affecting spin-mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite.

B800-to-B850 relaxation of excitation energy in bacterial light harvesting: All-state, all-mode path integral simulations

We report fully quantum mechanical simulations of excitation energy transfer within the peripheral light harvesting complex (LH2) of Rhodopseudomonas molischianum at room temperature. The exciton–vibration Hamiltonian comprises the 16 singly excited bacteriochlorophyll states of the B850 (inner) ring and the 8 states of the B800 (outer) ring with all available electronic couplings. The electronic states of each chromophore couple to 50 intramolecular vibrational modes with spectroscopically determined Huang–Rhys factors and to a weakly dissipative bath that models the biomolecular environment. Simulations of the excitation energy transfer following photoexcitation of various electronic eigenstates are performed using the numerically exact small matrix decomposition of the quasiadiabatic propagator path integral. We find that the energy relaxation process in the 24-state system is highly nontrivial. When the photoexcited state comprises primarily B800 pigments, a rapid intra-band redistribution of the energy sharply transitions to a significantly slower relaxation component that transfers 90% of the excitation energy to the B850 ring. The mixed character B850* state lacks the slow component and equilibrates very rapidly, providing an alternative energy transfer channel. This (and also another partially mixed) state has an anomalously large equilibrium population, suggesting a shift to lower energy by virtue of exciton–vibration coupling. The spread of the vibrationally dressed states is smaller than that of the eigenstates of the bare electronic Hamiltonian. The total population of the B800 band is found to decay exponentially with a 1/ e time of 0.5 ps, which is in good agreement with experimental results.

A simple improved low temperature correction for the hierarchical equations of motion

The study of open system quantum dynamics has been transformed by the hierarchical equations of motion (HEOM) method, which gives the exact dynamics for a system coupled to a harmonic bath at arbitrary temperature and system-bath coupling strength. However in its standard form the method is only consistent with the weak-coupling quantum master equation at all temperatures when many auxiliary density operators are included in the hierarchy, even when low temperature corrections are included. Here we propose a new low temperature correction scheme for the termination of the hierarchy based on Zwanzig projection which alleviates this problem, and restores consistency with the weak-coupling master equation with a minimal hierarchy. The utility of the new correction scheme is demonstrated on a range of model systems, including the Fenna-Metthews-Olson complex. The new closure is found to improve convergence of the HEOM even beyond the weak-coupling limit and is very straightforward to implement in existing HEOM codes.

Explaining the Efficiency of Photosynthesis: Quantum Uncertainty or Classical Vibrations?

Photosynthetic organisms are known to use a mechanism of vibrationally assisted exciton energy transfer to efficiently harvest energy from light. The importance of quantum effects in this mechanism is a long-standing topic of debate, which has traditionally focused on the role of excitonic coherences. Here, we address another recent claim: that the efficient energy transfer in the Fenna-Matthews-Olson complex relies on nuclear quantum uncertainty and would not function if the vibrations were classical. We present a counter-example to this claim, showing by trajectory-based simulations that a description in terms of quantum electrons and classical nuclei is indeed sufficient to describe the funneling of energy to the reaction center. We analyze and compare these findings to previous classical-nuclear approximations that predicted the absence of an energy funnel and conclude that the key difference and the reason for the discrepancy is the ability of the trajectories to properly account for Newton's third law.

A partially linearized spin-mapping approach for simulating nonlinear optical spectra

We present a partially linearized method based on spin-mapping for computing both linear and nonlinear optical spectra. As observables are obtained from ensembles of classical trajectories, the approach can be applied to the large condensed-phase systems that undergo photosynthetic light-harvesting processes. In particular, the recently derived spin partially linearized density matrix method has been shown to exhibit superior accuracy in computing population dynamics compared to other related classical-trajectory methods. Such a method should also be ideally suited to describing the quantum coherences generated by interaction with light. We demonstrate that this is, indeed, the case by calculating the nonlinear optical response functions relevant for the pump-probe and 2D photon-echo spectra for a Frenkel biexciton model and the Fenna-Matthews-Olsen light-harvesting complex. One especially desirable feature of our approach is that the full spectrum can be decomposed into its constituent components associated with the various Liouville-space pathways, offering a greater insight beyond what can be directly obtained from experiments.

On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation

Quantum master equations provide a general framework for describing the dynamics of electronic observables within a complex molecular system. One particular family of such equations is based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. In this paper, we show how different choices of projection operators, as well as whether one starts out with the time-convolution or the time-convolutionless forms of the generalized quantum master equation, give rise to four different types of such off-diagonal quantum master equations (OD-QMEs), namely, time-convolution and time-convolutionless versions of a Pauli-type OD-QME for only the electronic populations and an OD-QME for the full electronic density matrix (including both electronic populations and coherences). The fact that those OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrödinger picture electronic coherences from interaction picture populations recently introduced by Trushechkin in the context of time-convolutionless Pauli-type OD-QME to the other three types of OD-QMEs. The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar benchmark model for charge transfer in the condensed phase across a relatively wide parameter range. The results show that time-convolution OD-QMEs can be significantly more accurate than their time-convolutionless counterparts, particularly in the case of Pauli-type OD-QMEs, and that rather accurate Schrödinger picture coherences can be obtained from interaction picture electronic inputs.

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

A Road Map to Various Pathways for Calculating the Memory Kernel of the Generalized Quantum Master Equation

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronic energy, charge, and coherence transfer dynamics in molecular systems. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. However, the actual memory kernel depends on the choice of projection operator and is therefore not unique. Furthermore, calculating the memory kernel can be done in multiple ways that use different forms of projection-free inputs. Although the electronic dynamics is invariant to those choices when quantum-mechanically exact projection-free inputs are used, this is not the case when they are obtained via more feasible semiclassical or mixed quantum-classical approximate methods. Furthermore, the accuracy and numerical stability of the resulting electronic dynamics has been observed to be sensitive to the above-mentioned choices when approximate methods are used to calculate the projection-free inputs. In this article, we provide a systematic road map to 30 possible pathways for calculating the memory kernel and highlight how they are related as well as the ways in which they differ. We also compare the performance of different pathways in the context of the spin-boson benchmark model, with the projection-free inputs obtained via a mapping Hamiltonian linearized semiclassical method. In this case, we find that expressing the memory kernel with an exponential operator where the projection operator precedes the Liouvillian yields the most accurate and most numerically stable results.

A reciprocal-space formulation of mixed quantum-classical dynamics

We derive a formulation of mixed quantum-classical dynamics for modeling electronic carriers interacting with phonons in reciprocal space. For dispersionless phonons, we start by expressing the real-space classical coordinates in terms of complex variables. Taking these variables as a Fourier series then yields the reciprocal-space coordinates. Evaluating the electron-phonon interaction term through Ehrenfest's theorem, we arrive at a reciprocal-space formalism that is equivalent to mean-field mixed quantum-classical dynamics in real space. This equivalence is numerically verified for the Holstein and Peierls models, for which we find the reciprocal-space Hellmann-Feynman forces to involve momentum-derivative contributions in addition to the position-derivative terms commonly seen in real space. To illustrate the advantage of the reciprocal-space formulation, we present a proof of concept for the inexpensive modeling of low-momentum carriers interacting with phonons using a truncated reciprocal-space basis, which is not possible within a real-space formulation.

Simulating energy transfer dynamics in the Fenna-Matthews-Olson complex via the modified generalized quantum master equation

The generalized quantum master equation (GQME) provides a general and formally exact framework for simulating the reduced dynamics of open quantum systems. The recently introduced modified approach to the GQME (M-GQME) corresponds to a specific implementation of the GQME that is geared toward simulating the dynamics of the electronic reduced density matrix in systems governed by an excitonic Hamiltonian. Such a Hamiltonian, which is often used for describing energy and charge transfer dynamics in complex molecular systems, is given in terms of diabatic electronic states that are coupled to each other and correspond to different nuclear Hamiltonians. Within the M-GQME approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic density matrix is fully captured by a memory kernel superoperator, which can be obtained from short-lived (compared to the time scale of energy/charge transfer) projection-free inputs. In this paper, we test the ability of the M-GQME to predict the energy transfer dynamics within a seven-state benchmark model of the Fenna-Matthews-Olson (FMO) complex, with the short-lived projection-free inputs obtained via the Ehrenfest method. The M-GQME with Ehrenfest-based inputs is shown to yield accurate results across a wide parameter range. It is also found to dramatically outperform the direct application of the Ehrenfest method and to provide better-behaved convergence with respect to memory time in comparison to an alternative implementation of the GQME approach previously applied to the same FMO model.

On the advantages of exploiting memory in Markov state models for biomolecular dynamics

Biomolecular dynamics play an important role in numerous biological processes. Markov State Models (MSMs) provide a powerful approach to study these dynamic processes by predicting long time scale dynamics based on many short molecular dynamics (MD) simulations. In an MSM, protein dynamics are modeled as a kinetic process consisting of a series of Markovian transitions between different conformational states at discrete time intervals (called "lag time"). To achieve this, a master equation must be constructed with a sufficiently long lag time to allow interstate transitions to become truly Markovian. This imposes a major challenge for MSM studies of proteins since the lag time is bound by the length of relatively short MD simulations available to estimate the frequency of transitions. Here, we show how one can employ the generalized master equation formalism to obtain an exact description of protein conformational dynamics both at short and long time scales without the time resolution restrictions imposed by the MSM lag time. Using a simple kinetic model, alanine dipeptide, and WW domain, we demonstrate that it is possible to construct these quasi-Markov State Models (qMSMs) using MD simulations that are 5-10 times shorter than those required by MSMs. These qMSMs only contain a handful of metastable states and, thus, can greatly facilitate the interpretation of mechanisms associated with protein dynamics. A qMSM opens the door to the study of conformational changes of complex biomolecules where a Markovian model with a few states is often difficult to construct due to the limited length of available MD simulations.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Real-Time Path Integral Simulation of Exciton-Vibration Dynamics in Light-Harvesting Bacteriochlorophyll Aggregates

The mechanism of excitation energy transfer in photoexcited bacteriochlorophyll (BChl) aggregates poses intriguing questions, which have important implications for the observed efficiency of photosynthesis. We investigate this process through fully quantum mechanical calculations of exciton-vibration dynamics in chains and rings of BChl a molecules, with parameters characterizing the B850 ring of the LH2 complex of photosynthetic bacteria. The calculations are performed using the modular path integral methodology, which allows the exact treatment of 50 intramolecular vibrations on each pigment using parameters obtained from spectroscopic Huang-Rhys factors with computational effort that scales linearly with aggregate length. Our results indicate that the interplay between electronic and vibrational time scales leads to the rapid suppression but not the overdamping of electronic coherence, which facilitates the spreading of excitation energy throughout the aggregate.