Statistical and machine learning approaches for prediction of long-time excitation energy transfer dynamics Kimara Naicker1 2Ilya Sinayskiy1 2and Francesco Petruccione1 2 3

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Statistical and machine learning approaches for prediction of long-time excitation

energy transfer dynamics

Kimara Naicker,1, 2, ∗Ilya Sinayskiy,1, 2 and Francesco Petruccione1, 2, 3

1Quantum Research Group, School of Chemistry and Physics,

University of KwaZulu-Natal, Durban, KwaZulu-Natal, 4001, South Africa

2National Institute for Theoretical and Computational Sciences (NITheCS), South Africa

3School of Data Science and Computational Thinking and Department of Physics,

Stellenbosch University, Stellenbosch, 7600, South Africa

(Dated: October 28, 2022)

One of the approaches used to solve for the dynamics of open quantum systems is the hierarchical

equations of motion (HEOM). Although it is numerically exact, this method requires immense

computational resources to solve. The objective here is to demonstrate whether models such as

SARIMA, CatBoost, Prophet, convolutional and recurrent neural networks are able to bypass this

requirement. We are able to show this successfully by ﬁrst solving the HEOM to generate a data set

of time series that depict the dissipative dynamics of excitation energy transfer in photosynthetic

systems then, we use this data to test the models ability to predict the long-time dynamics when

only the initial short-time dynamics is given. Our results suggest that the SARIMA model can serve

as a computationally inexpensive yet accurate way to predict long-time dynamics.

I. INTRODUCTION

Time series analysis involves methods of analysing a

series of data points that are indexed in time order. The

objective of the analysis is to collect and study the past

observations of a time series to develop an appropriate

model which describes the inherent structure of the se-

ries. This model is then used to generate future values

for the series, i.e. to make forecasts [1]. In this work, the

data being analysed is relevant to the dissipative dynam-

ics of excitation energy transfer (EET) in systems similar

to the photosynthetic open quantum system regime.

In some cases, information about the underlying dy-

namical correlations in open quantum systems can be

encoded at the initial stages of their evolution. There-

fore, it may be possible to obtain long-time dynamics of

open quantum systems from the knowledge of their short-

time evolution. This conjecture allows the bypass of the

need for direct long-time simulations. The simulation of

numerically exact methods to describe the dynamics of

open quantum systems often require immense computa-

tional resources that scale exponentially with the size of

the system under study, hence, it is desirable to develop

an approach that can accurately predict long-time dy-

namics of open quantum systems along with eliminating

the need for direct calculations to some extent.

Various numerical solutions for the dynamics of open

quantum systems have been developed considering the

complexity of system-bath interactions. The dynamics

of an open quantum system that are dependent on the

Hamiltonian of the system can be described through den-

sity matrix-based approaches in the Liouville space of the

system. The numerically exact formalism adopted in this

study is the hierarchical equations of motion (HEOM) de-

∗kimaranaicker@gmail.com

veloped by Tanimura and Kubo, and later adapted to bi-

ological light harvesting complexes by Ishizaki and Flem-

ing [2–4]. Machine learning (ML) has been applied to this

focus area in many relevant cases [5–10]. L. E. Herrera

et al. conducted a comparative study where they bench-

marked ML models based on their eﬃciency in predict-

ing long-time dynamics of a two-level quantum system

linearly coupled to harmonic bath [9].

Successful time series forecasting depends on an appro-

priate model ﬁtting. The development of eﬃcient models

to improve forecasting accuracy has evolved in literature.

A comparison of the predictive capabilities of a standard

statistical, an additive regression and a tree-based model

against more structurally complex neural network models

to simulate the open quantum system dynamics is car-

ried out using Python. The ﬁrst stage of the dynamics

is obtained by solving the HEOM for a suﬃciently large

theoretical system, thereafter, we train and test suitable

models to determine the validity of our approach. That

is to predict a time series from that series past values

eﬃciently.

This paper contains several sections which are orga-

nized as follows: in Sec II we describe the formalism

used, Sec III describes the data pre-processing procedure,

Sec IV covers the various time series models used, Sec V

presents our experimental forecasting results in terms of

MSE obtained on relevant datasets and a brief conclusion

of our work as well as the prospective future aim in this

ﬁeld.

II. THE THEORY AND THE DATA

A time series is a sequential set of data points measured

over successive times. It is mathematically deﬁned as a

set of vectors x(t), t = 0,1,2, . . . where t represents the

time elapsed [11]. The variable x(t) can be treated as

a random variable. The measurements taken during an

arXiv:2210.14160v2 [quant-ph] 27 Oct 2022

event in a time series are arranged in chronological order.

Quantum systems faced in the real world are rarely

entirely isolated, hence, it is important to consider the

inﬂuence of the surrounding environment (bath) when

studying the dynamical behaviour of a system. In the

process of an open quantum system, such as a photosyn-

thetic pigment-protein complex, evolving over time we

can generate a set of time dependent observables that

depict the coherent movement of electronic excitations

through the system by solving the HEOM. This section

describes the theoretical background used to generate the

data sets used in the study.

The total Hamiltonian is composed of the Hamiltonian

of the system, bath and system-bath interaction,

HT ot =ˆ

HS+ˆ

HB+ˆ

HSB .(1)

We focus on the simplest electronic energy transfer sys-

tem, a dimer, where Hamiltonian of the system refers to

the electronic states of a complex containing 2 pigments,

HS=

j=1

|jijhj|+J12(|1ih2|+|2ih1|),(2)

where jis the excited state energy of the jth site and J12

denotes the electronic coupling between both sites. Here

we consider that each pigment is coupled to a separate

bath. The bath Hamiltonian represents the environmen-

tal phonons,

HB=

j=1

HBj,ˆ

HBj=X

~ωj,αˆp2

j,α + ˆq2

j,α

2,(3)

where pis the conjugate momentum, qis the dimension-

less coordinate and ωj,α is the frequency of the jth site

and αth phonon mode, respectively. The last term of

Eq. (1) represents the ﬂuctuations in the site energies

caused by the phonon dynamics,

HSB =

j=1

ˆuj|ji hj|,ˆuj=X

gj,α ˆqj,α,(4)

where gj,α is the coupling constant between the jth site

and αth phonon mode.

The spectral density Jj(ω) speciﬁes the coupling of

an electronic transition of the jth pigment to the envi-

ronmental phonons through the reorganization energy λj

and the timescale of the phonon relaxation γj. Here it

is expressed as the Ohmic spectral density with Lorentz-

Drude cut-oﬀ, Jj(ω)=2λjγjω/(ω2+γ2

j).

We focus on the application of this theory to EET at

physiological temperatures of around 300 K, hence, the

high-temperature condition characterized by ~γj/kBT

1 is imposed and the following hierarchically coupled

equations of motion are given [4],

∂

∂t ˆσ(n1,n2)(t) = −iˆ

Le+n1γ1+n2γ2ˆσ(n1,n2)(t)

+ˆ

Φ1ˆσ(n1+1,n2)(t) + n1ˆ

Θ1ˆσ(n1−1,n2)(t)

+ˆ

Φ2ˆσ(n1,n2+1)(t) + n2ˆ

Θ2ˆσ(n1,n2−1)(t).(5)

In Eq. (5), the element ˆσ(0, t) is identical to the reduced

density operator ˆρ(t), while the rest are auxiliary density

operators. The Liouvillian corresponding to the Hamil-

tonian ˆ

HSis denoted by ˆ

Leand the relaxation operators

Φjand and ˆ

Θjare given by Eqs. (6), (7) and (8),

Le= [ ˆ

HS,ˆρS],(6)

Φj=iV ×

j, V ×

jy= [Vj, y],(7)

Θj=i2λj

β~2V×

j−iλj

~γjV◦

j, V ◦

jy={Vj, y}.(8)

Formally the hierarchy in Eq. (5) is inﬁnite and cannot

be numerically integrated. In order to make this problem

tractable, the hierarchy can be terminated at a certain

depth. There are several methods of doing so and in this

work we have chosen the following termination condition

following Ishizaki and Fleming [12]. For the integers n=

(n1, n2) and for characteristic frequency ωeof ˆ

Lewhere

N ≡

j=1

njωe

min(γ1, γ2),(9)

Eq. (5) is replaced by

∂

∂t ˆσ(n, t) = −iˆ

Leˆσ(n, t).(10)

∂

∂t ˆσ(n, t) = −

iˆ

Le+

j=1

njγj

ˆσ(n, t)

j=1 ˆ

Φjˆσ(nj+, t) + njˆ

Θjˆσ(nj−, t).(11)

Eq. (11) is the general form of the reduced hierarchy

equation Eq. (5). The general form is employed in the

following section as it is solved to generate the time series

used in training and testing the models in the subsequent

sections for systems containing more than two sites.

III. DATA PRE-PROCESSING

Building a representative data set is an important ﬁrst

step in every machine learning project. Though the ap-

proach developed in this work can be generalized, we

discuss the simplest electronic energy transfer system -

FIG. 1: An example of a sequence generated by the

HEOM for a dimer to be split to form the input and

output data for the models.

a dimer (a spin-boson-type model [13]) or two-level sys-

tem as well as three- and four-level systems where lin-

ear chain conﬁgurations were imposed. The total system

under study can be fully determined by the ﬁve inde-

pendent energy scales: the site energy j, the coupling

strength Jjk, the reorganization energy λ, the cut-oﬀ fre-

quency ωcand thermal energy kBTof the bath. In or-

der to create a data set suitable for a framework aimed

at predicting quantum dynamics in all physically realiz-

able non-Markovian regimes, three parameters were ex-

tensively sampled while ﬁxing two parameters: the cut-

oﬀ frequency ωc= 53cm−1and thermal energy kBTof

the bath where T= 300K. These are the typical param-

eters of photosynthetic EET [14, 15] as seen in Table I.

Parameter Lower limit cm−1Upper limit cm−1

j-100 100

Jjk -100 100

λ1 100

TABLE I: The data set containing time-evolved reduced

density matrices is generated for all combinations of the

following parameters: the site energy j, the coupling

strength Jjk and the reorganization energy λ.

The HEOM method implemented in Python script is

used to solve equation (5). The hierarchy truncation is

set to 20 which is a suﬃcient depth based on the chosen

cut-oﬀ frequency [12]. To make sense of time series data,

it has to be collected over time in the same intervals.

The total propagation time is set to 1.0 ps as is suﬃcient

for observing coherent dynamics in photosynthetic EET

[12]. For each of the 40000 samples, observables based

on the diagonal elements of the time dependent density

matrices i.e. the time evolution of the site populations

are collected. Both of the generated observables are di-

vided into shorter trajectories or sequences to test the

capability of the models for varying output times to be

predicted.

One of our objectives is to determine the trade-oﬀ be-

tween how far ahead the model can forecast and the

shortest input time required to maintain high accuracy

in the forecast. As the total propagation time is ﬁxed to

1.0 ps, the actual size of the dataset used varies depend-

ing on the lengths of the input and output times. We

have performed a grid search in our model testing across

varying lengths of input times up to 0.2 ps and output

times between 0.01 ps and 0.6 ps.

Before testing, each sample time series was split into

multiple shorter slices. For example, a single series split

to produce inputs that are 0.2 ps long and outputs 0.6 ps

long would generate 1001 shorter sequences. This sliding

window technique is pictured in Figure 2

FIG. 2: The use of preceding time steps to predict the

following time steps is referred to as the sliding window

process. In the ﬁgure, the blue portion represents the

points in the series that will be used for training and

the yellow portion will be used in testing the models.

Each training/ testing set diﬀers as the ”window” is

moved forward over the entire time series by keeping

the length of the portions ﬁxed.

The data set is partitioned into a training set of 70%

of the data and a validation set of 10% of the data. Ad-

ditionally, 20% of the data is held out during the training

procedure and is used for testing.

IV. METHODOLOGY

In practice, a suitable model is ﬁtted to a given time

series and the corresponding parameters of the under-

lying are estimated using the known data values. The

procedure of ﬁtting a time series to a proper model is

known as Time Series Analysis. It comprises of methods

that attempt to understand the nature of the series and

is often useful for future forecasting and simulation. Past

observations are collected and analyzed to build a suit-

able mathematical model which captures the underlying

data generating process for the series. Then in time se-

ries forecasting, the future events are predicted using the

model [16, 17].

Competent time series analysis remains dominated by

traditional statistical methods as well as simpler machine

learning techniques such as ensembles of trees and linear

ﬁts.

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Statisticalandmachinelearningapproachesforpredictionoflong-timeexcitationenergytransferdynamicsKimaraNaicker,1,2,IlyaSinayskiy,1,2andFrancescoPetruccione1,2,31QuantumResearchGroup,SchoolofChemistryandPhysics,UniversityofKwaZulu-Natal,Durban,KwaZulu-Natal,4001,SouthAfrica2NationalInstituteforTheoret...

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Statistical and machine learning approaches for prediction of long-time excitation energy transfer dynamics Kimara Naicker1 2Ilya Sinayskiy1 2and Francesco Petruccione1 2 3

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