QuLTSF: Long-Term Time Series Forecasting with Quantum Machine
Learning
Hari Hara Suthan Chittoor
1 a
, Paul Robert Griffin
1 b
, Ariel Neufeld
2 c
, Jayne Thompson
3,4 d
and
Mile Gu
2,5 e
1
School of Computing and Information Systems, Singapore Management University, 178902, Singapore
2
Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
3
Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore
4
5
Centre for Quantum Technologies, National University of Singapore, Singapore
Keywords:
Quantum Computing, Machine Learning, Time Series Forecasting, Hybrid Model.
Abstract:
Long-term time series forecasting (LTSF) involves predicting a large number of future values of a time series
based on the past values. This is an essential task in a wide range of domains including weather forecasting,
stock market analysis and disease outbreak prediction. Over the decades LTSF algorithms have transitioned
from statistical models to deep learning models like transformer models. Despite the complex architecture
of transformer based LTSF models ‘Are Transformers Effective for Time Series Forecasting? (Zeng et al.,
2023)’ showed that simple linear models can outperform the state-of-the-art transformer based LTSF models.
Recently, quantum machine learning (QML) is evolving as a domain to enhance the capabilities of classical
machine learning models. In this paper we initiate the application of QML to LTSF problems by proposing
QuLTSF, a simple hybrid QML model for multivariate LTSF. Through extensive experiments on a widely used
weather dataset we show the advantages of QuLTSF over the state-of-the-art classical linear models, in terms
of reduced mean squared error and mean absolute error.
1 INTRODUCTION
Time series forecasting (TSF) is the process of pre-
dicting future values of a variable using its histori-
cal data. TSF is an import problem in many fields
like weather forecasting, finance, power management
etc. There are broadly two approaches to handle TSF
problems: statistical models and deep learning mod-
els. Statistical models, like ARIMA, are the tradi-
tional work horse for TSF since the 1970’s (Hynd-
man, 2018; Hamilton, 2020). Deep learning models,
like recurrent neural networks (RNN’s), often outper-
form statistical models in large-scale datasets (Lim
and Zohren, 2021).
Increasing the prediction horizon strain’s the mod-
els predictive capacity. The prediction length of more
a
https://orcid.org/0000-0003-1363-606X
b
https://orcid.org/0000-0003-2294-5980
c
https://orcid.org/0000-0001-5500-5245
d
https://orcid.org/0000-0002-3746-244X
e
https://orcid.org/0000-0002-5459-4313
than 48 future points is generally considered as long-
term time series forecasting (LTSF) (Zhou et al.,
2021). Transformers and attention mechanism pro-
posed in (Vaswani, 2017) gained a lot of attraction
to model sequence data like language, speech etc.
There is a surge in the application of transformers to
LTSF leading to several time series transformer mod-
els (Zhou et al., 2021; Wu et al., 2021; Zhou et al.,
2022; Liu et al., 2022; Wen et al., 2023). Despite
the complicated design of transformer based models
for LTSF problems, (Zeng et al., 2023) showed that a
simple autoregressive model with a linear fully con-
nected layer can outperform the state-of-the-art trans-
former models.
Quantum machine learning (QML) is an emerg-
ing field that combines quantum computing and ma-
chine learning to enhance tasks like classification, re-
gression, generative modeling etc., using the currently
available noisy intermediate-scale quantum (NISQ)
computers (Preskill, 2018; Schuld and Petruccione,
2021; Simeone, 2022). Hybrid models contain-
824
Chittoor, H. H. S., Griffin, P. R., Neufeld, A., Thompson, J. and Gu, M.
QuLTSF: Long-Term Time Series Forecasting with Quantum Machine Learning.
DOI: 10.5220/0013395500003890
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 17th International Conference on Agents and Artificial Intelligence (ICAART 2025) - Volume 1, pages 824-829
ISBN: 978-989-758-737-5; ISSN: 2184-433X
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.

ing classical neural networks and variational quan-
tum circuits (VQC’s) are increasingly becoming pop-
ular for various machine learning tasks, thanks to
rapidly evolving software tools (Bergholm et al.,
2018; Broughton et al., 2020). The existing QML
models for TSF focus on RNN’s (Emmanoulopoulos
and Dimoska, 2022; Ceschini et al., 2022). In time se-
ries analysis, recurrent quantum circuits have demon-
strated provable computational and memory advan-
tage during inference, however learning such models
remains challenging at scale (Binder et al., 2018).
In this paper we initiate the application of QML
to LTSF by proposing QuLTSF a simple hybrid QML
model. QuLTSF is a combination of classical linear
neural networks and VQC’s. Through extensive ex-
periments, on the widely used weather dataset, we
show that the QuLTSF model outperforms the state-
of-the-art linear models proposed in (Zeng et al.,
2023).
Organization of the Paper: Section 2 provides
background information on quantum computing and
quantum machine learning. Section 3 discusses the
problem formulation and related work. In Section 4
we introduce our QuLTSF model. Experimental de-
tails, results, discussion and future research directions
are given in Section 5 and Section 6 concludes the pa-
per.
2 BACKGROUND
2.1 Quantum Computing
The fundamental unit of quantum information and
computing is the qubit. In contrast to a classical bit,
which exist in either 0 or 1 state, the state of a qubit
can be 0 or 1 or a superposition of both. In the Dirac’s
ket notation the state of a qubit is given as a two-
dimensional amplitude vector
|ψ⟩ =

α
0
α
1

= α
0
|0⟩ + α
1
|1⟩,
where α
0
and α
1
are complex numbers satisfying the
unitary norm condition, i.e., |α
0
|
2
+ |α
1
|
2
= 1. The
state of a qubit is only accessible through measure-
ments. A measurement in the computational basis
collapses the state |ψ⟩ to a classical bit x ∈ {0, 1} with
probability |α
x
|
2
. A qubit can be transformed from
one state to another via reversible unitary operations
also known as quantum gates (Nielsen and Chuang,
2010).
Shor’s algorithm (Shor, 1994) and Grover’s al-
gorithm (Grover, 1996) revolutionized quantum al-
gorithms research by providing theoretical quantum
speedups compared to classical algorithms. The im-
plementation of these algorithms require larger num-
ber of qubits with good error correction (Lidar and
Brun, 2013). However, the current available quan-
tum devices are far from this and often referred to as
the noisy intermediate-scale quantum (NISQ) devices
(Preskill, 2018). Quantum machine learning (QML)
is an emerging field to make best use of NISQ devices
(Schuld and Petruccione, 2021; Simeone, 2022).
2.2 Quantum Machine Learning
The most common QML paradigm refers to a two
step methodology consisting of a variational quan-
tum circuit (VQC) or ansatz and a classical optimizer,
where VQC is composed of parametrized quantum
gates and fixed entangling gates. The classical data
is first encoded into a quantum state, using a suit-
able data embedding procedure like amplitude em-
bedding, angle embedding etc. Then, the VQC ap-
plies a parametrized unitary operation which can be
controlled by altering its parameters. The output of
the quantum circuit is given by measuring the qubits.
The parameters of the VQC are optimized using clas-
sical optimization tools to minimize a predefined loss
function. Often VQC’s are paired with classical neu-
ral networks, creating hybrid QML models. Sev-
eral packages, for instance (Bergholm et al., 2018;
Broughton et al., 2020), provide software tools to effi-
ciently compute gradients for these hybrid QML mod-
els.
3 PRELIMINARIES
3.1 Problem Formulation
Consider a multivariate time series dataset with M
variates. Let L be the size of the look back win-
dow or sequence length and T be the size of the
forecast window or prediction length. Given data
at L time stamps x
1:L
= {x
1
, ··· , x
L
} ∈ R
L×M
, we
would like to predict the data at future T time stamps
ˆ
x
L+1:L+T
= {
ˆ
x
L+1
, ··· ,
ˆ
x
L+T
} ∈ R
T ×M
using a QML
model. Predicting more than 48 future time steps is
typically considered as Long-term time series fore-
casting (LTSF) (Zhou et al., 2021). We consider the
channel-independence condition where each of the
univariate time series data at L time stamps x
m
1:L
=
{x
m
1
, ··· , x
m
L
} ∈ R
L×1
is fed separately into the model
to predict the data at future T time stamps
ˆ
x
m
L+1:L+T
=
{ ˆx
m
L+1
, ··· , ˆx
m
L+T
} ∈ R
T ×1
, where m ∈ {1, ··· , M}. To
measure discrepancy between the ground truth and
QuLTSF: Long-Term Time Series Forecasting with Quantum Machine Learning
825

prediction, we use Mean Squared Error (MSE) loss
function defined as
MSE = E
x
"
1
M
M
∑
m=1
∥x
m
L+1:L+T
−
ˆ
x
m
L+1:L+T
∥
2
2
#
. (1)
3.2 Related Work
LTSF is an extensive area of research. In this section,
we provide a concise overview of works most relevant
to our problem formulation. Since the introduction
of transformers (Vaswani, 2017), there has been an
increase in transformer based models for LTSF (Wu
et al., 2021; Zhou et al., 2021; Liu et al., 2022; Zhou
et al., 2022; Li et al., 2019). (Wen et al., 2023) pro-
vided a comprehensive survey on transformer based
LTSF models. Despite the sophisticated architec-
ture of transformer based LTSF models, (Zeng et al.,
2023) showed that simple linear models can achieve
superior performance compared to the state-of-the-art
transformer based LTSF models.
(Zeng et al., 2023) proposed three models: Lin-
ear, NLinear and DLinear. Linear model is just a one
layer linear neural network. In NLinear, the last value
of the input is subtracted before being passed through
the linear layer and the subtracted part is added back
to the output. DLinear first decomposes the time se-
ries into trend, by a moving average kernel, and sea-
sonal components which is a famous method in time
series forecasting (Hamilton, 2020) and is extensively
used in the literature (Wu et al., 2021; Zhou et al.,
2022; Zeng et al., 2023). Two similar but distinct lin-
ear models are trained for trend and seasonal com-
ponents. Adding the outputs of these two models
gives the final prediction. We adapt the simple Linear
model to propose our QML model in the next section.
4 QuLTSF
Figure 1: QuLTSF model architecture.
In this section, we propose QuLTSF a hybrid QML
model for LTSF and is illustrated in Fig. 1. It is a
hybrid model consisting of input classical layer, hid-
den quantum layer and an output classical layer. The
input classical layer maps the L input features in to
a 2
N
length vector. Specifically, the input sequence
x
m
1:L
∈ R
L×1
is given to the input classical layer with
trainable weights W
in
∈ R
2
N
×L
and bias b
in
∈ R
2
N
×1
,
and it outputs a 2
N
length vector
y
1
= W
in
x
m
1:L
+ b
in
. (2)
The output of the input classical layer, y
1
∈ R
2
N
×1
,
is given as input to the hidden quantum layer which
consists of N qubits. We use amplitude embedding
(Schuld and Petruccione, 2021) to encode 2
N
real
numbers in y
1
to a quantum state |φ
in
⟩. We use hard-
ware efficient ansatz (Simeone, 2022) as a VQC, and
is composed of K layers each containing a trainable
parametrized single qubit gate on each qubit and a
fixed circular entangling circuit with CNOT gates as
shown in Fig. 1. Every single qubit gate has 3 param-
eters and the total number of parameters in K layers
is 3NK. The output of VQC is given as
|φ
out
⟩ = (VQC)|φ
in
⟩. (3)
We consider the expectation value of Pauli-Z observ-
able for each qubit, which serves as the output of hid-
den quantum layer and is denoted as y
2
∈ R
N×1
. Fi-
nally, y
2
is passed through output classical layer with
trainable weights W
out
∈ R
T ×N
and bias b
out
∈ R
T ×1
,
which maps N length quantum hidden layer output to
predicted T length vector
ˆ
x
m
L+1:L+T
= W
out
y
2
+ b
out
. (4)
The parameters of the hidden quantum layer and two
classical layers can be jointly trained, similar to clas-
sical machine learning, using software packages like
PennyLane (Bergholm et al., 2018).
5 EXPERIMENTS
In this section, we validate the superiority of the pro-
posed QuLTSF model through extensive experiments.
The code for experiments is publicly available on
GitHub
1
. All experiments are conducted on SMU’s
Crimson GPU cluster
2
.
5.1 Dataset Description
We evaluate the performance of our proposed
QuLTSF model on the widely used Weather dataset. It
1
https://github.com/chariharasuthan/QuLTSF
2
https://violet.scis.dev/
QAIO 2025 - Workshop on Quantum Artificial Intelligence and Optimization 2025
826

Table 1: Multivariate long-term time series forecasting (LTSF) results in terms of MSE and MAE between the proposed
QuLTSF model and the state-of-the-art on the widely used weather dataset. Sequence length L = 336 and prediction length
T ∈ {96, 192, 336, 720}. The best results are in bold and the second best results are underlined.
Methods QuLTSF* Linear NLinear DLinear FEDformer Autoformer Informer
Metric MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE
96 0.156 0.211 0.176 0.236 0.182 0.232 0.176 0.237 0.217 0.296 0.266 0.336 0.300 0.384
192 0.199 0.253 0.218 0.276 0.225 0.269 0.220 0.282 0.276 0.336 0.307 0.367 0.598 0.544
336 0.248 0.296 0.262 0.312 0.271 0.301 0.265 0.319 0.339 0.380 0.359 0.395 0.578 0.523
720 0.315 0.346 0.326 0.365 0.338 0.348 0.323 0.362 0.403 0.428 0.419 0.428 1.059 0.741
*QuLTSF is implemented by us; Other results are from (Zeng et al., 2023).
is recorded by Max-Planck Institute of Biogeochem-
istry
3
and consists of 21 meteorological and environ-
mental features like air temperature, humidity, carbon
dioxide concentration in parts per million etc. This is
recorded in 2020 with granularity of 10 minutes and
contains 52, 696 timestamps. 70 percent of the avail-
able data is used for training, 20 percent for testing
and the remaining data for validation.
5.2 Baselines
We choose all three state-of-the-art linear models
namely Linear, NLinear and DLinear proposed in
(Zeng et al., 2023) as the main baselines. We also
consider a few transformer based LTSF models FED-
former (Zhou et al., 2022), Autoformer (Wu et al.,
2021), Informer (Zhou et al., 2021) as other baselines.
Moreover (Wu et al., 2021; Zhou et al., 2021) showed
that transformer based models outperform traditional
statistical models like ARIMA (Box et al., 2015) and
other deep learning based models like LSTM (Bai
et al., 2018) and DeepAR (Salinas et al., 2020), thus
we do not include them in our baselines.
5.3 Evaluation Metrics
Following common practice, in the state-of-the-art we
use MSE (1) and Mean Absolute Error (MAE) (5) as
evaluation metrics
MAE = E
x
"
1
M
M
∑
m=1
||x
m
L+1:L+T
−
ˆ
x
m
L+1:L+T
||
1
#
. (5)
5.4 Hyperparameters
The number of qubits N = 10 and number of VQC
layers K = 3 in the hidden quantum layer. Adam op-
timizer (Kingma and Ba, 2017) is used to train the
model in order to minimize the MSE over the training
set. Batch size is 16 and initial learning rate is 0.0001.
For more hyperparameters refer to our code.
3
https://www.bgc-jena.mpg.de/wetter/
5.5 Results
Figure 2: MSE comparison with fixed prediction
length T = 96 and varying sequence length L ∈
{48, 96, 192, 336, 504, 720}.
Figure 3: MSE comparison with fixed prediction
length T = 720 and varying sequence length L ∈
{48, 96, 192, 336, 504, 720}.
For a fair comparison, we choose the fixed sequence
length L = 336 and 4 different prediction lengths
T ∈ {96, 192, 336, 720} as in (Zeng et al., 2023; Zhou
et al., 2022; Wu et al., 2021; Zhou et al., 2021). Table
1 provides comparison of MSE and MAE of QuLTSF
QuLTSF: Long-Term Time Series Forecasting with Quantum Machine Learning
827

with 6 baselines. The best and second best results are
highlighted in bold and underlined respectively. Our
proposed QuLTSF outperform all the baseline models
in all 4 cases.
To further validate QuLTSF against the base-
line linear models we conduct experiments for the
QuLTSF and classical models with varying sequence
lengths L ∈ {48, 96, 192, 336, 504, 720} and plot the
MSE results for a fixed smaller prediction length T =
96 in Fig. 2, and for a fixed larger prediction length
T = 720 in Fig. 3. In all cases QuLTSF outperforms
all the baseline linear models.
5.6 Discussion and Future Work
QuLTSF uses generic hardware-efficient ansatz. Sim-
ilar to classical machine learning in QML we need to
choose the ansatz, if possible, based on dataset and
domain expertise. Searching for optimal ansatz is a
research direction by itself (Du et al., 2022). Finding
better ans
¨
atze for QML based LTSF models for dif-
ferent datasets is an open problem. One potential way
is to use parameterized two qubit rotation gates (You
et al., 2021).
Other possible future direction is to use efficient
data preprocessing, for example reverse instance nor-
malization (Kim et al., 2021) to mitigate the distribu-
tion shift between training and testing data. This is
already being used in the state-of-the-art transformer
based LTSF models like PatchTST (Nie et al., 2023)
and MTST (Zhang et al., 2024). These models also
show better performance than the linear models in
(Zeng et al., 2023). Interestingly, our simple QuLTSF
model outperforms or comparable to these models in
limited settings. For instance, for the setting (L =
336, T = 720) MSE of PatchTST, MTST and QuLTSF
are 0.320, 0.319 and 0.315 respectively. For the set-
ting (L = 336, T = 336) MSE of PatchTST, MTST
and QuLTSF are 0.249, 0.246 and 0.248 respectively
(see Table 2 in (Zhang et al., 2024) for PatchTST and
MTST; and Table 1 for QuLTSF). QML based LTSF
models with efficient data preprocessing may lead to
improved results.
Implementation of QML based LTSF models on
the quantum hardware poses significant challenges.
As quantum systems grow, maintaining qubit state
coherence and minimizing noise become increasingly
difficult, leading to errors that degrade computational
accuracy and performance (Lidar and Brun, 2013).
Additionally, the barren plateau phenomenon, where
the gradient of the cost function vanishes as system
size or circuit depth increases, further complicates
optimization of VQC’s (McClean et al., 2018). Ad-
dressing these challenges requires innovations in er-
ror correction, problem-specific circuit designs, and
alternative optimization strategies, all of which are
critical for enabling scalable, noise-resilient, and ef-
fective QML based LTSF models.
6 CONCLUSIONS
We proposed QuLTSF, a simple hybrid QML model
for LTSF problems. QuLTSF combines the power
of VQC’s with classical linear neural networks to
form an efficient LTSF model. Although simple lin-
ear models outperform more complex transformer-
based LTSF approaches, incorporating a hidden quan-
tum layer yielded additional improvements. This is
demonstrated by extensive experiments on a widely
used weather dataset showing QuLTSF’s superiority
over the state-of-the-art classical linear models. This
opens up a new direction of applying hybrid QML
models for future LTSF research.
ACKNOWLEDGMENTS
This research is sponsored by the Singapore Quan-
tum Engineering Programme (QEP), project ID :
NRF2021-QEP2-02-P06.
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