The Standard Model of Particle Physics (SM) provides a remarkable description of the fundamental constituents of matter and their interactions, being in excellent agreement with the collider data accumulated so far. Nonetheless, there are still important open questions, unaddressed by the SM, such as gravity, dark matter, dark energy, or the matter-antimatter asymmetry in the universe (Ellis, 2012), motivating a comprehensive search program for new physics phenomena beyond the SM (BSM) at the Large Hadron Collider (LHC) at CERN.

The search for BSM phenomena at colliders poses specific challenges in data processing and analysis, given the extremely large datasets involved and the low signal to background ratios expected. In this context, the analysis of the collision data obtained by the LHC experiments often relies on machine learning (ML), a field in computer science that can harness large amounts of data to train generalizable algorithms for a variety of applications (Guest et al., 2018; Feickert and Nachman, 2021), such as classification tasks. These techniques have shown an outstanding ability to find correlations in high-dimensional parameter spaces to discriminate between potential signal and background processes. They are known to scale with data, and usually rely on a large number of learnable parameters to achieve their remarkable performance.

In order to train these large models, classical¹ machine learning (CML) takes advantage of hardware accelerators, such as graphics processing units (GPUs), for efficient, parallel, and fast matrix multiplications. On the other hand, a new class of hardware is becoming available, with the advent of noisy intermediate-scale quantum (NISQ) computing devices. This accelerated the development of new quantum algorithms targeted at exploiting the capacity and feasibility of this new technology for ML applications.

Quantum machine learning (QML) is an emerging research field aiming to use quantum circuits to tackle ML tasks. One of the motivations for using this new technology in high energy physics (HEP) relates to the intrinsic properties of quantum computations, namely representing the data in a Hilbert space where the data can be in a superposition of states or in entangled states, which can allow to explore additional information in data analysis and, eventually, contribute to better classification of HEP events, namely in the context of the search for BSM phenomena. Recently, this new technology has been applied to various HEP problems (Guan et al., 2021). Namely, in event reconstruction (Das et al., 2019; Shapoval and Calafiura, 2019; Bapst et al., 2020; Tüysüz et al., 2020; Wei et al., 2020; Zlokapa et al., 2021a; Funcke et al., 2022), classification tasks (Mott et al., 2017; Belis et al., 2021; Blance and Spannowsky, 2021; Terashi et al., 2021; Wu et al., 2021; Zlokapa et al., 2021b; Araz and Spannowsky, 2022; Chen et al., 2022; Gianelle et al., 2022), data generation (Chang et al., 2021a,b; Delgado and Hamilton, 2022; Borras et al., 2023; Rehm et al., 2023), and anomaly detection problems (Ngairangbam et al., 2022; Alvi et al., 2023; Schuhmacher et al., 2023; Woźniak et al., 2023).

Despite the promising potential of quantum computation, NISQ processors have important limitations, such as the qubit quality (i.e., the accuracy with which it is possible to execute quantum gates), the qubit lifetime and the limited depth of quantum circuits, since for large circuits the noise overwhelms the signal (Li et al., 2018; Preskill, 2018). This necessarily limits the complexity of the circuits and the size of the datasets used to train them.

In this paper, we perform a systematic comparison of the performance of QML and shallow CML algorithms in HEP. The choice to focus on shallow methods rather than state-of-the-art architectures based on deep neural networks is to provide a fair comparison between methodologies, since neural networks are known to require large datasets (both in terms of sample size and dimension) to achieve good performance, something that is not feasible with current quantum computers. By choosing CML algorithms suited for smaller datasets, we will add to the on-going discussion regarding potential advantages of quantum computing by comparing QML and CML in the same footing.

The use of QML algorithms in this context is studied by targeting a common binary classification task in HEP: classifying a BSM signal against SM background. A benchmark BSM signal leading to the Zt final state is considered, in events with multiple leptons and b-tagged jets, which can be used to achieve a reasonable signal to background ratio. Variational quantum classifiers (VQC) are trained and optimized via a grid search. The use of reduced data is explored, considering both the number of features and the number of events, via different strategies: ranking of features, data transformations aiming for a richer reduced set of features, use of random samples, and choice of representative data samples.

The QML algorithms are implemented using a quantum circuit, i.e., a collection of quantum gates applied to an n-qubit quantum state, followed by a measurement (or multiple measurements) that represent the output of the circuit. In order to implement a learning algorithm, the quantum circuit can be parameterized with parameters that can be learned by confronting the measurement to a loss function.

QML is effectively an extension of CML techniques to the Hilbert space, where instead of representing data as vectors in a high-dimensional real space, we encode it in state vectors of a Hilbert space. A QML algorithm, such as a quantum neural network, can be implemented using the quantum equivalent of a perceptron, one of the building blocks of CML. A problem arises from the realization that the activation functions used in CML can not be expressed using a linear operation, which is inherently required from the quantum evolution of a state. Ideas have been proposed to imitate an activation function in the quantum space (Gupta and Zia, 2001; Schuld et al., 2015), but, in the current paper, only variational quantum classifiers (Farhi and Neven, 2018; Schuld et al., 2020) are used for binary classification.

A VQC is a parameterized quantum circuit, a circuit type containing adjustable gates with tunable parameters. These gates are a universal set of quantum gates and, in the current study, rotation [R_X(w), R_Y(w), R_Z(w)] and CNOT gates are used.²

The considered VQC pipeline used has the following components:

Throughout this work, the PennyLane package (Bergholm et al., 2018) was used as a basis for the hybrid quantum-classical machine learning applications. Leveraging PennyLane's default.qubit quantum simulator, a straightforward tool for quantum circuit simulations, we trained and assessed the performance of various QML algorithms. Subsequently, the performance of the algorithms trained on IBM's quantum computers was gauged by integrating PennyLane with IBM's quantum computing framework, Qiskit (Anis et al., 2023).

Shallow CML methods are used to provide a baseline comparison to the QML models. The specific methods chosen for the comparison are Logistic Regression (LR) and Support Vector Machines (SVM), with these algorithms being trained with the same data as the QML algorithms.

All the classical methods were implemented using scikit-learn (Pedregosa et al., 2011) library and, if not specified otherwise, the default parameters were used.

The dataset used in this work (Crispim Romão et al., 2021) is comprised of simulated events of pp collisions at 13 TeV, in final states with two leptons, at least 1 b-jet, at least 1 large-R jet, and large scalar sum of transverse³ momentum (p_T) of all reconstructed particles in the event (H_T > 500 GeV). Such basic selection corresponds to a topology commonly used in different searches for BSM events at the LHC (Crispim Romão et al., 2021). The dominant SM background for this topology, Zbb̄, and the BSM signal corresponding to tt̄ production with one of the top-quarks decaying via a flavor changing neutral current decay t→qZ (q = c, u) (Durieux et al., 2015), were considered. Such signal was chosen given the kinematic similitude to the background, thus providing a good benchmark for the present study.

Both samples were generated with MadGraph5 2.6.5 (Alwall et al., 2014) and Pythia 8.2 (Sjöstrand et al., 2015), and the detector was simulated using Delphes 3 (Selvaggi, 2014) with the default CMS card. Jets were clustered using the anti-k_t algorithm (Cacciari et al., 2008), implemented via FastJet (Cacciari et al., 2012), with R-parameters of 0.5 and 0.8 (the latter for the large-R jets).

The following features were used for training of both the classical and quantum machine learning algorithms:

The proportion of signal and background events was kept the same as the original simulated data during training, being 13 and 87%, respectively. Additionally, the Monte-Carlo weights, corresponding to the theoretical prediction for each process at target luminosity of 150 fb⁻¹, were taken into account in the evaluation of all the considered metrics and loss functions.

As described in the previous section, a total of 47 features are available for training. Considering the type of data embedding chosen, 47 qubits would be needed to train a VQC using all the dataset features. Such number of qubits is impractical given the currently available quantum computers and thus it is not feasible to train a VQC using all the features in our dataset. For the purposes of the current study, quantum computers with only five qubits were considered and two methods for feature selection were implemented: principal component analysis (PCA) and sequential feature selection (SFS).

A relative comparison of the best five features⁴ is shown in Table 1 while the best performance obtained with state-of-the-art CML methods without any features or data points restrictions can be seen in Figure 2.

The present paper addresses the use of reduced datasets to overcome the limitation of NISQ processors while minimizing the loss of information and thus avoiding a performance loss of the QML algorithms in the HEP context. The primary method used for this purpose in the current study is KMeans, where the kth most representative points, i.e., a set of centroids, is obtained from the original dataset. Although these centroids are the most representative data points, they are not necessarily contained in the original dataset and, consequently, a resampling process, allowing to choose points of the original dataset (centrus), is required.

A study of the performance of the proposed dataset reduction method will be done by training a logistic regression model with the original dataset and comparing the results with those obtained when Kmeans and randomly undersampled datasets are used.

The training of the QML algorithms used in the current paper requires the use of optimizers. Two different ones were considered: Adam (Kingma and Ba, 2014) and tree-structured Parzen estimator sampler (TPE; Bergstra et al., 2011, 2013).

The Adam optimizer uses an extension of stochastic gradient descent, leveraging techniques such as adaptive moment estimation, being extensively used in optimization problems, namely in the context of machine learning. Nonetheless, since there is no reason to expect, a priori, that it will work equally well in the context of QML, where specific challenges are expected, the TPE optimizer was also tested.

The TPE is a Bayesian optimization algorithm first developed for HP tuning in the context of machine learning. In the current study, it will be used to optimize VQC weights in a way very similar to what is typically done for HP tuning. TPE is implemented using Optuna (Akiba et al., 2019), a library focused on HP optimization for machine learning models. TPE works by choosing a parameter candidate that maximizes the likelihood ratio between a Gaussian Mixture Model (GMM) fitted to the set of parameters associated with the best objective values, with another GMM being fitted to the remaining parameter values. In the context of HEP, TPE has also been used to explore parameter spaces of BSM models (de Souza et al., 2022).

Different machine learning methods were optimized, namely a LR, a SVM, and a VQC. The corresponding training was done for the set of HP summarized in Table 5, where the scanned values are also listed. For each set of HP, 5 models were trained on 5 different subsets of the initial dataset (random sampling).

For both optimizers, the considered cost function used is the squared error, with the individual Monte Carlo samples being properly weighted. During the training of VQCs, the inference was done on the validation dataset at five epoch intervals, the AUC computed and only the best-performing model, according to the previously mentioned metric, was considered.

Until this point, only simulated quantum environments were used. In order to test the performance in real quantum computers, and thus validate the simulation results, the Pennylane framework was used as the integration layer with Qiskit, which works effectively as a direct API to the quantum computers provided by IBM.

In this study, only the best performant model HP-set was used, i.e., the TPE-trained VQC. This VQC was implemented and its test set was inferred on six different quantum systems with identical architectures, all freely available. Evaluating our model in multiple identical quantum systems allows us to get an idea of the scale of the associated systematic uncertainty via the variability of the observed results. Since the implemented circuits are small, no error mitigation techniques were implemented. IBM's transpiler optimization level was set to 3⁷ (Anis et al., 2023) and, for each event, the final expectation value was computed by averaging 20k shots on the quantum computer. The obtained results, shown in Figure 13, are compatible with the simulated ones (Figure 5).

In this paper, we assessed the feasibility of using large datasets in QML applications by exploring data reduction techniques. To allow for a fair comparison between CML and QML models, we opted to use shallow classical methods as opposed to deep methods, which require large datasets that are not viable given the limitations of the current quantum computers. Our results indicate that there is comparable performance between CML and QML when tested on the same small dataset regime.

To achieve this, our study first compared feature selection techniques, showing that while SBS can produce the best performant QML model, it generally yielded worse and more unstable results than PCA. Additionally, we found this was produced by using discrete variables in VQCs, highlighting the suitability of PCA-transformed data for QML applications in the HEP context, where discrete variables are commonly used to describe collider events.

Our grid search over different HP combinations of VQC ran in simulation provided no evidence of quantum advantage in our study. We confirmed the results by running the best performing configuration on real-world quantum systems, obtaining compatible performances and therefore validating our conclusions. We compared the performance of TPE and Adam optimizers in QML and found that TPE achieves competitive results. Being a gradient-free optimizer, TPE offers the advantage that it can lead to faster training with a smaller memory usage when compared to Adam, which in principle can further facilitate the application of QML in current quantum computers.

We then explored data reduction techniques, finding that reducing the dataset size with the KMeans algorithm produces results that are similar to those obtained from random undersampling. This finding is significant in that it means that the model can achieve similar performance with fewer accesses to a quantum computer during training, which is a considerable bottleneck in current QML.

In conclusion, while our study found no evidence of quantum advantage in the current state of QML within the context of large HEP datasets, the performance of QML models was comparable to that of classical machine learning models when restricted to small dataset regimes. Our findings suggest that using dataset reduction techniques enables us to use large datasets more efficiently to train VQCs, facilitating the usage of current quantum computers in large datasets often found in HEP.

The dataset used in the studies reported in the current paper can be found in https://zenodo.org/doi/10.5281/zenodo.5126746, while the computational code used to obtain the present results is publicly available via https://github.com/mcpeixoto/QML-HEP.

MP: Conceptualization, Data curation, Formal analysis, Investigation, Project administration, Software, Validation, Visualization, Writing—original draft, Writing—review & editing. NC: Conceptualization, Data curation, Funding acquisition, Investigation, Methodology, Resources, Supervision, Writing—original draft, Writing—review & editing. MR: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Supervision, Validation, Writing—original draft, Funding acquisition, Writing—review & editing. MO: Conceptualization, Formal analysis, Investigation, Project administration, Software, Validation, Visualization, Writing—original draft, Writing—review & editing. IO: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Resources, Supervision, Writing—review & editing.