††institutetext:  bDepartment of Physics, University of California, Berkeley,
CA 94720, USA
bPhysics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA
cBerkeley Institute for Data Science, University of California, Berkeley, CA
94720, USA

# Quantum Anomaly Detection for Collider Physics

Sulaiman Alvi,a,b Christian Bauer,b and Benjamin Nachman,b,c
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum Machine Learning (QML) is an exciting tool that has received
significant recent attention due in part to advances in quantum computing
hardware. While there is currently no formal guarantee that QML is superior to
classical ML for relevant problems, there have been many claims of an
empirical advantage with high energy physics datasets. These studies typically
do not claim an exponential speedup in training, but instead usually focus on
an improved performance with limited training data. We explore an analysis
that is characterized by a low statistics dataset. In particular, we study an
anomaly detection task in the four-lepton final state at the Large Hadron
Collider that is limited by a small dataset. We explore the application of QML
in a semi-supervised mode to look for new physics without specifying a
particular signal model hypothesis. We find no evidence that QML provides any
advantage over classical ML. It could be that a case where QML is superior to
classical ML for collider physics will be established in the future, but for
now, classical ML is a powerful tool that will continue to expand the science
of the LHC and beyond.

##  1 Introduction

Quantum computing is a promising tool for a variety of problems because an
exponentially large Hilbert space can be described by polynomially many
qubits. In high energy physics, there is particular promise for simulations of
quantum field theories, where every spacetime point has quantum degrees of
freedom, but polynomial algorithms exist for state preparation and time
evolution Jordan:2012xnu ; Bauer:2022hpo . However, not all classically hard
algorithms are more efficient on a quantum computer.

One particular class of algorithms that has received significant attention in
high energy physics (HEP) is Quantum Machine Learning (QML). In this paper,
QML refers to machine learning tasks that are executed on quantum computing
hardware. While QML is not known to be more efficient than classical machine
learning (CML), there have been many empirical studies to explore the
potential of QML for HEP Mott:2017xdb ; Zlokapa:2019lvv ; Blance:2020nhl ;
Terashi:2020wfi ; Chen:2020zkj ; Wu:2020cye ; Chen:2021ouz ; Heredge:2021vww ;
Wu:2021xsj ; Belis:2021zqi ; Araz:2021ifk ; Bravo-Prieto:2021ehz ;
Blance:2021gcs ; Ngairangbam:2021yma ; Araz:2022haf ; Gianelle:2022unu ;
Humble:2022vtm  (see also Ref. Guan:2020bdl  for a recent review).

A common theme that has emerged from these studies is that QML seems to
outperform CML with small training datasets. While there is no rigorous
explanation for this observation, it could be that QML provides a superior
inductive bias and/or more expresivity with a smaller number of parameters. In
nearly all studies, CML outperforms QML when there are more than
$\mathcal{O}(100)$ examples. There are almost no problems in collider HEP that
have such small numbers of events for training. The goal of this paper is to
explore a realistic use case for near-term QML for collider physics. See also
Ref. Schuld:2022lss  for the broader context of QML versus CML.

Most analyses at the Large Hadron Collider (LHC) make use of simulation for
training classifiers. Since these simulations can be used to generate more
events independent of collider operations, they are not in a regime where QML
is expected to outperform CML with near-term quantum hardware. Therefore,
analyses that have the potential for QML to outperform CML should require data
for training. Since we do not know the origin of any particular data event,
such methods are called less than supervised Karagiorgi:2021ngt .

One particularly promising class of less-than-supervised methods is signal
model-independent anomaly detection. Such approaches are characterized by the
comparison of data in a particular region of phase space (signal region) with
a reference sample. There have been many proposals for doing this comparison
using machine learning Collins:2018epr ; Collins:2019jip ; DAgnolo:2018cun ;
DAgnolo:2019vbw ; Andreassen:2020nkr ; Nachman:2020lpy ; Hallin:2021wme ;
Farina:2018fyg ; Heimel:2018mkt ; Roy:2019jae ; Cerri:2018anq ; Blance:2019ibf
; Hajer:2018kqm ; DeSimone:2018efk ; Mullin:2019mmh ; 1809.02977 ;
Dillon:2019cqt ; Aguilar-Saavedra:2017rzt ; Romao:2019dvs ; Romao:2020ojy ;
knapp2020adversarially ; collaboration2020dijet ; 1797846 ; 1800445 ;
Amram:2020ykb ; Cheng:2020dal ; Khosa:2020qrz ; Thaprasop:2020mzp ;
Alexander:2020mbx ; aguilarsaavedra2020mass ; Aguilar-Saavedra:2021utu ;
1815227 ; pol2020anomaly ; Mikuni:2020qds ; vanBeekveld:2020txa ; Park:2020pak
; Faroughy:2020gas ; Stein:2020rou ; Kasieczka:2021xcg ; Batson:2021agz ;
Blance:2021gcs ; Bortolato:2021zic ; Collins:2021nxn ; Dillon:2021nxw ;
Finke:2021sdf ; Shih:2021kbt ; Atkinson:2021nlt ; Kahn:2021drv ;
Aarrestad:2021oeb ; Chakravarti:2021svb ; Dorigo:2021iyy ; Caron:2021wmq ;
Govorkova:2021hqu ; Kasieczka:2021tew ; dAgnolo:2021aun ; Volkovich:2021txe ;
Govorkova:2021utb ; Ostdiek:2021bem ; Fraser:2021lxm ; Raine:2022hht ;
Krzyzanska:2022mto ; Letizia:2022xbe ; Letizia2021 . A highly sensitive
approach is to train a classifier to distinguish data from a precise
prediction of the background (semi-supervised learning) Collins:2018epr ;
Collins:2019jip ; DAgnolo:2018cun ; DAgnolo:2019vbw ; Andreassen:2020nkr ;
dAgnolo:2021aun ; Chakravarti:2021svb ; Nachman:2020lpy ; Hallin:2021wme ;
Krzyzanska:2022mto ; Letizia:2022xbe ; Letizia2021 ; Raine:2022hht . If the
background is well-understood theoretically, then the reference sample could
be simulation DAgnolo:2018cun ; DAgnolo:2019vbw ; dAgnolo:2021aun ;
Chakravarti:2021svb ; Krzyzanska:2022mto ; Letizia:2022xbe ; Letizia2021 .
This has the advantage that the background prediction does not need to be
learned, but has the disadvantage of being strongly background-model
dependent.

There are few final states at the LHC for which the background is known
precisely enough to be used directly for background estimation. One exception
is the final state with four charged leptons. Both ATLAS ATLAS:2020tlo ;
ATLAS:2018coo ; ATLAS:2020wny  and CMS CMS:2016ilx ; CMS:2020bni ; CMS:2021nnc
directly use Monte Carlo (MC) simulations to estimate the background and ATLAS
even uses machine learning to isolate particular signals ATLAS:2020tlo . While
powerful, this approach is signal model-specific and does not readily extend
to models with multidimensional parameters. Reference Krzyzanska:2022mto
recently proposed to use machine learning as an alternative approach. It was
shown that training classifiers to distinguish data from background-only
simulation provides a complementary approach to direct searches and has broad
signal sensitivity. As the four lepton final state also has a small cross
section, this is a natural target for studying QML.

The prospect of QML for anomaly detection was first studied in Ref.
Ngairangbam:2021yma  in the context of autoencoders. These unsupervised tools
can be trained without any simulation, but are not as effective as semi-
supervised methods when there is a good background model and/or when the new
physics is not the lowest density events Collins:2021nxn ; Fraser:2021lxm .
For this reason, our focus is on semi-supervised learning. We also assume an
idealized situation where there are no systematic uncertainties. Nuisance
parameters will likely not change the qualitative conclusions of the QML
versus CML comparison and have been discussed in Ref. dAgnolo:2021aun  for
anomaly detection.

This paper is organized as follows. Section 2 briefly introduces various QML
approaches. The simulated samples to be used for the machine learning are
described in Sec. 3. Numerical results are presented in Sec. 5 and the paper
ends with conclusions and outlook in Sec. 6.

##  2 Quantum Machine Learning

There are a number of ways quantum computers can be used for machine learning.
One possibility is that quantum algorithms can reduce the computational
complexity of linear algebra operations core to a number of ML approaches (see
e.g. Refs. Rebentrost_2014 ; PhysRevLett.103.150502 ; Biamonte_2017 ). Another
possibility is to use quantum circuits as flexible function approximators
similar to neural networks and other classical ML techniques. Quantum-
classical variational methods used to optimize parameteric circuits are the
analog of tuning the weights and biases of a classical neural network. There
have been a number of claims in the literature that QML methods of this kind
outperform CML for limited training data. While there is no formal proof of
this claim, it could be justified intuitively as a result of the improved
inductive bias of quantum circuits. In other words, the class of functions
that QML represent with a limited set of parameters are more relevant /
tailored to HEP problems. We aim to explore this claim in the context of two
variational algorithms in the four lepton anomaly detection search111As a
cross-check, we also explored the Supersymmetry example of Ref.
Terashi:2020wfi . We were unable to reproduce the QML superior performance at
low sample size and the CML was also significantly improved by adding more
parameters..

The two QML methods we study are called Variational Quantum Circuits (VQC) and
Quantum Circuit Learning (QCL). These are both implementations of
parameterized quantum circuits where the various rotation angles are optimized
via classical methods. Each algorithm is composed of multiple components:
state preparation, which encode classical data into quantum states, the model
circuit, which contains the parameters that are optimized during the training
process, and the measurement and output, which are used to evaluate
performance of the circuit. VQC and QCL differ only in the structure of the
parameterized circuit, as detailed in Sec. 4. As the examples we study in this
paper are relatively small in terms of quantum resources, all circuits are
simulated on classical computers.

Note that we also studied Quantum Support Vector Machines (QSVMs)
PhysRevLett.103.150502 , but initial tests suggested that they are strictly
less effective than other methods so they were not included in the final
tests. See also Ref. Paul-Aymeric  for a broader perspective on QVSMs. We have
also explored the quantum gradient descent studied in Ref. Blance:2020nhl . We
tested the setup using Pennylane Pennylane  and found that the learning rates
were unreliable for Gaussian classification as well as our HEP application, so
this was not pursued further.

##  3 Simulation

The simulated datasets are the same as in Ref. Krzyzanska:2022mto  and are
briefly summarized in the following. All events are generated with
MadGraph5_aMC@NLO 2.8.0 Alwall:2014hca . Both signal and background events are
generated using the Higgs Boson Effective Field Theory (heft) in which the
heavy top quark limit is used for the gluon-gluon-Higgs vertex. We focus on
the $e^{+}e^{-}\mu^{+}\mu^{-}$ final state to avoid combinatoric ambiguity.
After the matrix element calculations, the outgoing particles are processed
with Pythia 8.244 Sjostrand:2006za ; Sjostrand:2007gs ; Sjostrand:2014zea
with its default settings for parton showering and hadronization. Pythia also
handles the decay of the anomalies, which are Higgs-like scalar particles with
a mass of 125 GeV decaying asymmetrically into two lighter-mass bosons, one of
which decays to electrons and the other which decays into muons. This is
accomplished technically by generating Higgs bosons and then replacing the
PDGID Zyla:2020zbs  of 25 (SM Higgs) with 35 (2HDM heavier Higgs) in the Les
Houches Event (LHE) files Alwall:2006yp  and then setting the decay of this
particle into particles with PDGIDs 23 (Z boson) and 36 (2HDM pseudoscalar).
Subsequently, the $Z$ boson (pseudoscalar) is forced to decay into electrons
(muons). All three BSM particles are set to a narrow width. The detector
response is emulated with Delphes 3.4.2 deFavereau:2013fsa ; Mertens:2015kba ;
Selvaggi:2014mya  using the default CMS card. We will make the simplifying
assumption that there is no system uncertainty in the background estimation
and so the ‘data’ and ‘simulation’ are statistically identical when no signal
events are injected. Recent studies exploring the integration of systematic
uncertainties in this setup can be found in Ref. dAgnolo:2021aun . The number
of events in background corresponds to the LHC Run 2 dataset (about 150 fb-1).

In this work we focus only on the leptoninc final states. Other event
properties could also be useful, however, information about the hadronic final
state is known with less precision and thus may introduce the need to go
beyond a pure simulation-based background estimation. Each event is
characterized four three-momenta (12 numbers in total). For our target models,
$pp\rightarrow A\rightarrow B(\rightarrow
e^{+}e^{-})C(\rightarrow\mu^{+}\mu^{-})$, the three masses
$m_{e^{+}e^{-}\mu^{+}\mu^{-}},m_{e^{+}e^{-}},m_{\mu^{+}\mu^{-}}$ are nearly
sufficient statistics for characterizing the new physics. In this paper, we
consider signals of this form, where $A$ is a non-SM Higgs boson that decays
to two different mass bosons. The model is specified by three parameters:
$m_{A},m_{B}$, and $m_{C}$. There is also an overall cross section set by the
coupling of the $A$ particle to the rest of the Standard Model. This cross
section will be varied in the subsequent analysis by considering different
numbers of signal events. Given the three-dimensional parameter space, we
focus on the three-dimensional problem in this paper. Non-resonant signals and
signals with non-trivial spin structures could benefit from using more of the
phase space in the future. While there are currently LHC searches for the case
that $B=C$ (or $B$ is a BSM particle and $C$ is a $Z$ boson), there is
currently on search where all three masses could be different. This is not
because physics motivation is lacking Robens:2019kga , but instead that even a
three-dimensional traditional search strategy is too complicated.

The spectra of the three invariant masses
($m_{e^{+}e^{-}\mu^{+}\mu^{-}},m_{e^{+}e^{-}},m_{\mu^{+}\mu^{-}}$) for the
background and our representative signal are presented in Fig. 1. As expected,
the di-electron and di-muon invariant masses peak near the $Z$ boson mass of
90 GeV Zyla:2020zbs  and there are peaks in the four-lepton invariant mass at
the $Z$ peak and the Higgs boson mass of about 125 GeV Zyla:2020zbs . The
signal is resonant in all three observables with peaks at the masses of the
particles. The parent particle (125 GeV mass) decays to two children, with
masses of 25 and 15 GeV for electrons and muons, respectively. Note that these
parameters are not known to the neural network.

Figure 1: The three dimensions used for machine learning: $m_{e^{+}e^{-}}$
(left), $m_{\mu^{+}\mu^{-}}$ (right) and $m_{4\ell}$ (bottom).

##  4 (Quantum) Machine Learning Setup

As stated earlier, our QML approaches are both variational circuits that are
analogous to classical neural networks with a large number of tunable
parameters. We consider two flavors: VQC and QCL. These two approaches are
described in more detail below and only differ in how the classical data are
encoded and what parameterized circuits are used in the learning. VQC uses a
simpler encoding with a multi-qubit rotation followed by a series of CNOT
gates for the variational part. Instead, QCL uses a more complex encoding and
then time evolution of a certain Hamiltonian for the variational component.

###  4.1 VQC

Figure 2 shows the VQC circuits used in this study. The input features
($x_{i}$) are min-max scaled so that the argument of the initial $R_{y}$ gates
are valid angles. The rotational gate $R(\theta)$ is given by
$R_{Z}(\alpha)R_{Y}(\omega)R_{Z}(\phi)$, where $\alpha$, $\omega$, and $\phi$
are the trainable weights of the circuit. These angles are unique for each
qubit, leading to a total of 18 parameters for the one-dimensional setup and
27 for the three-dimensional setup. The expectation value $z_{i}$ of the Pauli
Z operator for the first two qubits is measured and the loss function is
cross-entropy:

$\displaystyle-\sum\left[\log\left(\frac{e^{z_{0}}}{e^{z_{0}}+e^{z_{1}}}\right)+\log\left(\frac{e^{z_{1}}}{e^{z_{0}}+e^{z_{1}}}\right)\right],$
(1)

where the exponential terms represent the soft-max activation function.

Figure 2: VQC circuit for the one-dimensional (left) and three-dimensional
(right) datasets.

There are a number of hyperparameters that must be selected and the ones
described above were chosen after a detailed study. By varying the number of
qubits, we observed that two qubits resulted in an optimal accuracy for the
one-dimensional classification task. Using more than six qubits caused issues
related to overfitting of the data while also increasing runtime. As such we
chose three qubits for the three-dimensional case. We observed that we needed
a minimum of three rotation gates in order to achieve an effective level of
expressivity of the circuit. Between three and five rotation gates, we did not
see a significant increase in accuracy, and beyond five rotation gates, the
time to run the simulation became prohibitive while also not resulting in an
increase in performance. The default encoding gate included a $Y$ and $Z$
rotation but testing different combinations of input encoding gates showed no
performance difference so the final circuit just included a $Y$ rotation gate.
Batch learning was also implemented for the three-dimensional dataset tests,
which greatly improved performance. The circuit was optimized using vanilla
gradient descent, with the quantum gradient calculated using Pennylane’s
Pennylane  implementation of the parameter shift rules for quantum circuits.

###  4.2 QCL

The circuit’s parameterized gates follow Terashi:2020wfi  in a way that allows
for access of the entire Bloch sphere given an arbitrary input state. The gate
that introduces entanglement involves a time evolution operation following the
Ising model Hamiltonian $H$ as in Figure 3. We did not experiment with
different Hamiltonians as their work remarks that changing this did not
achieve a different result. As with VQC, the output is the expectation value
of $Z$. The number of qubits to encode parameters in the 1D case was chosen
after a brief exploration to be two. Each input value was duplicated and
stored in two separate qubits which were transformed by the same circuit. For
3D testing we used six qubits. For all tests the circuits had three layers,
which are outlined in Figure 3. The optimization was completed using the
COBYLA method Powell1994  with the parameter shift rule as defined in
Terashi:2020wfi  and binary cross-entropy loss.

The $\theta_{i}$ are the trainable weights of the circuit. The three-
dimensional case follows the same structure as Figure 2 and the alternative
$U_{\textbf{in}}$ in Figure 10 from Terashi:2020wfi , except the three inputs
are duplicated so that six qubits can be used. There were 18 trainable
parameters for the one-dimensional setup and 54 trainable parameters for the
three-dimensional setup.

Figure 3: QCL circuit for 1 dimensional datasets.

###  4.3 CML

The performance of the quantum machine learning setups are compared against
two neural networks implemented in TensorFlow tensorflow2015-whitepaper . The
first network (‘NN Low’) has one layer with 32 nodes and the second network
(‘NN High’) had two 32 node layers, which corresponds to 97 and 1,217
trainable parameters, respectively. The output was passed through a sigmoid
activation and the NN High model used rectified linear activation (ReLU)
functions for the intermediate layer.

The networks were optimized using the binary cross entropy with Adam adam .
Each network was run for a fixed number of epochs, determined by the
convergence of the loss curves: 50 epochs for the one-dimensional example and
100 for three-dimensional example.

##  5 Results

We perform a weakly/semi-supervised search where a classifier is trained to
distinguish a background sample from another, statistically independent and
identical sample, that has some number of signal events added to it. Due to
the small number of injected signal events, it is important to study the
sensitivity to different random sets of signal events. The results presented
below are averaged over 80 different random selections of signal and
background (with error bars representing the standard deviation). The
performance is quantified in terms of the number of standard deviations222This
is approximated by the number of signal events divided by the square root of
the number of background events. We do not expect that more precise
calculations/approximations will qualitatively change the results. achieved
after applying the optimal (maximum significance improvement) threshold on the
network333Other papers have used the Area Under the Receiver Operator
Characteristic Curve (ROC AUC). While the AUC is a standard metric in machine
learning, it is not representative for HEP applications where typically one
working point (threshold) on the classifier is used instead of integrating
across the entire ROC curve.. The maximum significance improvement is model
dependent, but it is chosen here to bound the achievable performance. In
practice, the challenge of selecting the threshold is the same for both QML
and CML methods. Two relevant benchmark significance are $2\sigma$ and
$5\sigma$ which approximately correspond to the community standards for
excluding and discovering a model, respectively.

Numerical results are presented in Fig. 4. As a first test, we fix the
background at 1000 events and scan the number of signal events. For reference,
the naive significance with 30 signal events is about unity. All of the
methods are better than doing nothing and surpass $2\sigma$ by 40 events. The
QML models do not outperform the classical approaches and in fact appear to be
systematically worse except perhaps for the lowest number of signal events
where the significances themselves are below unity (and thus irrelevant).
Similar trends hold for the three-dimensional data, except that the QCL
performs relatively worse and the NN High performs relatively better. We note
that the error bars on the plots are not small, which illustrates the
importance of ensembling over the random parts of the training as well as over
the random injetion of signal (and background) events.

As a second test, we fix the signal fraction and vary the number of signal
events (which also changes the number of background events). At 1% fraction,
for reference, 10 signal events would then correspond to the 1000 background
events in the lefthand plots of Fig. 4. The trends for the fixed signal
fraction are similar to the fixed background fraction, with slightly worse
performance due to the larger number of background events beyond 10 signal
events. Additional performance metrics including the relative significance
improvement and the Area Under the ROC curve can be found in Appendix 7. These
additional statistics corroborate the story presented in Fig. 4. The AUC is
difficult to interpret as it integrates across the entire ROC curve when in
practice, we typically operate at a fixed working point. Small changes in the
AUC could correspond to regions of the ROC curve that are physically useful,
so the maximum significance improvement is instead chosen as the default
statistic.

Figure 4: Results plotted against increasing signal events. The horizontal
grey lines indicate thresholds for exclusion (2$\sigma$) and discovery
(5$\sigma$).

##  6 Conclusions

Motivated by promising numerical studies from the HEP literature on quantum
machine learning for low-event count applications, we have explored where
these techniques could be practically useful for collider physics. Given that
most machine learning methods are trained using simulation, we concluded that
a task relying directly on data should be the target application. An important
topic in this area is anomaly detection, where machine learning methods are
used to reduce model dependence. One strategy that reduces signal model
dependence is to train a classifier to distinguish data from background-only
simulation. This will be fundamentally limited by the number of events in data
and thus may be a good target for exploring an advantage of quantum machine
learning over its classical counterpart.

There are not many final states that are modeled precisely enough for the data
versus simulation strategy to be effective at finding small signals. Following
Ref. Krzyzanska:2022mto , we consider anomaly detection in the four-lepton
final state, where the Standard Model is well-known, yet there is plenty of
phase space for new physics. We consider a low-dimensional version of the
problem within a model framework that has three free parameters (masses),
which already extends beyond the current searches at the LHC that focus on one
and sometimes two-dimensional versions of the problem. We do not find any
advantage of quantum machine learning over classical machine learning.

It could be that this particular problem is not well-suited for quantum
machine learning or that we have not picked exactly the right quantum machine
learning architecture or training process. We do not claim that our results
are general for all of QML and all of HEP, but we hope that our process and
numerical results will be useful to put existing and future studies of QML for
HEP in context.

## Acknowledgments

This work was supported by the Department of Energy, Office of Science under
contract number DE-AC02-05CH11231. S.A. was also supported by a University of
California Summer Undergraduate Research Fellowship. We thank Vinicius Mikuni
for computing help and Andrew Blance, Koji Terashi, and Michael Spannowsky for
useful discussions about QML.

##  7 Appendix

Figures 5 and 6 quantify the performance of the same studies as in Sec. 4, but
using different metrics (Area Under the ROC curve and maximum significance
improvement, respectively).

Figure 5: AUC plotted against increasing signal events

Figure 6: The maximum significance improvement as a function of the number of
signal events.

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