Improving Quantum Classiﬁer Performance in NISQ Computers by V oting Strategy from Ensemble Learning

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Improving Quantum Classiﬁer Performance in

NISQ Computers by Voting Strategy from

Ensemble Learning

Ruiyang Qin1, Zhiding Liang2, Jinglei Cheng3, Peter Kogge4, and Yiyu Shi5

1, 2, 4, 5University of Notre Dame

3Purdue University

Abstract—Due to the immense potential of quantum computers

and the signiﬁcant computing overhead required in machine

learning applications, the variational quantum classiﬁer (VQC)

has received a lot of interest recently for image classiﬁcation.

The performance of VQC is jeopardized by the noise in Noisy

Intermediate-Scale Quantum (NISQ) computers, which is a

signiﬁcant hurdle. It is crucial to remember that large error

rates occur in quantum algorithms due to quantum decoherence

and imprecision of quantum gates. Previous studies have looked

towards using ensemble learning in conventional computing to

reduce quantum noise. We also point out that the simple average

aggregation in classical ensemble learning may not work well for

NISQ computers due to the unbalanced conﬁdence distribution in

VQC. Therefore, in this study, we suggest that ensemble quantum

classiﬁers be optimized with plurality voting. On the MNIST

dataset and IBM quantum computers, experiments are carried

out. The results show that the suggested method can outperform

state-of-the-art on two- and four-class classiﬁcations by up to

16.0% and 6.1% , respectively.

Index Terms—Quantum Machine Learning, Variational Quan-

tum Circuits, Quantum Ensemble Learning, Noise Mitigation

I. INTRODUCTION

In the past decade, machine learning models have achieved

consistent success in many practical applications such as

natural language processing [1]–[3], image classiﬁcation [4],

[5], and medical diagnosis [6]. However, as dataset size and

model complexity increase, computation power in classical

computing hardware has become the bottleneck. To alleviate

the problem, various efforts have been made to leverage the

power of quantum computers to speed up machine learning

tasks, which have opend a research area known as quantum

machine learning (QML). One of the most popular QML

models is Variational Quantum Classiﬁer (VQC) [7], which

deploys parameterized quantum circuits and trains them on

the target classiﬁcation tasks [8], [9].

Nevertheless, despite the great potentials of quantum com-

puting, current Noisy Intermediate-Scale Quantum (NISQ)

computers are extremely sensitive to their surrounding envi-

ronments and risk losing their quantum state due to quantum

decoherence, which contributes to high quantum noise. And

it is theoretically impossible to accurately predict or eliminate

quantum noise [10] in NISQ computers. As a result, numerous

studies have been done in the literature to reduce the noise

by constructing noise-resistant circuits [11]–[13] or control-

Fig. 1: Conceptual illustration of EQV. Different variational

quantum circuits and quantum computers are used to produce

the conﬁdence numbers for input data. A plurality voting

system is proposed to generate the ﬁnal predictions

ling lower-level quantum hardware [14]–[16]. Many of these

methods are task-speciﬁc, i.e., they are developed with the

implementation of certain quantum algorithms in mind.

More speciﬁcally, for VQC, current noise mitigation tech-

niques include mitigating noise-induced gradient vanishment

[17], adopting hybrid optimization [18], pre-processing data

[19], and developing kernel matrix [20]. Recently, QUILT [21]

was proposed for VQC tasks. It was inspired by the idea of

ensemble learning, where a mixture of various models make

a prediction collectively for greater accuracy. To generate

the ﬁnal prediction, it deploys several VQCs and averages

their output conﬁdence, which is deﬁned as the possibilities

arXiv:2210.01656v3 [quant-ph] 17 Dec 2022

that the prediction being correct. Although quantum noise is

unpredictable, as will be shown in Section II, we note that

the impact of wrong predictions is typically much more than

that of correct predictions. As a result, utilizing averaged

conﬁdence may result in incorrect prediction.

In this paper, we present EQV, an Ensemble Quantum

classiﬁers with plurality Voting, to address this issue. As

illustrated in Fig. 1, EQV deploys numerous VQCs onto

multiple quantum computers and integrates the outcoming

results through plurality voting to produce the ﬁnal prediction.

If the number of quantum computers producing accurate

predictions predominate, such an approach can successfully

eliminate inaccurate predictions, even if they have very low

conﬁdence numbers. Experimental results on real-world quan-

tum computers with MNIST dataset demonstrate that EQV can

increase the accuracy by 16.0% and 6.1% over the state-of-

the-art method for two-class and four-class classiﬁcation tasks

respectively.

The rest of the paper is structured as follows. Section II pro-

vides background information and introduces the motivation

for our work. The detailed framework of quantum ensemble

learning for VQC is articulated in Section III. Section IV

contains the experimental results as well as the concluding

remarks.

II. BACKGROUND AND MOTIVATION

A. Background

The quantum bit (qubit) is the fundamental building block

of quantum information. Similar to how a conventional bit

may store either a 0 or 1, a qubit can also be used to store

information. The states of a qubit can be represented by two

vectors: |0iand |1i. The linear combination of these two state

vectors is superposition, which can be represented as

|ψi=α|0i+β|1i,(1)

where αand βshould satisfy |α|2+|β|2= 1.

Besides superposition, qubits can also be entangled, which

is impossible for classical bits. Using two-qubit gates such as

CNOT gates to connect two qubits is a typical method for

entangling qubits.

A quantum circuit is a collection of various quantum gates

that can perform quantum operations efﬁciently. Parameterized

quantum circuit is a type of quantum circuit that can be

parameterized to enable trainability by changing the angles

of the rotation gates. We are able to obtain results that

are analogous to those obtained from training a classical

neural network if we design and train the variational quantum

classiﬁer. For instance, variational quantum classiﬁers (VQC)

are adequate for solving image classiﬁcation problems in an

effective manner [22], [23].

The leading quantum computers in the NISQ era contain

around one hundred qubits, but they are not advanced enough

to achieve fault-tolerance nor large enough to demonstrate

quantum supremacy on practical problems. The performance

of these computers is restricted by their high quantum noise.

Fig. 2: Probability density vs. Impact factor for two-class

classiﬁcation with two-qubit ansatz. It illustrates that wrong

prediction has higher impact than correct prediction, if a

simple averaging aggregation is adopted.

Therefore, powerful QML applications such as quantum neu-

ral network [8], [24], which is believed to better simulate

human neurons than classical neural networks, cannot be

implemented.

B. Motivation and Related Work

The idea of quantum ensemble learning was brought up

in 2017, where the ensemble corresponds to state preparation

routines, and the quantum classiﬁers can be evaluated in

parallel [25]. Schuld et al theoretically proves the feasibility of

quantum ensemble learning. In another work, Macaluso et al

[26] propose a quantum algorithm which takes the advantages

of quantum superposition, entanglement and interference to

build an ensemble classiﬁcation. Macaluso et al [26] achieves

quantum ensemble learning by using the bagging strategy

[27], and it mainly discusses how to apply classical ensemble

learning methods to quantum computing. Chen et al [28]

combines quantum ensemble learning with supervised learning

by recasting quantum ensemble classiﬁcation as a super-

vised quantum learning problem, and using a sampling-based

learning control to present quantum discrimination. There

are also researchers working on building an exponentially

larger ensemble size of classiﬁers, aiming to explore the

scalability problem [29]. In hierarchical quantum classiﬁers

[30], researchers use the combination of different quantum

binary classiﬁers to complete classiﬁcation tasks.

Most recently, QUILT [21] deploys ﬁve core classiﬁers and

Nbinary classiﬁers for N-class classiﬁcation tasks. In QUILT,

accuracy-based weights are used to aggregate outputs from

core classiﬁers. The binary classiﬁers in QUILT are One-Vs-

All classiﬁers, which are trained to discriminate one class from

other classes. In QUILT, binary classiﬁers are used to tell apart

two outputs with lowest conﬁdence numbers. For example, if

the lowest conﬁdence classes are one and f ive, the binary

classiﬁer will be adopted to make the ﬁnal decision.

We conduct a toy experiment and present its results in

Fig. 2 to demonstrate the distribution of “impact numbers”

for wrong and correct prediction. For an input image, two

numbers are generated by multiple models to determine the

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ImprovingQuantumClassierPerformanceinNISQComputersbyVotingStrategyfromEnsembleLearningRuiyangQin1,ZhidingLiang2,JingleiCheng3,PeterKogge4,andYiyuShi51,2,4,5UniversityofNotreDame3PurdueUniversityAbstractDuetotheimmensepotentialofquantumcomputersandthesignicantcomputingoverheadrequiredinmachinelear...

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Improving Quantum Classiﬁer Performance in NISQ Computers by V oting Strategy from Ensemble Learning

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