An ecient combination strategy for hybird quantum ensemble classier Xiao-Ying Zhang1and Ming-Ming Wang1 1Shaanxi Key Laboratory of Clothing Intelligence School of Computer Science

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An eﬃcient combination strategy for hybird quantum ensemble classiﬁer

Xiao-Ying Zhang1and Ming-Ming Wang1, ∗

1Shaanxi Key Laboratory of Clothing Intelligence, School of Computer Science,

Xi’an Polytechnic University, Xi’an 710048, China

(Dated: today)

Quantum machine learning has shown advantages in many ways compared to classical machine

learning. In machine learning, a diﬃcult problem is how to learn a model with high robustness

and strong generalization ability from a limited feature space. Combining multiple models as base

learners, ensemble learning (EL) can eﬀectively improve the accuracy, generalization ability, and

robustness of the ﬁnal model. The key to EL lies in two aspects, the performance of base learners and

the choice of the combination strategy. Recently, quantum EL (QEL) has been studied. However,

existing combination strategies in QEL are inadequate in considering the accuracy and variance

among base learners. This paper presents a hybrid EL framework that combines quantum and

classical advantages. More importantly, we propose an eﬃcient combination strategy for improving

the accuracy of classiﬁcation in the framework. We verify the feasibility and eﬃciency of our

framework and strategy by using the MNIST dataset. Simulation results show that the hybrid EL

framework with our combination strategy not only has a higher accuracy and lower variance than

the single model without the ensemble, but also has a better accuracy than the majority voting and

the weighted voting strategies in most cases.

I. INTRODUCTION

Based on the basic principles of quantum mechanics, quantum computing provides new models for accelerating

solutions of some classical problems [1–3]. With the great success of machine learning [4, 5], quantum machine learning

[6, 7] have been developed to show characteristics of quantum acceleration. Typical examples include quantum neural

networks (QNNs) [8, 9], quantum principal component analysis [10], quantum support vector machine [11], quantum

unsupervised learning [12], quantum linear system algorithm for dense matrices [13], etc.

Neural networks are at the center of machine learning. As its counterpart, a variety types of QNN models have

been proposed since its ﬁrst appearance [8, 9], which include models based on quantum dot [14], on superposition [15],

on quantum gate circuits [16], on quantum-walk [17], and quantum analogue of classical neurons [18], etc. In the last

few years, quantum deep learning [19], quantum convolutional neural networks (QCNNs) [20–22], quantum generative

adversarial network [23], quantum autoencoders [24], have also been developed. Most of QNNs are constructed by

parameterized quantum circuit (PQC) [25, 26]. In a PQC model, the number of parameters, the types of quantum

gates, and the width and depth of the circuit have a great impact on the required resources, the diﬃculty of solving

gradients, and whether the optimal model can be obtained [27, 28].

Ensemble learning (EL), also known as the multi-classiﬁer system, is an important algorithm that combines multiple

learning models to achieve better performance. In 1990, Schapire pointed out that it is possible to surpass one strong

learner by combining several weak learners (base learners) in a roughly correct sense [29]. It lays the foundation for

the Adaboost algorithm [30]. EL has shown advantages in avoiding over-ﬁtting, reducing the risk of decision error,

and reducing the acquisition of local minimum [30]. It has been widely used in object detection [31], education [32],

malware detection [33], etc. The performance of a EL mainly depends on the diversity and the prediction performance

of base learners. The diversity of the EL can be realized by using diﬀerent model structures, training sets, method of

subsetting, and others [30]. While the prediction performance of the EL is correlated with the uncorrelated degree of

error among base learners [34]. An important part of a EL is the combination strategy for combinating the predictions

of base learners. Currently, combination strategies can be roughly divided into weight combination and meta-learning

methods. In terms of a EL in binary classiﬁcation tasks, the classiﬁcation accuracy of each base learner must be

better than random guessing, such that the EL can be eﬀective. In addition, diversity among models should still be

maintained. There are many EL methods related to classical learning, such as AdaBoost, bagging, stacking, random

forest, and so on [35, 36].

For quantum computing, some studies have been performed on quantum ensemble learning (QEL) [37–40]. In Ref.

[37], Schuld et al. proposed a framework to construct ensembles of quantum classiﬁers, which evaluate the predictions

of exponentially large ensembles with parallelism. In Refs. [38, 39], a QEL scheme using the bagging was proposed

∗bluess1982@126.com

arXiv:2210.06785v1 [quant-ph] 13 Oct 2022

by using quantum cosine classiﬁer as base learners, and numerical simulation experiments were carried out. However,

their quantum base learner is untrained. The implementation of the classiﬁer is just for one test at one time, which

is not suitable for practical applications. Based on Ref. [37], Araujo et al. [40] further proposed the QEL of trained

classiﬁers, which shown the advantage of QEL for quantum classiﬁers. But the increasing of data size optimization

steps has a great impact on the model since it is diﬃcult to execute on a quantum computer. Their combination

process can not eﬀectively diﬀerentiate the diﬀerence among base learners, which will impair the accuracy of the

result.

In this paper, a quantum-classical hybrid EL model is proposed. That is, quantum classiﬁers are used as base

learners, and the bagging EL method [41] is used to assemble quantum classiﬁers in a classical computer. Since the

previous QEL framework [40] can not clearly distinguish the diﬀerences among base learners, while classical combi-

nation strategies can not distinguish the similarity among base learners, we propose a new combination strategy that

considers the diﬀerence among base learners and the performance of each learner in our framework. The MindQuan-

tum platform [42] is used as the training platform for a single quantum base learner. The MNIST handwritten digits

[43] are used as the dataset. The feasibility of our strategy is veriﬁed by using the homogenous ensemble method that

uses the same structural base learner as ensemble members.

II. ENSEMBLE LEARNING

EL mainly divided into the “homogeneous” ensemble and the “heterogeneous” ensemble, where their main diﬀerence

lies in whether the models of classiﬁers adopt the same model structure. A single learner in EL is called a base learner.

The key to the ﬁnal eﬀectiveness of the ensemble lies in whether the characteristic with “good but diﬀerent” can be

maintained among the base learners. That is, each base learner should have a certain accuracy that is better than

random guess and there are certain diﬀerences between them [30, 44, 45]. For a binary classiﬁcation problem, assuming

that the accuracy rate of each base classiﬁer is pand the base classiﬁers are independent of each other, a simple voting

method is adopted to combine Nbase classiﬁers, then the error rate of the ensemble result is [45]

N/2

k=0 N

kpk(1 −p)N−k.(1)

Under ideal conditions, the error rate will gradually decrease and eventually approach 0 with the increase of the

number of base classiﬁers. Assuming the accuracies of the two classiﬁers are acc1and acc2with 1

2≤acc1≤acc2≤1.

The interval of similarity of two classiﬁers is [acc1−(1 −acc2), acc1+ (1 −acc2)]. The higher the accuracy of the base

classiﬁer is, the higher the similarity between models will be, and the smaller the diﬀerence will be, in which case the

ensemble is likely to be invalid. So it is almost impossible to reach ideal conditions.

In addition to the performance requirements among base learners and their models, the combination strategy is

also an important factor aﬀecting the results. In general, models need to be pruned before using the combination

strategy, i.e., to determine which models can participate in the combination. For example, the sorting or search-based

strategies can be used to ﬁlter models [41]. The models’ predictions are then combined. Commonly used combination

strategies include the averaging strategy, voting strategy, and learning strategy [46]. Among them, the voting strategy

is divided into absolute majority voting, pluraity voting and weighted voting [45].

III. HYBRID QUANTUM-CLASSICAL ENSEMBLE LEARNING

We present a hybrid EL framework that combines quantum computation with classical computation. It takes

advantage of the parallelism of quantum computing and the convenience of classical computer processing parameters.

The hybrid learning framework is shown in Fig. 1. Taking the image classiﬁcation task as an example, the image

dataset is ﬁrstly reduced in the classical computer to preserve 2nfeatures, where nis the number of qubits used by the

quantum base learner. The dataset is divided into a training set and a test set. By random sampling, Nsubsets are

got from the training set, where Nis the number of the quantum base learner. These data are encoded into quantum

states before the quantum base learners are trained in a quantum computer. The training of a quantum base learner

can be completed in parallel by multiprocessing. Each quantum base learner predicts for the test set and outputs the

decision result as the ﬁnal result on the classical computer according to the combination strategy. ﬁg3

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AnecientcombinationstrategyforhybirdquantumensembleclassierXiao-YingZhang1andMing-MingWang1,1ShaanxiKeyLaboratoryofClothingIntelligence,SchoolofComputerScience,Xi'anPolytechnicUniversity,Xi'an710048,China(Dated:today)Quantummachinelearninghasshownadvantagesinmanywayscomparedtoclassicalmachinelear...

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An ecient combination strategy for hybird quantum ensemble classier Xiao-Ying Zhang1and Ming-Ming Wang1 1Shaanxi Key Laboratory of Clothing Intelligence School of Computer Science

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