Variational quantum one-class classier Gunhee Park1Joonsuk Huh2 3 4and Daniel K. Park5 6y 1Division of Engineering and Applied Science California Institute of Technology Pasadena CA 91125 USA

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Variational quantum one-class classiﬁer

Gunhee Park,

Joonsuk Huh,

2, 3, 4, ∗

and Daniel K. Park

5, 6, †

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

Department of Chemistry, Sungkyunkwan University, Suwon, 16419, Republic of Korea

SKKU Advanced Institute of Nanotechnology, Sungkyunkwan University, Suwon, 16419, Republic of Korea

Institute of Quantum Biophysics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

Department of Applied Statistics, Yonsei University, Seoul, 03722, Republic of Korea

Department of Statistics and Data Science, Yonsei University, Seoul, 03722, Republic of Korea

One-class classiﬁcation is a fundamental problem in pattern recognition with a wide range of

applications. This work presents a semi-supervised quantum machine learning algorithm for such a

problem, which we call a variational quantum one-class classiﬁer (VQOCC). The algorithm is suitable

for noisy intermediate-scale quantum computing because the VQOCC trains a fully-parameterized

quantum autoencoder with a normal dataset and does not require decoding. The performance

of the VQOCC is compared with that of the one-class support vector machine (OC-SVM), the

kernel principal component analysis (PCA), and the deep convolutional autoencoder (DCAE) using

handwritten digit and Fashion-MNIST datasets. The numerical experiment examined various

structures of VQOCC by varying data encoding, the number of parameterized quantum circuit layers,

and the size of the latent feature space. The benchmark shows that the classiﬁcation performance of

VQOCC is comparable to that of OC-SVM and PCA, although the number of model parameters

grows only logarithmically with the data size. The quantum algorithm outperformed DCAE in most

cases under similar training conditions. Therefore, our algorithm constitutes an extremely compact

and eﬀective machine learning model for one-class classiﬁcation.

I. INTRODUCTION

With the growing demand for eﬃcient and eﬀective

methods to extract useful knowledge from data, Quantum

Machine Learning (QML) has emerged as a promising

application of quantum technology [

]. Many pattern

recognition problems in data science can be formulated

as a classiﬁcation problem, which can be addressed via

supervised machine learning. Several theoretical works

showed that QML can be advantageous for classiﬁcation in

terms of runtime [

–

], trainability and model capacity [

11], and prediction accuracy [12].

While the majority of existing works on QML for clas-

siﬁcation addresses binary problems, this work focuses on

one-class classiﬁcation (OCC). One-class classiﬁcation has

a wide range of applications, such as anomaly detection in

ﬁnance [

], bioinformatics [

], manufacturing [

], and

computer vision [

]. The goal of OCC is to train a ma-

chine learning (ML) model that distinguishes normal data

from anomalous ones. In OCC, instead of having input-

output example pairs as in the usual setup for supervised

learning, only the input information is provided. Since

the training example does not contain the class labels, the

OCC is often called semi-supervised learning and is more

diﬃcult than the binary or multinomial classiﬁcation with

the label information. Moreover, a multinomial classiﬁer

can be constructed with multiple one-class classiﬁers.

One-class classiﬁcation problems have been tackled by

statistical machine learning approaches, such as princi-

pal component analysis (PCA) [

], one-class support

∗joonsukhuh@gmail.com

†dkd.park@yonsei.ac.kr

vector machine (OC-SVM) [

–

], and deep learning

based algorithms [

]. In particular, an autoencoder,

a feed-forward neural network that aims to copy its input

to its output [

–

], is widely used in one-class classi-

ﬁcation. An autoencoder consists of an encoder, which

extracts the essential feature of data and reduces dimen-

sion, and a decoder, which reconstructs the data. Given

a training dataset, an autoencoder is trained to act as

an identity function with respect to the training dataset

and the mean squared reconstruction error is subject to

minimization. For one-class classiﬁcation, after training

a neural network as an autoencoder with normal class

data, the reconstruction error can be used as a decision

function [

]. Alternatively, the autoencoder can be

used as a feature extractor of other statistical machine

learning techniques like OC-SVM [28–30].

As a classical autoencoder is able to learn the eﬃcient

representation of low dimensional latent space, a quantum

autoencoder (QAE) is proposed for eﬃcient quantum data

compression. The QAE utilizes a Parameterized Quan-

tum Circuit (PQC) [

], which is central in variational

quantum algorithms [

]. In addition to quantum data

compression, several applications of QAE have been ex-

plored including denoising quantum data [

], quantum

error correction [

], quantum error mitigation [

], and

quantum metrology [

]. The QAE has also been explored

for detecting anomalous phases in the context of quantum

Hamiltonian problems [37].

Motivated by the success of classical autoencoders for

OCC problems, we present a Variational Quantum One-

Class Classiﬁer (VQOCC) algorithm based on the QAE

that applies to classical data. The VQOCC is composed

of data encoding, PQC, and quantum measurements for

classical post-processing. In the past, anomaly detection

arXiv:2210.02674v1 [quant-ph] 6 Oct 2022

algorithms based on the quantum OC-SVM and quan-

tum PCA that could achieve exponential speedup were

proposed [

]. However, these quantum algorithms re-

quire expensive subroutines, such as the quantum linear

solver [

] and matrix exponentiation [

] that are not suit-

able for Noisy Intermediate-Scale Quantum (NISQ) com-

puting [

]. In contrast, training a shallow-depth PQC

with a classical optimizer is regarded as a promising ap-

proach for near-term quantum machine learning [

]. This

work focuses on taking the NISQ-friendly approach that

constructs a variational quantum algorithm for one-class

classiﬁcation with classical data, and verifying whether a

quantum advantage can be attained.

Numerical experiments are performed on handwritten

digits and the Fashion-MNIST dataset with open-source

Python API Qibo [

] for quantum circuit simulation.

The performance of VQOCC is evaluated via the area

under a receiver operating characteristic (ROC) curve

(AUC), and compared to classical methods including OC-

SVM, Kernel PCA, and deep convolutional autoencoder

(DCAE). We benchmark the performance of VQOCC

with various structures of the quantum autoencoder. The

structure of the QAE is determined by selecting data

encoding, the number of PQC layers, and the size of

the latent feature space. The general result of VQOCC

shows comparable performance to the classical methods

despite having the number of model parameters grow

only logarithmically with the data feature size. Notably,

the performance of VQOCC is better than DCAE under

similar training conditions.

The remainder of the paper is organized as follows. Sec-

tion II describes the one-class classiﬁcation and reviews

some of the well-known approaches to the problem. Sec-

tion III explains the quantum autoencoder, which is the

basis of the quantum one-class classiﬁer proposed in this

work. Section IV explains the application of quantum

autoencoder for one-class classiﬁcation and constructing

diﬀerent models via modifying the ansatz (i.e. structure

of the PQC) and cost functions. Numerical experiments

performed using scikit-learn and Qibo with handwrit-

ten digits and Fashion-MNIST datasets are explained

in Sec. V. This section also compares the AUC of ROC

curves of our algorithm with a one-class SVM, a kernel

PCA, and a deep convolutional autoencoder. Section VI

provides conclusion and suggestions for future work.

II. ONE-CLASS CLASSIFICATION

Assigning an input data to one of a given set of classes

is a canonical problem in pattern recognition and can be

formally described as a classiﬁcation problem. Classiﬁ-

cation aims to predict the class label of an unseen (test)

data ˜

x∈RN, given a labelled (training) dataset

D={(x1, y1),...,(xM, yM)} ⊂ RN×Zl,

where

is the number of classes. The one-class classiﬁ-

cation is a special case of the aforementioned problem

when

= 1 [

]. In this case, the training dataset is

{x1,x2,...,xM}

, which is treated as a normal class,

and the goal is to identify whether a test data

is in the

normal class or not. Since anomalous data is not used

in training, this is known as semi-supervised learning. It

is also possible to perform one-class classiﬁcation with

unsupervised methods with an unlabelled dataset under

the assumption that most of the test dataset is composed

of normal data [21, 22].

Given a training dataset of normal class

, a decision

function

(

;

x1,x2,...,xM

) is attained from a one-class

classiﬁcation algorithm, which expresses how far the input

data is from the training dataset. If the decision function

(

;

x1,x2,...,xM

) has a value smaller than a certain

threshold value

Cth

(i.e.

(

)

< Cth

), then

is classiﬁed

as normal. Otherwise, if

(

)

> Cth

, then the test data

is classiﬁed as anomalous. If

(

) =

Cth

, the decision can

be made at random.

Two well-known statistical approaches for addressing

one-class classiﬁcation problems are principal compo-

nent analysis (PCA) [

] and support vector machine

(SVM) [

–

]. PCA is a dimensionality reduction tech-

nique that projects data

into a lower dimensional sub-

space such that the projections have the largest variances.

The projected space provides reconstructed data

. The

lower dimensional subspace is determined to minimize

the reconstruction error

Pikxi−ˆ

xik2

. Once the lower

dimensional subspace is chosen, the reconstruction error

(

) =

kx−ˆ

xk2

can be considered as a decision function

for one-class classiﬁcation, since it will be small for normal

data and large for anomalous one. The kernel trick can

be utilized in PCA to include non-linearity [43].

The support vector machine is a supervised learning

model that aims to ﬁnd a hyperplane that separates two

classes of training data with the maximum margin. Thus

it is commonly used in binary classiﬁcation. The SVM

can be modiﬁed for one-class classiﬁcation by ﬁnding a

maximum-margin hyperplane that separates normal data

from the origin. This is known as the one-class SVM

(OC-SVM) [

]. The decision function of OC-SVM is

f(x) = hw,Φ(x)i − b, (1)

where

and

describes the hyperplane and Φ is the

feature map. If the decision function is positive (nega-

tive), the corresponding test data is classiﬁed as normal

(anomalous).

Alternatively, the SVM can be modiﬁed for one-class

classiﬁcation by ﬁnding the smallest hypersphere that

encapsulates normal data. This is known as the support

vector data description (SVDD) [

]. After ﬁnding the

optimal hypersphere, the data located outside of the

hypersphere is classiﬁed as anomalous. In this case, the

decision function can be expressed as

f(x) = kΦ(x)−ak2−R, (2)

where

is a center of the hypersphere, and

is a radius of

the hypersphere. Note that when the data is normalized

Decoder

Encoder Data

Encoding

Parameterized

Quantum Circuit

One-Class

Classification

FIG. 1. Graphical representation of (a) QAE and (b) VQOCC with number of trash qubits

= 2 and total qubits

= 6.

Quantum autoencoder is composed of encoder and decoder parts, which are represented as parameterized quantum circuit

(

)

and

U†

(

), respectively. The VQOCC quantum circuit consists of three parts: data encoding (blue rectangle), parameterized

quantum circuit

(

), and measurement of trash qubits for one-class classiﬁcation. Note that the parameterized quantum

circuit and measurement from the VQOCC quantum circuit is directly taken from the encoder part of quantum autoencoder.

to unit norm, the OC-SVM and SVDD become equiva-

lent [

]. Intuitively, these methods can be understood as

a process of learning the boundary for the normal data

and identifying the data outside of the boundary to be

anomalies.

III. QUANTUM AUTOENCODER

QAE is the quantum-analog of classical autoencoder,

for which a PQC learns to reduce the dimensionality of

data [

]. The training is carried out through a classical

optimization process; hence, it is the classical-quantum

hybrid algorithm. The dimensionality reduction means

that a quantum autoencoder compresses quantum data

into a smaller number of qubits than the input qubits.

Following the convention used in classical machine learn-

ing, we refer to the set of qubits to which the data is

compressed as latent qubits. A QAE is composed of an

encoding part and a decoding part as depicted in Fig. 1

(a). The encoding part applies a parameterized unitary

gate

(

), where

is a set of trainable parameters, aim-

ing to compress data into latent qubits. Other qubits are

discarded after this step (i.e. traced out) and are called

trash qubits. The number of trash qubits is denoted by

. For example, the QAE circuit in Fig. 1 (a) uses four

latent qubits and two trash qubits. The decoder applies

U†

(

) on the latent qubits and a reference state

|0i⊗nt

reconstruct the initial quantum state. For a QAE to be

successful, the parameterized unitary gates for encoding

should learn to disentangle latent qubits and trash qubits

to put them into a product state. This guarantees the

reconstruction of the initial quantum state via decoding

with a proper ancillary state. The PQC is trained by min-

imizing a cost function deﬁned with the quantum state

ﬁdelity or Hamming distance between the trash qubit sys-

tem and the target state

|0i⊗nt

[

]. More details

on the cost function used in this work will be described

in the next section.

IV. VARIATIONAL QUANTUM ONE-CLASS

CLASSIFIER

The QAE lays the ground for variational quantum

one-class classiﬁcation. The structure of a QAE can

be simpliﬁed if it is applied to a one-class classiﬁcation.

Namely, only the encoder part of the QAE is needed.

In Romero et al. [

], two cost functions based on the

trash state ﬁdelity and the decoded state ﬁdelity were

analyzed. It shows that the trash state ﬁdelity is the upper

bound of the decoded state ﬁdelity, and when the trash

state ﬁdelity equals one, the decoded state ﬁdelity also

equals one. The cost function based on the decoded state

requires the access of two identical copies of the input

state, whereas the cost function based on the trash state

does not. Thus formulating the optimization problem

with the cost function that only uses the trash state is

more advantageous in terms of computational resources.

After training the PQC to minimize the cost function

for the normal class training dataset, the cost function for

anomalous data is expected to yield values far from zero.

Hence by setting a threshold value to the cost function,

normal and anomalous data can be discriminated. In this

case, the cost function can be understood as a decision

function

of one-class classiﬁcation, which is analogous

to using the reconstruction error as a decision function in

the classical autoencoder.

In the following, the essential steps of the QAE-based

VQOCC, namely data encoding, parameterized unitary

gates, and measurement of

qubits from which the cost

function is evaluated, are explained in detail. Hereinafter,

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Variationalquantumone-classclassierGunheePark,1JoonsukHuh,2,3,4,andDanielK.Park5,6,y1DivisionofEngineeringandAppliedScience,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA2DepartmentofChemistry,SungkyunkwanUniversity,Suwon,16419,RepublicofKorea3SKKUAdvancedInstituteofNanotechnology,Sungkyunkw...

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