Hardware-efficient quantum principal component analysis for medical image recognition

doi:10.1007/s11467-024-1391-x

Front. Phys.

2024, Vol. 19

Issue (5) : 51202 https://doi.org/10.1007/s11467-024-1391-x

Hardware-efficient quantum principal component analysis for medical image recognition

Zidong Lin¹, Hongfeng Liu¹, Kai Tang¹, Yidai Liu², Liangyu Che¹, Xinyue Long¹, Xiangyu Wang¹, Yu-ang Fan¹, Keyi Huang¹, Xiaodong Yang^1,^3,⁴, Tao Xin^1,^3,⁴, Xinfang Nie^1,^4,⁵(

), Dawei Lu^1,^3,^4,⁵(

)

¹. Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
². Department of Physics, Hong Kong University of Science and Technology, ClearWaterBay, Kowloon, Hong Kong, China
³. International Quantum Academy, Shenzhen 518055, China
⁴. Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁵. Quantum Science Center of Guangdong–HongKong–Macao Greater Bay Area, Shenzhen–HongKong International Science and Technology Park, No. 3 Binlang Road, Futian District, Shenzhen 518045, China

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Abstract

Principal component analysis (PCA) is a widely used tool in machine learning algorithms, but it can be computationally expensive. In 2014, Lloyd, Mohseni & Rebentrost proposed a quantum PCA (qPCA) algorithm [Nat. Phys. 10, 631 (2014)] that has not yet been experimentally demonstrated due to challenges in preparing multiple quantum state copies and implementing quantum phase estimations. In this study, we presented a hardware-efficient approach for qPCA, utilizing an iterative approach that effectively resets the relevant qubits in a nuclear magnetic resonance (NMR) quantum processor. Additionally, we introduced a quantum scattering circuit that efficiently determines the eigenvalues and eigenvectors (principal components). As an important application of PCA, we focused on classifying thoracic CT images from COVID-19 patients and achieved high accuracy in image classification using the qPCA circuit implemented on the NMR system. Our experiment highlights the potential of near-term quantum devices to accelerate qPCA, opening up new avenues for practical applications of quantum machine learning algorithms.

Keywords quantum simulation quantum principal component analysis nuclear magnetic resonance

Corresponding Author(s): Xinfang Nie,Dawei Lu

About author:

#usheng Xing, Yannan Jian and Xiaodan Zhao contributed equally to this work.]]>

Issue Date: 30 May 2024

Cite this article:

Zidong Lin,Hongfeng Liu,Kai Tang, et al. Hardware-efficient quantum principal component analysis for medical image recognition[J]. Front. Phys. , 2024, 19(5): 51202.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-024-1391-x
https://academic.hep.com.cn/fop/EN/Y2024/V19/I5/51202

Fig.1 PCA-based medical image recognition and the corresponding qPCA circuits. (a) Flowchart illustrating the application of PCA for identifying lung CT images. The information of the CT images is encoded in the covariance matrix, which captures the interdependencies among different data components. Classically, principal components are obtained using iterative eigen-algorithms, e.g., power iteration, allowing for dimension reduction by projecting the data onto the subspace spanned by these principal components. (b) Quantum circuit for extracting the eigenvalues of the data matrix

ρ

. The Hadamard gate prepares the probe qubit in an equal superposition state, while the controlled-

e − i ρ t

gate encodes the eigenvalues of

ρ

into the phase of the probe qubit. The trial qubits are initialized to an arbitrary pure state that must have non-zero overlaps with the eigenvectors of

ρ

. (c) Quantum circuit for eigenvectors extraction of

ρ

. The evolution time is set as

τ = π / (λ 2 − λ 1)

, assuming that

λ 1 < λ 2

. The single-qubit gate

U 0

prepares the probe qubit in

| 0 ? + e i λ 1 τ | 1 ?

. (d) Complete circuit incorporating the realization of the controlled-

e − i ρ t

gate. It requires

N

copies of the data matrix

ρ

, and the key component is the

e − i S Δ t

gate between the trial qubits and each copy, where

S

is the standard swap gate and

Δ t = t / N

Fig.2 Data loading and the experimental quantum circuit in the 4-qubit NMR system. (a) Training dataset comprises one positive and one negative CT image from the iCTFT database. Each image is flattened into a

49400 × 1

vector based on grayscale values, resulting in a training dataset represented by a

49400 × 2

matrix

X

. After centering the data, instead of the covariance matrix

C = X X T

, we compute the commuted covariance matrix

D = X T X

and load it into the quantum register, denoted as

ρ D

. (b) Experimental quantum circuit for performing the iterative qPCA algorithm on the 4-qubit NMR quantum processor. We prepare two copies of

ρ D

and implement the controlled-

e − i ρ D t

operation in

N

steps, where each step consists of two controlled-

e − i S Δ t

gates applied between the trial qubit (

C 2

) and the data qubits (

C 3

C 4

). In each step, the probe and trial qubits are reinitialized to their final state from the previous iteration using single- and two-qubit gates determined by the state of the previous iteration. Gradient-field pulses are applied to destroy instantaneous quantum coherence, serving as non-unitary operations.

Fig.3 Experimental eigenvalues obtained by the hardware-efficient qPCA approach. (a) Training set images for Group 1 experiment. The left and right images are randomly sampled from lung CT images of negative and positive patients, respectively. Clear textures are observed in the healthy lung CT image, while the diseased lung CT image exhibits a hazy shadow. (b) Magnitude of the probe qubit’s

x −

component

M x (t)

[cf. Eq. (2)] for Group 1 at different time instants. (c) Discrete Fourier analysis of

M x (t)

in Group 1. The spectral lines of different colors represent the Fourier analysis of

M x (t)

at every 200 iterations. (d−f) Results for Group 2. (g) Two eigenvalues obtained from the Fourier analysis every 50 iterations in Group 1 and Group 2. As the number of iterations increases, the eigenvalues

λ 1, 2 e x p

tend to converge towards the theoretical values. (h) Precision of eigenvalue measurement is quantified by the half-widths of the two peaks, denoted as

Δ λ 1, 2

. The values in both groups gradually decrease as the number of iterations increases.

Fig.4 Experimental eigenvectors and the iterative process. (a) Experimental results of the eigenvectors for Group 1 (left panel) and Group 2 (right panel). The coefficients of

| e 1 ?

and

| e 2 ?

, represented in the computational basis, are illustrated. Both theoretical (green bars) and experimental (yellow bars) coefficients are shown for comparison, and the fidelities of the experimental eigenvectors exceed 0.9998. (b) Change in state fidelity (probe and trial qubits) as a function of the number of iterations. The fidelity is plotted at every 50 iterations for better visualization, while the solid line represents the fitting result. As a reference, we display the density matrices of the probe and trial qubits at the 200th and 800th iterations, as shown in the insets.

Fig.5 Classification of lung CT images after qPCA. (a) Classification results for Group 1. After performing qPCA, each CT image in the dataset is projected onto a two-dimensional space spanned by the two experimental eigenvectors

| e 1 e x p ?

and

| e 2 e x p ?

. The two training images are denoted by circles, while the test images are represented by crosses. The test dataset consists of 39 virus-negative images (blue) and 39 virus-positive images (red). The decision boundary is depicted by the dashed line. Out of the 78 test images, 72 are correctly identified, resulting in a success rate of 92.31%. (b) Classification results for Group 2. Among the 78 test images, 68 are accurately classified, yielding a success rate of 87.17%.

Fig. A1 Molecular structure and relevant parameters. (a) Molecular structure of ¹³C-labeled trans-crotonic acid. C₁, C₂, C₃ and C₄ in the green dashed line are used as four qubits in the experiment. (b) Molecular properties and the Hamiltonian relevant parameters of the sample. Chemical shifts (diagonal, Hz), scalar coupling strengths (off-diagonal, Hz), and relaxation times (

T 1

and

T 2

) are all listed in the table.

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