1. Introduction – quantum machine learning

Machine learning (ML) is a sub-area of artificial intelligence, enabling computers to learn from data and improve tasks over time. ML has a very wide range of applications, including quantum many-body problems [1], phase transition [2], classification and regression [3], engineering [4], bioinformatics analysis [5], anomaly detection [6], bibliometric analysis of quantum k-means clustering [7] etc.

Quantum machine learning (QML) [8] is an emerging and popular research field that integrates the concepts of quantum computing and machine learning. This integration leverages the unique capabilities of quantum computers to enhance and accelerate machine learning tasks. The most commonly used model in quantum computing is the gate-based quantum computation model [9]. Essentially the model can be described in three steps: (i) Initialization: Qubits are initialized in a specific quantum state, often the |0> state. (ii) Encoding: Input data is encoded into the quantum amplitudes of these qubits. (iii) Quantum Gates: Quantum unitary gates are applied to transform the quantum states of the qubits. These gates can operate on multiple qubits simultaneously, a property known as quantum parallelism.

By designing and implementing quantum circuits (quantum algorithms) appropriately, the quantum states undergo interference. This interference enhances the amplitudes of the desired solutions. The final solution is obtained by performing measurements on the quantum states. Due to the inherent randomness in quantum measurements, multiple repetitions are necessary to determine the required solution with high probability.

Quantum computing excels in solving specific types of problems, such as quantum simulations in physics and chemistry. However, it is not expected to replace classical computing entirely. Instead, a hybrid approach that combines classical and quantum computing is often used to tackle complex problems more efficiently [10–12].

Leveraging quantum computing in machine learning may be essential for solving complex problems for chemistry and physics [13], in molecular biology, such as: cancer classification [14–17], non-small cell lung cancer classification [18], COVID-19 [19], biochemical systems [20], medical imaging [21,22], malignant breast cancer diagnosis [23], pancreatic cancer data classification [24], cancer gene expression biomarker [25], oncology application [26], clinical decision [27], drug discovery [28–30] etc.

Renowned ML algorithms, including the support vector machine classifier (SVC), random forest, neural network (NN), ensemble methods etc. Among the many well-known ML algorithms, SVC can achieve a better performance in certain real-world applications [31–34].

A quantum version of the SVC algorithm (QSVC) was proposed by Rebentrost et al., at 2014 [35]. Later, the kernel-based classifier was implemented on a superconducting quantum machine [36]. At 2020, Suzuki et al., [37] suggested to combine different kernels to construct a more efficient feature map for classification using QSVC. QSVC stand out as innovative method that combine the power of quantum mechanics with domain of machine learning [38].

In addition to QSVC, quantum versions of several machine learning algorithms have been developed, including QNN (quantum neural network) [39] and QCNN (quantum convolutional neural network) [40]. Both QSVC and QNN classifiers are considered in the work.

The use of QCNN combined with NQE [41] for classifying the digits 0 and 1 within the MNIST dataset has been reported. It was found that the classification accuracy of QCNN trained with NQE significantly improves compared to QCNN without NQE training. However, there are no studies on the use of NQE+QSVC and NQE+QNN for classification problems. We explore this possibility using biomedical data. We noted that the quantum version of random forest (QRF) is currently not available.

For both QSVC and QNN, the key factor that determines the success of classification performance is the choice of an appropriate feature map. If the feature map is not well-suited to the data, it will hinder the performance of the entire machine learning model [42].

2. Materials and methods

In this study, we improve the methodology used in our prior research [53], aiming to overcome specific shortcomings and enhance both the robustness and precision of the QML approach. Previous analyses revealed that the QSVC algorithm had a prolonged execution time and suffered from lower classification accuracy. Enhanced classification accuracy for metastatic patients can improve treatment efficacy and outcomes, and streamline resource allocation in healthcare.

The workflow diagram for the gene expression biomarker study is depicted in Fig 1.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. The workflow diagram for the gene expression biomarker study which involve using principal component analysis (PCA) to reduce the number of features (principal components), thus requiring fewer qubits.

We utilize the SMOTE package for data balancing and optimize SVC and QSVC models through the NQE approach. This hybrid method combines quantum data encoding with neural network training to separate the classes of data into orthogonal subspaces, followed by performance comparison.

https://doi.org/10.1371/journal.pone.0327928.g001

The miRNA classification study does not require PCA. Data imbalance handling is also unnecessary because the smaller number of features and both the disease and control groups have similar sample sizes.

2.2. Data embedding by using Neural Quantum Embedding (NQE)

The goal of data embedding is to train the input features in order to maximize data separability. Improving data separability in higher dimensions enhances training performance. Quantum embedding uses unitary operators to encode classical data non-linearly into high-dimensional quantum state spaces [66,67]. However, achieving this through quantum embedding necessitates introduce more noise and resources, leading to increased circuit depth and gates.

In quantum machine learning, quantum embedding is important because it maps classical data into high-dimensional quantum states, enabling the exploitation of quantum advantages in pattern recognition and classification. The Neural Quantum Embedding (NQE) approach [41] utilizing a hybrid classical quantum approach, that is, combining neural network (NN) and a quantum circuit, to train the input data enables the generation of the feature maps capable of separating the classes of data into orthogonal subspaces. This process is crucial as it enhances classification performance, and the use of a hybrid architecture can also help reduce computation time.

In layman terms, the input features are learned by a NN with trainable weights and a loss function (mean squared error) designed to maximize the separation between the two classes. The outputs are non-linearly mapped to a higher-dimensional Hilbert space using a quantum feature mapping function, which consists of an exponential operator made up of non-linear functions and the rotational Pauli Z operators.

The initial two classes of input samples are encoded into quantum states, and the resulting states are measured to determine whether the two samples share the same class label. This overlap is quantified by the fidelity function. The fidelity result is then used to update the NN’s weighting factors, optimizing the separability of the two classes of data into orthogonal subspaces, thereby maximizing their difference. Once iterations complete, save the optimized model, apply it to training and test sets, then pass to the classifier.

Technically, the NQE method involve the following steps. (I) Input feature x is trained by a NN, the output function of the NN is denoted by ,where w represents r trainable parameters. (II) The output function () is inserted into a unitary quantum circuit, which is denoted by . The unitary operator is defined by

(1)

where , and Z denote the mapping function for feature x_k, the mapping function for features x_j and x_k, and Pauli Z operator, respectively.

In general, there are no limits on the type of quantum embedding circuit that can be used. In this case, the operator focuses on improving the ZZ feature embedding by training an Instantaneous Quantum Polynomial (IQP) function, [68] through multiple steps. IQP is a model of quantum computing where all the gates in the circuit can be applied in any order without changing the outcome of the computation.

(III) Given the embedding, the initial input feature is encoded to a quantum state, that is, x → . The resulting quantum state |x > is measured, the post-process data is used to update the trainable parameter w, by computing a loss function, which is derived from a fidelity measure function F,

(2)

where denotes the class label of the data . The fidelity loss function can be computed using the swap test [69]. It is noted that if and are similar and share the same class label, the loss function is minimized ( and , so ) Conversely, if they belong to different classes, the loss function is maximized. Therefore, the loss function offers a means to infer the separability of the data. (IV) The result from (III) is used as input for the NN. Then, the whole process is re-iterate again, and the quantum state is updated to . (V) In this step, the algorithm returns back to step (I) to update . Fig 2 illustrates each step of NQE in detail. The NQE algorithm is available as an open-source program, accessible for download from https://takh04.github.io/NQE_tutorial.html.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Data embedding by using Neural Quantum Embedding (NQE).

https://doi.org/10.1371/journal.pone.0327928.g002

The authors of reference [41] verify NQE’s efficacy by experimenting on IBM quantum devices with the MNIST data, elevating classification accuracy significantly from 0.52 to 0.96.

2.3. NQE with QML and hyperparameter optimization

2.3.1. NQE with SVC, QSVC and hyperparameter optimization.

After the two classes of data are successfully separated into different subspaces, we can test how well different machine learning classifiers perform. Fig 3 shows how NQE is combined with QML classifiers, QSVC and QNN, for classifying biomedical data. Fig 3 also illustrates the workflow of integrating NQE with QSVC and QNN classification.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. The workflow of integrating NQE with QSVC and QNN classifiers.

https://doi.org/10.1371/journal.pone.0327928.g003

The key design choices are as follows: (i) PCA is used to reduce the number of qubits because simulating quantum circuits on a classical computer requires a lot of computing power. (ii) NQE separates the two classes of data into different subspaces based on a loss function. (iii) Quantum versions of SVC, NN, k-nearest neighbors, and CNN are available, but quantum random forest and other classifiers are not yet supported at the moment. (iv) QSVC and QNN can be implemented using IBM Qiskit. The ZZ feature map and combinations of layers, including rotation and CNOT gates, are used for the QSVC and QNN classifiers, respectively. These quantum classifiers excel in representing complex functions due to their superior expressive power, surpassing the capabilities of classical classifiers [70].

SVC is a powerful algorithm used for classification tasks. It maps features from a lower dimension space to a higher dimension space using a kernel function, subject to a constraint that maximizes the margin between two classes [71,72]. The margin is the distance between the decision boundary and the closest data point. Mathematically, maximizing the margin is equivalent to minimizing the following function,

(3)

subject to the constraint

(4)

where y_k, |w| and t denote the class label for the kth feature, 1-norm of the normal vector to the decision boundary (hyperplane) and the decision threshold (bias), respectively. The above quadratic constraint optimization problem, also known as the primal problem, can be solved by using the method of Lagrange multipliers.

The primal problem is a convex problem; it can be solved efficiently and has the property that any local minimum is also a global minimum. It is possible to derive the dual problem from the primal problem by eliminating w and t, then formulating the Lagrange function entirely in terms of the Lagrange multipliers. The dual problem is to maximize the following Lagrange function

(5)

subject to the positivity constraints and an equality constraint:

(6)

where denotes the Lagrange multipliers. The dual formulation leads to the identification of support vectors. Support vectors are data points that are closest to the decision boundary and are critical for determining the decision boundary. Under the conditions known as the Karush-Kuhn-Tucker conditions, the solutions to the primal and dual problems become equal [73].

The dual formulation is useful because it expresses the Lagrange function using the dot product or kernel function between feature vector pairs. This allows flexibility to choose different kernels, like linear, polynomial, or Gaussian RBF, based on the data and problem. This flexibility enables SVC algorithms to capture complex patterns in data that might not be linearly separable in the original feature space. Solving the dual problem has a time complexity of O(N³) [74].

The QSVC in the IBM Qiskit package is used to handle the classification problem by leveraging the computational advantages of quantum computers. The QSVC inherits methods like ‘fit’ and ‘predict’ from Scikit-learn. The QSVC uses a quantum kernel, named ‘FidelityQuantumKernel’, for computing the kernel matrix, which can capture complex patterns in the data by leveraging quantum computation.

In QSVC, the quantum kernel is found by calculating the inner product of the quantum states that encode the data, which is given by

(7)

where represents the unitary operation derived from the application of a layer of Hadamard gates followed by the quantum feature map using NQE method. The kernel is computed by selecting and iterating through the other state . When is close to , the circuit exhibits a large transition amplitude between states and , where is the computational z-basis. When the measurement is repeated, the frequency of the all state approximates the quantum kernel . Once this kernel is trained, it can be used to compute the kernel function between the training and test set, facilitating classification.

We note that the potential advantage of using quantum kernels lies in their ability to leverage computational features that are difficult to simulate efficiently on classical systems. However, this advantage alone is a necessary condition but it is not sufficient [75].

Note that we perform the quantum computing part by using a classical computer to simulate the quantum circuit, rather than executing the code on actual quantum computing hardware.

In order to ensure a fair comparison between the SVC and QSVC classification results, we employ grid search twice to identify the optimal gamma and C parameters in both the SVC and QSVC algorithms. We start with a grid search over a wide range of gamma and C values to get initial estimates. Then, we perform a second grid search to refine these results and find the optimal gamma and C values. The optimal result is determined using the F1 score with five-fold cross-validation.

2.3.2. NQE with QNN and hyperparameter optimization.

To evaluate the impact of NQE on classification performance other than SVC, we have included studies on NQE + NN and NQE + QNN in this work, utilizing 5-fold cross-validation. The workflow diagram of this study is similar to that in Fig 1, with SVC/QSVC replaced by NN/QNN; therefore, the diagram is omitted. For the NN classifier, the following three hyperparameters were optimized: the number of hidden layers, learning rate, and epochs. For the QNN classifier (SampleQNN), the following three hyperparameters were optimized: the number epochs, the number of ZZfeatureMap repetitions, and the number of ansatz layers. The NQE + NN and NQE + QNN models were then applied to the test set.

2.4. Performance metrics

Lastly, the performance metrics results entail the comparison of SVC classifier and the LS-QSVC. To assess the performance of NQE+SVC and NQE+QSVC, we will compute the following 10 performance metrics. It’s important to highlight that lower values of the FPR, FDR, and FNR indicate better performance.

(8)(9)(10)(11)(12)(13)(14)(15)(16)

Matthews Correlation Coefficient (MCC):

(17)

3. Results

After completing the PCA for the gene expression profile study, the original log₂(Fold change), that is, log₂FC, values were projected onto the new coordinates (PCs). Assuming we keep nine PCs, the data distribution projected onto two principal components (S1 Fig). The high degree of overlap between the two classes in the PCA plots suggests that the PCs do not provide enough discriminatory power to separate the classes. The overlap between the two labelled classes is still persists even when considering 10–12 principal components (PCs), as illustrated in S2–S4 Figs. This suggests the original features do not capture the underlying differences between the classes effectively; hence, that is a non-trivial classification problem. Nevertheless, we will compare the classical and quantum versions to determine which performs better.

In the next step, we divided 2/3 of total sample as training set: 334 samples (286 M0 and 48 M1) and 1/3 as testing set: 168 samples (138 M0 and 30 M1). After employing SMOTE for oversampling the ccRCC gene expression training dataset, we obtain 286 samples for both M0 and M1 as the training set. Next, we perform hyperparameter tuning by dividing the oversampling set into the training set and the validation set using 5-fold cross-validation.

For the miRNA expression profiles, only four features are selected by using multi-variable Cox analysis in our previous study [76]. In the present case, there is no need to use the PCA method for dimension reduction. The data distribution projected onto the four features is shown in S5 Fig. Unlike the mRNA case study, the extent of overlap between the classes is relatively minor; therefore, we anticipate that the classification performance would be better. Next, we divided 2/3 of total sample as training set: 113 primary samples and 27 M1 samples, and 1/3 as testing set: 70 samples.

We utilized the NQE algorithm to pre-process the three input transcriptomic datasets, aiming to optimize data embedding by training the NN part of the NQE algorithm. The number of hidden layers, nodes per layer, the learning rate, the number of iterations and batch sizes are set equal to 1, two times the number of features (=8 nodes), 0.01, 1000 and 10, respectively. The number of qubits corresponds to the number of the input features.

Fig 4 illustrates the quantum embedding circuit. Its objective is to train the input features in a way that maximizes the separability between the M0 and M1 class labels. The number of qubits corresponds to the number of input features (or PCs). The embedding process employs Hadamard (H) gates, rotation (RZ) gates, and control-NOT (CNOT) gates, applied three times. These gates serve the following purposes: (i) creating superposition in the input quantum qubits, (ii) parametrizing the states by applying phase shifts, which are essential for generating interference among the quantum states, and (iii) introducing entanglement between the qubits.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. The quantum embedding circuit designed for the miRNA dataset, featuring three layers of repeated Hadamard (H), RZ, and control-NOT unitary gates.

To conserve space, a partial depiction of the inverse RZ gates applied in reverse order is shown.

https://doi.org/10.1371/journal.pone.0327928.g004

To reverse the embedding, a layer of the inverse of the RZ gates is applied in reverse order. The two mapping functions, φ_k and φ_jk (as shown in the function V(g(x, w)) in Fig 2) are defined to be 2x_k and (π – x_j)(π – x_k) respectively.

In other words, the input features are mapped non-linearly to a higher-dimensional Hilbert space using a feature mapping function. This function includes an exponential operator composed of non-linear functions and rotation Z operators. The initial two input samples are encoded into quantum states, and the resulting states are measured to determine whether the two samples share the same class label. This overlap is quantified by the fidelity function. The fidelity result is then used to update the neural network’s weights, improving the separability of the two data classes into orthogonal subspaces and maximizing their difference.

Fig 5 shows the training loss plots (using PyTorch mean squared error package, MSEloss) as a function of the number of iterations for 9–12 principal components (PCs), during the training phase of the NQE method using the stochastic gradient descent algorithm to update the weights. This process resulted in an updated weight for the function g(x, w).

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Training loss results plotted against the number of iterations for (a) 9, (b) 10, (c) 11, and (d) 12 principal components.

https://doi.org/10.1371/journal.pone.0327928.g005

Once the weight for the function g(x, w) is updated, it is used in operator V [Eq (1)], which changes the quantum circuit’s parameters to better separate the two classes by maximizing their distance [Eq (2)]. A high training loss means the two classes are not well separated in the quantum space, so further adjustments are needed.

The training loss function plotting shows high training loss at the beginning which decreases during the training process and then flattens gradually. A high initial training loss indicates that the initial parameters of the NN are not good for modelling the data. As training progresses, the NN parameters are updated to minimize this loss, improving the model’s performance. As the training loss flattens, it indicates that the NN is approaching convergence, meaning that further training may not result in significant improvements.

After achieving minimal training loss, one can generate a heatmap of the kernel matrix to visualize similarities between pairs of feature vectors mapped into a higher-dimensional space. Instead of presenting this intermediate result, our focus is on reporting the classification performance using the gene expression profile study. Visualization of the heatmap are deferred for the miRNA case study.

We utilize both SVC and QSVC on a quantum simulator within the IBM cloud through its quantum machine learning library, Qiskit, to assess and compare their respective performances.

The performance metrics comparison between NQE + SVC and NQE+QSVC for 9, 10, 11, and 12 PCs is illustrated in Fig 6. The first and second bar plots indicate the NQE + SVC and NQE+QSVC result, respectively. For recall (TPR), specificity (SPC), precision (PPV), F1 score, accuracy (ACC), negative predictive value (NPV), and Matthews Correlation Coefficient (MCC), higher scores indicate better performance. Conversely, for False Positive Rate (FPR), False Discovery Rate (FDR), and False Negative Rate (FNR), lower scores suggest better performance. As we can observe from Fig 6, NQE+QSVC performs better for 9 and 10 PCs for almost all the performance metric scores, on the other hand, classical SVC achieves better performance for 11 and 12 PCs.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. The results of the ten performance metrics for NQE+SVC vs NQE+QSVC using (a) 9, (b) 10, (c) 11, and (d) 12 principle components, respectively.

https://doi.org/10.1371/journal.pone.0327928.g006

We assessed the classification performance of SVC and QSVC models utilizing gene expression datasets from ccRCC cases. Table 1 presents a comparison of performance metric score of NQE + SVC and NQE+QSVC for each PC. Numbers highlighted in bold indicate the better performance. As shown in Table 1, NQE+QSVC outperforms NQE + SVC in two cases; i.e., both 9 and 10 PCs. When considering 9 and 10 PCs, NQE+QSVC achieved scores of 10 and 8 points, respectively, across the 10 performance metrics. Alternatively, NQE + SVC achieves better performance in two cases: 11 and 12 PCs. When evaluating 11 and 12 PCs, SVC attained scores of 8 and 6 points, respectively. Notably, NQE+QSVC achieves the highest F1-score and MCC of 0.447 and 0.304, respectively, for the 10 PCs case.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. Comparison of performance metrics, SVC vs QSVC after using NQE for data embedding. Bold-faced fonts denote that the classifier achieves better performance.

https://doi.org/10.1371/journal.pone.0327928.t001

Next, we presented the results concerning the training loss and classification performance, employing the dataset comprising microRNA (miRNA) expression profiles of ccRCC patients.

We assess the classification performance of both SVC and QSVC with and without the utilization of NQE for data embedding. Without NQE, QSVC outperforms SVC. Specifically, in the comparison between SVC and QSVC, QSVC achieves scores of 8 points, one tier, across the 10 performance metrics (S3 File).

In addition, we have conducted a more comprehensive study by selecting the number of principal components (PCs) in the range of 4–6, for the classification analysis. Table 2 presents the results of the 10 performance metrics for the training set, evaluated using five-fold cross-validation on the balanced training set, along with their standard deviations, providing a comprehensive view of the model’s robustness.

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. Classification performance results using 5-fold cross-validation for the training datasets after applying the data imbalance tool, SMOTE and NQE for data embedding. The values without the ‘±’ signs represent the results for the ‘test’ set.

https://doi.org/10.1371/journal.pone.0327928.t002

The last row of Table 2 presents the comparative scores obtained by evaluating the 10 metrics, where higher values indicate better performance. We observed that NQE+QSVC outperforms NQE + SVC when using 5 and 6 PCs ‘test’ set results (comparative score is 0 vs. 10). However, NQE + SVC achieves superior performance with 4 PCs. These results suggest that NQE has the potential to improve the performance of QSVC classifiers. Additionally, all standard deviation values are relatively small in comparison to the mean values, except in five cases (NQE + SVC, 4PCs – SPC, FPR, FNR, and MCC; and NQE + SVC, 5PCs – SPC), indicating that the results for the 10 performance metrics are generally robust.

We also observed a significant difference between the performance metric values for the training and test sets, particularly for PPV, F1, FDR, FNR, and MCC. However, this difference is less pronounced when data balancing techniques are applied, compared to the scenario without data balancing, where the discrepancy between training and test set results is much more pronounced (Table 2). This suggests that the issue of data imbalance remains an inherently challenging problem.

For the miRNA case study using NQE, Fig 7 illustrates the loss function plot as a function of the number of iterations. The training loss function plot depicts initially high loss that diminishes over training, plateauing eventually. High initial loss suggests inadequate NN parameters for data embedding, gradually refined through training to enhance model performance. Flattening loss indicates convergence, implying further training may yield limited improvements. This trend aligns with what is depicted in Fig 7. The interpretation of this plot is essentially identical to that of Fig 5.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. Training loss results plotted against the number of iterations.

https://doi.org/10.1371/journal.pone.0327928.g007

Having achieved a minimal training loss, we proceed to create a kernel matrix, which is instrumental in the classification of the dataset. To accomplish this, we utilize the ‘FidelityQuantumKernel’ class. During its instantiation, we provide two key parameters: first, the feature map, a two-qubit ZZFeatureMap, and second, the fidelity measure, employing the ‘ComputeUncompute’ fidelity method that utilizes the ‘Sampler’ primitive. The kernel matrix’s elements reflect the similarity levels between pairs of feature vectors within a higher-dimensional space. This is achieved without the need to directly compute the mapping functions, thereby enabling us to identify a linear hyperplane that can be used for classification tasks within this transformed feature space. The train and test kernel matrices can be visualized as shown in Fig 8. Darker colored dots mean higher level of similarity of two feature vectors. The kernel matrix for the training set is a square, with both the x and y axes representing the size of the training set. For the testing set kernel matrix, the x-axis shows the training set size, and the y-axis shows the test set size.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 8. The train and test kernel matrix matrices.

Darker colored dots mean higher level of similarity.

https://doi.org/10.1371/journal.pone.0327928.g008

As we can see from Fig 9, by using miRNA expression, NQE+QSVC performs better in every performance metric. The F1-score and MCC for NQE+QSVC is 0.92 and 0.68, whereas NQE + SVC achieves a score of 0.9 and 0.62. Again, lower scores for FPR, FDR, and FNR indicate better performance. In summary, NQE+QSVC attained higher scores of 9 points across all 10 performance metrics (S1 File), which is slightly better comparing to the case study where NQE is not used, that is, QSVC vs NQE+QSVC.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 9. The results of the ten performance metrics for NQE+SVC vs NQE+QSVC based on classifying miRNA expression profiles.

https://doi.org/10.1371/journal.pone.0327928.g009

Table 3 presents the results of the 10 performance metrics for the training set, including standard deviation, after selecting the optimal cross-entropy value via 5-fold cross-validation. The final row of Table 3 provides the comparative scores across the 10 metrics using the ‘test’ set results, where higher values correspond to better performance. We observed that NQE + NN outperforms NQE + QNN when using 4 and 5 PCs. However, NQE + QNN demonstrates superior performance with 6 PCs, as indicated by the comparative score of 0 versus 10. These findings suggest that NQE has the potential to enhance the performance of the QNN classifiers. Furthermore, all standard deviation values are relatively small compared to the mean values, except in two instances (NQE + NN, 6 PCs, FPR, and FNR), indicating that the results for the 10 performance metrics are robust.

Download:

• PNG
  larger image
• TIFF
  original image

Table 3. Classification performance results using 5-fold cross-validation for the training datasets after applying the data imbalance tool, SMOTE and NQE for data embedding. The values without the ‘±’ signs represent the results for the ‘test’ set.

https://doi.org/10.1371/journal.pone.0327928.t003

We also found a significant difference between the performance metric values for the training and test sets, particularly for PPV, F1, FDR, FNR, and MCC. As suggested earlier, the issue of data imbalance remains an intrinsically challenging problem.

To evaluate how well our method works, we applied it to a third dataset. Our results show that among the three case studies NQE+QSVC outperforms NQE + SVC in two cases, while NQE + NN performs better than NQE + QNN in two cases (Table 4). Since the analysis process is the same as with the first two datasets, we only highlight the main findings and provide the full results in the supplementary material.

Download:

• PNG
  larger image
• TIFF
  original image

Table 4. Comparison of NQE combined with SVC, NN, and QCNN classifiers. The table shows the number of test cases, comparison with classical classifiers, and data imbalance issues in this and previous work.

https://doi.org/10.1371/journal.pone.0327928.t004

S4 File as well as S5–S7 Figs show the results of 10 performance measurements for the training and test sets of dataset (iii), evaluated using NQE + SVC and NQE+QSVC. These results include standard deviation and are based on choosing the best cross-entropy value using 5-fold cross-validation. The last row of the table shows the test set results across the 10 metrics. Higher values mean better performance, except for FPR, FDR, and FNR, where lower values are better. We found that NQE+QSVC did better than NQE + SVC when using 5 and 6 PCs. For example, the scores were 0 vs. 10 and 3 vs. 7. This suggests that NQE can help improve the QSVC model. Most standard deviation values are small compared to their averages, which means the results are quite stable. Only FNR and MCC had larger variation. We also noticed a big difference between training and test scores, especially for MCC. This shows that the issue of unbalanced data is still a hard problem to solve.

S5 File and S8–S10 Figs present the results of 10 performance metrics for the training and test sets of dataset (iii), analysed using NQE + NN and NQE + QNN. We found that NQE + QNN did better than NQE + SVC when using 6 PCs, and both NQE + NN and NQE + QNN perform equally well when using 4 PCs.

4. Discussion

The application of NQE to enhance machine learning classification was initially explored with non-biomedical data, specifically the MNIST dataset. The results indicate that combining NQE with QCNN improves classification performance. However, there is no comparison with NQE + CNN to determine if it yields inferior results compared to the NQE+QCNN architecture.

We explored the application of NQE to enhance machine learning classification in four ways: (i) using three types of biomedical data, (ii) addressing data imbalance issues, (iii) comparing NQE+QSVC with NQE + SVC, and (iv) comparing NQE + QNN with NQE + NN. This allows us to determine whether machine learning classification algorithms yield inferior results without the use of NQE compared to when NQE is utilized. Our results indicate that the NQE approach can be extended to molecular medicine data, demonstrating its potential as a general method to enhance classification performance. This study does not aim to improve the data embedding process or the quantum machine learning algorithms.

We evaluate the performance of SVC/NN compared to QSVC/QNN after applying NQE for classifying patients with ccRCC. This classification is based on their normal, metastasis and primary status using three independent datasets.

From our results, the following four findings were established. First, by utilizing the PCA method, the required qubits decreases; hence, allow us to complete the classification task within reasonable time. This process effectively dealing with high-dimensional datasets enhancing the feature extraction process. Second, upon performance comparison, NQE+QSVC (NQE + QNN) demonstrates better prediction capabilities compared to NQE + SVC (NQE + NN) in certain cases. Without the use of NQE, QSVC/QNN performs relatively poorly compared to SVC/NN. We demonstrate the effectiveness of the NQE approach across three biomedical datasets. Third, these results have potential clinical implication in metastasis biomarker study for precision medicine diagnosis. It can aid physicians in making accurate diagnoses and minimizing the wastage of medical resources. Fourth, we demonstrate the effectiveness and generalizability of the current approach by analysing three independent biomedical datasets. This suggests that the approach could be applicable to other datasets as well.

Some might be concerned about comparing our hybrid approach to other machine learning classifiers to evaluate NQE’s effectiveness. As mentioned earlier, only the quantum versions of SVC, NN, and CNN are available in IBM Qiskit. Table 4 presents a comparison of NQE combined with these three classifiers, highlighting the strengths and weaknesses of each method. We tested QSVC thoroughly using three datasets and seven different test cases (using different PCs). This means our method was checked more carefully than the others.

We also compared how well two QML models worked against classical machine learning models. The results showed that combining NQE with the quantum advantage of QML is an effective approach. On the other hand, the work of QCNN has not been compared to CNN to see if it performs better [41]. Furthermore, we address the issue of data imbalance, which is not considered in the previous work as well. Out of the 17 test cases, NQE+QSVC and NQE + QNN produce better results in 10 cases and are tied in one case compared to NQE + SVC and NQE + NN. We note that out of the 10 winning cases, NQE+QSVC accounts for seven, suggesting that it performs better overall.

Over several decades, clear cell renal cell carcinoma (ccRCC) has been the predominant subtype of renal cancer, accounting for 80–90% of renal cases and remaining a highly lethal urological malignancy. Metastasis, the final stage of cancer, remains a challenging area of study, with the precise mechanisms behind cancer cell dissemination and establishment at distant sites still largely unknown. In this context, both gene and microRNA (miRNA) expression profiles are used to classify tumor metastasis, offering a more accurate and comprehensive patient classification. This approach aligns with the principles of precision medicine, tailoring treatments based on molecular profiles to improve outcomes. By combining both mRNA and miRNA data, this study helps us understand cancer metastasis better. It also supports creating more personalized treatments for ccRCC and other types of cancer.

There is a limitation of utilizing the gene expression dataset for patients with ccRCC from the UCSC Xena database. The most obvious limitation is that there is data imbalance problem, where the M0 samples being 5.44 times larger than the M1 samples. In case of using the miRNA expression profiles, the overall performance for the 10 metrics is better than the use of the gene expression data. For example, using miRNA expression data, the NQE+QSVC model achieved an F1-score of 0.92 and an MCC of 0.68. In comparison, with gene expression data, the scores were lower—0.447 for F1 and 0.304 for MCC. This variance could be attributed to the utilization of the data imbalance package, SMOTE, which likely has a significant influence on classification performance. In fact, the imbalance between the number of M0 and M1 cases is a prevalent issue observed in at least 14 cancer types documented by the Xena database. This limitation affects classification performance even if one chooses a different cancer cohort.

Another limitation is that simulating quantum circuits on classical computers is constrained by scalability issues. Even when encoding features with only a few quantum qubits, the process remains highly time-consuming. A promising solution involves using tensor networks to represent quantum circuits. To address this, we plan to accelerate the QSVC implementation with NVIDIA’s cuQuantum package [77]. We aim to enhance computational efficiency by integrating Tensor Network (TN) technique into the QSVC implementation. TN technique enables efficient simulation by converting quantum circuits into tensor-network representations and optimizing the contraction sequence to reduce both time complexity and memory usage. This future work could demonstrate the potential for scaling to more than 30 qubits using tensor-based optimization, enabling efficient and accurate analysis of high-dimensional biomedical data.

5. Conclusion

We propose a classical-quantum approach for the binary classification problem. To ensure a fair comparison between the SVC/NN and QSVC/QNN classification results, we perform hyperparameter optimization to eliminate any bias towards a particular classification method.

Additionally, we aim to assess whether combining NQE with QSVC works better than combining NQE with SVC or NN on three biomedical datasets. Our results show that the proposed method is effective. To summarize, NQE+QSVC/NQE + QNN exhibits better performance than NQE + SVC/NQE + NN across a range of performance metrices. These results could have potential clinical implication in metastasis biomarker study for precision medicine diagnosis.

Supporting information