Quantum Kernels Research Articles

Combined influence of quantum iterative reconstruction level and kernel sharpness on image quality in photon counting CT angiography of the upper leg.

Aim was to evaluate the influence of different quantum iterative reconstruction (QIR) levels on the image quality of femoral photon-counting CT angiographies (PCD-CTA).Ultra-high resolution PCD-CTA were obtained from both extremities of five extracorporeally-perfused cadavers using constant tube voltage and maximum radiation dose (71.2 ± 11.0 mGy). Images were reconstructed with three kernels (Bv48, Bv60, Bv76) and the four available levels of QIR. Signal attenuation in the arterial lumen, muscle, and fat were measured. Contrast-to-noise ratios (CNR) and blurring scores were calculated for objective assessment. Six radiologists evaluated the subjective image quality using a pairwise comparison tool.Higher QIR level resulted in a decisive image noise reduction, especially with sharper convolution kernels (Bv60: Q1 11.5 ± 6.3 HU vs. Q4 8.4 ± 2.6 HU; p < 0.001). Largest improvement of CNR was recorded with ultra-sharp reconstructions (Bv76: Q1 20.2 ± 4.4 vs. Q4 28.0 ± 3.5; p < 0.001). Blurring decreased with higher QIR levels for soft Bv48, remained constant for medium Bv60, and increased for sharp Bv76 reconstructions. Subjective QIR level preference varied kernel depending, preferred combinations were: Bv48/Q4, Bv60/Q2, Bv76/Q3. Interrater agreement was excellent.Sharp kernels benefited most from noise reduction of higher QIR levels in lower extremity PCD-CTA. In sum, QIR level 3 provided the best objective and subjective image quality results.

Near-optimal quantum kernel principal component analysis

Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, $D$, and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in $\log(D)$, with $D$ being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.

Phase-space negativity as a computational resource for quantum kernel methods

Phase-space negativity as a computational resource for quantum kernel methods

Quantum support vector machines: theory and applications

Abstract. Quantum Support Vector Machines (QSVMs) combine the fundamental principles of quantum computing and classical Support Vector Machines (SVMs) to improve machine learning performance. In this paper, the author will further explore QSVM. Firstly, introduce the basics of classical SVM, including hyperplane, margin, support vector, and kernel methods. Then, introduce the basic theories of quantum computing, including quantum bits, entanglement, quantum states, superposition, and some related quantum algorithms. Focuses on the concept of QSVM, quantum kernel methods, and how SVM runs on a quantum computer. Key topics include quantum state preparation, measurement, and output interpretation. Theoretical advantages of QSVMs, such as faster computation speed, stronger performance to process high-dimensional data, and kernal computation. In addition, the author discussed the implementation of QSVM, quantum algorithms, quantum gradient descent, and optimization techniques for SVM training. The article also discusses practical issues such as error mitigation and quantum hardware requirements. The purpose of this paper is to show the advantages of QSVM by comparing SVM and QSVM.

Enhanced QSVM with elitist non‐dominated sorting genetic optimisation algorithm for breast cancer diagnosis

AbstractMachine learning has emerged as a promising method for predicting breast cancer using quantum computation techniques. Quantum machine learning algorithms, such as quantum support vector machines (QSVMs), are demonstrating superior efficiency and economy in tackling complex problems compared to traditional machine learning methods. When compared with classical support vector machine, the quantum machine produces remarkably accurate results. The suggested quantum SVM model in this study effectively resolved the binary classification problem for diagnosing malignant breast cancer. This work introduces an enhanced approach to breast cancer diagnosis by integrating QSVM with elitist non‐dominated sorting genetic optimization (ENSGA), leveraging the strengths of both techniques to achieve more accurate and efficient classification results. ENSGA plays a crucial role in optimising QSVM parameters, ensuring that the model attains the best possible classification accuracy while considering multiple objectives simultaneously. Moreover, the quantum kernel estimation method demonstrated exceptional performance by achieving high accuracy within an impressive execution time of 0.14 in the IBM QSVM simulator. The seamless integration of quantum computation techniques with optimisation strategies such as ENSGA highlights the potential of quantum machine learning in revolutionising the field of healthcare, particularly in the domain of breast cancer diagnosis.

An advanced quantum support vector machine for power quality disturbance detection and identification

Quantum algorithms have demonstrated extraordinary potential across numerous fields, offering significant advantages in solving practical problems. Power Quality Disturbances (PQDs) have always been a critical factor affecting the stability and safety of electrical power systems, and accurately detecting and identifying PQDs is crucial for ensuring reliable system operation. This paper explores the application of quantum algorithms in the field of power quality and proposes a novel method using Quantum Support Vector Machines (QSVM) to detect and identify PQDs, which marks the first application of QSVM in PQD analysis. The QSVM model employed involves three main stages: quantum feature mapping, quantum kernel computation, and model training. Quantum feature mapping uses quantum circuits to map classical data into a high-dimensional Hilbert space, enhancing feature separability. Quantum kernel computation calculates the inner products between features for model training. Rigorous theoretical and experimental analyses validate our approach. This method achieves a time complexity of O(N2log(N))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$O(N^{2} \\log (N))$\\end{document}, superior to classical SVM algorithms. Simulation results show high accuracy in PQDs detection, achieving a 100% detection rate and a 96.25% accuracy rate in single PQD identification. Experimental outcomes demonstrate robustness, maintaining over 87% accuracy even with increased noise levels, confirming its effectiveness in PQDs detection and identification.

Efficient Parameter Optimisation for Quantum Kernel Alignment: A Sub-sampling Approach in Variational Training

Quantum machine learning with quantum kernels for classification problems is a growing area of research. Recently, quantum kernel alignment techniques that parameterise the kernel have been developed, allowing the kernel to be trained and therefore aligned with a specific dataset. While quantum kernel alignment is a promising technique, it has been hampered by considerable training costs because the full kernel matrix must be constructed at every training iteration. Addressing this challenge, we introduce a novel method that seeks to balance efficiency and performance. We present a sub-sampling training approach that uses a subset of the kernel matrix at each training step, thereby reducing the overall computational cost of the training. In this work, we apply the sub-sampling method to synthetic datasets and a real-world breast cancer dataset and demonstrate considerable reductions in the number of circuits required to train the quantum kernel while maintaining classification accuracy.

Metaheuristic optimization scheme for quantum kernel classifiers using entanglement‐directed graphs

AbstractEntanglement is crucial for achieving quantum advantages. However, in the context of quantum machine learning, existing optimization strategies for generating quantum classifier circuits often result in unentangled circuits, indicating an underutilization of the entanglement effect needed to learn complex patterns. In this study, we proposed a novel metaheuristic approach—genetic algorithm—for designing a quantum kernel classifier that incorporates expressive entanglement. This classifier utilizes a loopless entanglement‐directed graph, where each directed edge represents the entanglement between the target and control qubits. The proposed method consistently outperforms classical and quantum baselines across various artificial and actual datasets, achieving improvements up to 32.4% and 17.5%, respectively, compared with the best model among all other baselines. Moreover, this method successfully reconstructs the hidden entanglement structures underlying artificial datasets. The results also demonstrate that the optimized circuits exhibit diverse entanglement variations across different datasets, indicating the versatility of the proposed approach.

Quantum inspired kernel matrices: Exploring symmetry in machine learning

Quantum machine learning, an emerging field at the intersection of quantum computing and classical machine learning, has shown great promise in enhancing computational capabilities beyond classical bounds. A key element in this area of research is the utilization of Quantum Kernel Estimators, traditionally grounded in the symmetries of SU(2) groups associated with qubits. This study extends the conceptual framework of Quantum Kernel Estimators to incorporate a broader spectrum of symmetry groups. By harnessing the various structures of Lie groups, we develop novel quantum-inspired feature maps that offer more flexible and potentially powerful ways to encode and compress classical data into quantum states. We present a comprehensive theoretical introduction for this approach, followed by a methodology that integrates the developed feature maps into quantum-inspired kernel classifiers. Our results, derived from a series of computational experiments across various datasets, demonstrate the efficacy of this approach in comparison to traditional quantum and classical machine learning models. The findings not only underline the versatility of Lie-group theory in potentially enhancing quantum machine learning algorithms but also open new avenues for exploring complex symmetries in quantum information processing. This research bridges a gap between the study of symmetries and machine learning, paving the way for more sophisticated quantum algorithms capable of tackling complex, high-dimensional data in ways previously unattainable.

Benchmarking of quantum fidelity kernels for Gaussian process regression

Abstract Quantum computing algorithms have been shown to produce performant quantum kernels for machine-learning classification problems. Here, we examine the performance of quantum kernels for regression problems of practical interest. For an unbiased benchmarking of quantum kernels, it is necessary to construct the most optimal functional form of the classical kernels and the most optimal quantum kernels for each given data set. We develop an algorithm that uses an analog of the Bayesian information criterion to optimize the sequence of quantum gates used to estimate quantum kernels for Gaussian process models. The algorithm increases the complexity of the quantum circuits incrementally, while improving the performance of the resulting kernels, and is shown to yield much higher model accuracy with fewer quantum gates than a fixed quantum circuit ansatz. We demonstrate that quantum kernels thus obtained can be used to build accurate models of global potential energy surfaces (PES) for polyatomic molecules. The average interpolation error of the six-dimensional PES obtained with a random distribution of 2000 energy points is 16 cm−1 for H3O+, 15 cm−1 for H2CO and 88 cm−1 for HNO2. We show that a compositional optimization of classical kernels for Gaussian process regression converges to the same errors. This indicates that quantum kernels can achieve the same, though not better, expressivity as classical kernels for regression problems.

Quantum tangent kernel

The quantum kernel method is one of the key approaches to quantum machine learning, which has the advantage of not requiring optimization and its theoretical simplicity. By virtue of these properties, several experimental demonstrations and discussions of the potential advantages have been developed so far. However, as is the case in classical machine learning, not all quantum machine learning models could be regarded as kernel methods. In this work, we explore a quantum machine learning model with a deep parametrized quantum circuit and aim to go beyond the conventional quantum kernel method. In this case, the expressive power and performance are expected to be enhanced, while the training process might be a bottleneck because of the barren plateaus. Moreover, the high computational cost of gradient-based optimization and the large search space of the gradient-free optimization directly make the training intractable. However, we find that parameters of a deep enough quantum circuit do not move much from their initial values during training, allowing for a first-order expansion with respect to the parameters. This behavior is similar to that of the neural tangent kernel in classical machine learning, and such a quantum machine learning with deep variational quantum circuits can be described by another emergent kernel, the . We show that the proposed quantum tangent kernel has the potential to outperform the conventional quantum kernel method by performing a classification task on an ansatz-generated dataset. This work provides a different direction beyond the conventional quantum kernel method and explores the potential power of quantum machine learning with deep parametrized quantum circuits. Published by the American Physical Society 2024

Continuous-variable quantum kernel method on a programmable photonic quantum processor

Continuous-variable quantum kernel method on a programmable photonic quantum processor

QKSAN: A Quantum Kernel Self-Attention Network.

The Self-Attention Mechanism (SAM) excels at distilling important information from the interior of data to improve the computational efficiency of models. Nevertheless, many Quantum Machine Learning (QML) models lack the ability to distinguish the intrinsic connections of information like SAM, which limits their effectiveness on massive high-dimensional quantum data. To tackle the above issue, a Quantum Kernel Self-Attention Mechanism (QKSAM) is introduced to combine the data representation merit of Quantum Kernel Methods (QKM) with the efficient information extraction capability of SAM. Further, a Quantum Kernel Self-Attention Network (QKSAN) framework is proposed based on QKSAM, which ingeniously incorporates the Deferred Measurement Principle (DMP) and conditional measurement techniques to release half of quantum resources by mid-circuit measurement, thereby bolstering both feasibility and adaptability. Simultaneously, the Quantum Kernel Self-Attention Score (QKSAS) with an exponentially large characterization space is spawned to accommodate more information and determine the measurement conditions. Eventually, four QKSAN sub-models are deployed on PennyLane and IBM Qiskit platforms to perform binary classification on MNIST and Fashion MNIST, where the QKSAS tests and correlation assessments between noise immunity and learning ability are executed on the best-performing sub-model. The paramount experimental finding is that the QKSAN subclasses possess the potential learning advantage of acquiring impressive accuracies exceeding 98.05% with far fewer parameters than classical machine learning models. Predictably, QKSAN lays the foundation for future quantum computers to perform machine learning on massive amounts of data while driving advances in areas such as quantum computer vision.

A hyperparameter study for quantum kernel methods

Quantum kernel methods are a promising method in quantum machine learning thanks to the guarantees connected to them. Their accessibility for analytic considerations also opens up the possibility of prescreening datasets based on their potential for a quantum advantage. To do so, earlier works developed the geometric difference, which can be understood as a closeness measure between two kernel-based machine learning approaches, most importantly between a quantum kernel and a classical kernel. This metric links the quantum and classical model complexities, and it was developed to bound generalization error. Therefore, it raises the question of how this metric behaves in an empirical setting. In this work, we investigate the effects of hyperparameter choice on the model performance and the generalization gap between classical and quantum kernels. The importance of hyperparameters is well known also for classical machine learning. Of special interest are hyperparameters associated with the quantum Hamiltonian evolution feature map, as well as the number of qubits to trace out before computing a projected quantum kernel. We conduct a thorough investigation of the hyperparameters across 11 datasets, and we identify certain aspects that can be exploited. Analyzing the effects of certain hyperparameter settings on the empirical performance, as measured by cross validation accuracy, and generalization ability, as measured by geometric difference described above, brings us one step closer to understanding the potential of quantum kernel methods on classical datasets.

Quadratic Speed‐ups in Quantum Kernelized Binary Classification

AbstractClassification is at the core of data‐driven prediction and decision‐making, representing a fundamental task in supervised machine learning. Recently, several quantum machine learning algorithms that use quantum kernels as a measure of similarities between data have emerged to perform binary classification on datasets encoded as quantum states. The potential advantages of quantum kernels arise from the ability of quantum computers to construct kernels that are more effective than their classical counterparts in capturing patterns in data or computing kernels more efficiently. However, existing quantum kernel‐based classification algorithms do not harness the capability of having data samples in quantum superposition for additional enhancements. This work demonstrates how such capability can be leveraged in quantum kernelized binary classifiers (QKCs) through Quantum Amplitude Estimation (QAE) for quadratic speed‐up. Additionally, new quantum circuits are proposed for the QKCs in which the number of qubits is reduced by one, and the circuit depth is reduced linearly with respect to the number of sample data. The quadratic speed‐up over previous methods is verified through numerical simulations on the Iris dataset.

Leveraging Quantum Kernel Support Vector Machine for breast cancer diagnosis from Digital Breast Tomosynthesis images

Leveraging Quantum Kernel Support Vector Machine for breast cancer diagnosis from Digital Breast Tomosynthesis images

Exponential concentration in quantum kernel methods

Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model’s parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the input data. We identify four sources that can lead to concentration including expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.

Quantum Fisher kernel for mitigating the vanishing similarity issue

Quantum kernel (QK) methods exploit quantum computers to calculate QKs for the use of kernel-based learning models. Despite a potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; the exponential decay of the expectation value and the variance of the QK deteriorates implementation feasibility and trainability of the model with the increase of the number of qubits. This implies the need to design QKs alternative to the fidelity-based one. In this work, we propose a new class of QKs called the quantum Fisher kernels (QFKs) that take into account the geometric structure of the data source. We analytically and numerically demonstrate that the QFK can avoid the issue when shallow alternating layered ansatzes are used. In addition, the Fourier analysis numerically elucidates that the QFK can have the expressivity comparable to the fidelity-based QK. Moreover, we demonstrate synthetic classification tasks where QFK outperforms the fidelity-based QK in performance due to the absence of vanishing similarity. These results indicate that QFK paves the way for practical applications of quantum machine learning toward possible quantum advantages.

Quantum kernels for classifying dynamical singularities in a multiqubit system

Dynamical quantum phase transition is a critical phenomenon involving out-of-equilibrium states and broken symmetries without classical analogy. However, when finite-sized systems are analyzed, dynamical singularities of the rate function can appear, leading to a challenging physical characterization when parameters are changed. Here, we report a quantum support vector machine algorithm that uses quantum Kernels to classify dynamical singularities of the rate function for a multiqubit system. We illustrate our approach using N long-range interacting qubits subjected to an arbitrary magnetic field, which induces a quench dynamics. Inspired by physical arguments, we introduce two different quantum Kernels, one inspired by the ground state manifold and the other based on a single state tomography. Our accuracy and adaptability results show that this quantum dynamical critical problem can be efficiently solved using physically inspiring quantum Kernels. Moreover, we extend our results for the case of time-dependent fields, quantum master equation, and when we increase the number of qubits.

Dequantizing quantum machine learning models using tensor networks

Ascertaining whether a classical model can efficiently replace a given quantum model——is crucial in assessing the true potential of quantum algorithms. In this work, we introduced the dequantizability of the function class of variational quantum-machine-learning (VQML) models by employing the tensor network formalism, effectively identifying every VQML model as a subclass of matrix product state (MPS) model characterized by constrained coefficient MPS and tensor product-based feature maps. From this formalism, we identify the conditions for which a VQML model's function class is dequantizable or not. Furthermore, we introduce an efficient quantum kernel-induced classical kernel which is as expressive as given any quantum kernel, hinting at a possible way to dequantize quantum kernel methods. This presents a thorough analysis of VQML models and demonstrates the versatility of our tensor-network formalism to properly distinguish VQML models according to their genuine quantum characteristics, thereby unifying classical and quantum machine-learning models within a single framework. Published by the American Physical Society 2024