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CRC Press eBooks

Quantum Linear Algebra for Machine Learning

Hamid D. Ismail

Published Nov 4, 2026• 0 citations
Publisher

Abstract

This chapter examines quantum linear algebra as a foundational framework for quantum machine learning. Because many core learning tasks, including regression, dimensionality reduction, kernel evaluation, recommendation systems, and representation learning, depend on linear algebraic operations, this chapter explores how these operations can be reformulated in the quantum setting. Emphasis is placed on the shift from explicit numerical computation to coherent manipulation of quantum states and measurement-based access to information. This chapter introduces quantum inner product estimation as a primitive for similarity and kernel evaluation, highlighting the swap test and related overlap-estimation methods. It then develops quantum matrix inversion and the HHL algorithm, emphasizing the preparation of solution states rather than explicit solution vectors and discussing the importance of conditioning and regularization. Quantum singular value decomposition is presented through Hermitian embeddings, phase estimation, block encoding, and quantum singular value transformation, providing a unified framework for spectral filtering and low-rank approximation. This chapter concludes with quantum principal component analysis, in which covariance structure is represented by a density matrix and accessed through quantum phase estimation and density matrix exponentiation. Throughout, practical demonstrations in Qiskit and PennyLane illustrate the algorithms, while careful attention is given to the limitations imposed by state preparation, measurement, noise, and classical data loading.

DOI
10.1201/9781042002818-13

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This chapter examines quantum linear algebra as a foundational framework for quantum machine learning. Because many core learning tasks, including regression, dimensionality reduction, kernel evaluation, recommendation systems, and representation learning, depend on linear algebraic operations, this chapter explores how these operations can be reformulated in the quantum setting. Emphasis is placed on the shift from explicit numerical computation to coherent manipulation of quantum states and measurement-based access to information. This chapter introduces quantum inner product estimation as a primitive for similarity and kernel evaluation, highlighting the swap test and related overlap-estimation methods. It then develops quantum matrix inversion and the HHL algorithm, emphasizing the preparation of solution states rather than explicit solution vectors and discussing the importance of conditioning and regularization. Quantum singular value decomposition is presented through Hermitian embeddings, phase estimation, block encoding, and quantum singular value transformation, providing a unified framework for spectral filtering and low-rank approximation. This chapter concludes with quantum principal component analysis, in which covariance structure is represented by a density matrix and accessed through quantum phase estimation and density matrix exponentiation. Throughout, practical demonstrations in Qiskit and PennyLane illustrate the algorithms, while careful attention is given to the limitations imposed by state preparation, measurement, noise, and classical data loading.
This chapter introduces the principal quantum algorithms that form the computational foundation of quantum machine learning (QML). It presents the Quantum Fourier Transform and quantum phase estimation as the core spectral tools for extracting eigenvalue and phase information and then examines Grover’s algorithm, amplitude amplification, and amplitude estimation as search- and probability-based primitives relevant to optimization, expectation estimation, and probabilistic inference. This chapter also develops phase kickback as a unifying mechanism for transferring information into measurable relative phases, highlighting its role in phase oracles, eigenvalue estimation, and phase-based data encoding. This chapter further discusses the Harrow–Hassidim–Lloyd (HHL) algorithm for quantum linear systems and places it in the broader setting of modern quantum linear algebra, including block encoding, quantum singular value transformation, and improved Hamiltonian-simulation techniques. These methods are connected to machine learning tasks such as regression, kernel methods, dimensionality reduction, and graph-based learning. Throughout, the discussion emphasizes that these algorithms function as reusable primitives within hybrid quantum-classical pipelines rather than as standalone learning systems and that their practical value depends critically on assumptions about state preparation, matrix access, conditioning, and measurement overhead.
Most quantum algorithms offering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. In spite of the linear algebraic nature of the postulates of quantum mechanics, until recent work by Lloyd and coauthors cite{LMR13, LMR13a, LMR13b} no quantum algorithms achieving speedups for linear algebra or machine learning had been proposed. A quantum machine learning algorithm must address three issues: encoding of classical data into a succinct quantum representation, processing the quantum representation and extraction of classically useful information from the processed quantum state. In this dissertation, we make progress on all three aspects of the quantum machine learning problem and obtain quantum algorithms for low rank approximation and regularized least squares. The oracle QRAMQRAMQRAM, the standard model studied in quantum query complexity, requires time O(sqrtn)O(sqrt{n})O(sqrtn) to encode vectors vinRnv in R^{n}vinRn into quantum states. We propose simple hardware augmentations to the oracle QRAMQRAMQRAM, that enable vectors vinRnv in R^{n}vinRn to be encoded in time O(logn)O(log n)O(logn), with pre-processing. The augmented QRAMQRAMQRAM incurs minimal hardware overheads, the pre-processing can be parallelized and is a flexible model that allows storage of multiple vectors and matrices. It provides a framework for designing quantum algorithms for linear algebra and machine learning. Using the augmented QRAMQRAMQRAM for vector state preparation, we present two different algorithms for singular value estimation where given singular vector ketvket{v}ketv for AinRmtimesnA in R^{mtimes n}AinRmtimesn, the singular value sigmaisigma_{i}sigmai​ is estimated within additive error epsilonnormAFepsilon norm{A}_{F}epsilonnormAF​. The first algorithm requires time wt1/epsilon3wt{1/epsilon^{3}}wt1/epsilon3 and uses the approach for simulating e−irhoe^{-i rho}e−irho in cite{LMR13}. However, the analysis cite{LMR13} does not establish the coherence of outputs, we provide a qualitatively different analysis that uses the quantum Zeno effect to establish coherence and reveals the probabilistic nature of the simulation technique. The second algorithm has a running time wt1/epsilonwt{1/epsilon}wt1/epsilon and uses Jordan's lemma from linear algebra and the augmented QRAMQRAMQRAM to implement reflections. We use quantum singular value estimation to obtain algorithms for low rank approximation by column selection, the algorithms are based on importance sampling from the leverage score distribution. We obtain quadratic speedups for a large class of linear algebra algorithms that rely on importance sampling from the leverage score distribution including approximate least squares and CXCXCX and CURCURCUR decompositions. Classical algorithms for these problems require time O(mnlogn+poly(1/epsilon))O(mn log n + poly(1/epsilon))O(mnlogn+poly(1/epsilon)), the quantum algorithms have running time O(sqrtmpoly(1/epsilon,k,Delta))O(sqrt{m}poly(1/epsilon, k, Delta))O(sqrtmpoly(1/epsilon,k,Delta)) where k,Deltak, Deltak,Delta are the rank and spectral gap. The running time of the quantum CXCXCX decomposition algorithm does not depend on mmm, it is polynomial in problem parameters. We also provide quantum algorithms for ell2ell_{2}ell2​ regularized regression problems, the quantum ridge regression algorithm requires time wt1/mu2deltawt{1/mu^{2} delta}wt1/mu2delta to output a quantum state that is deltadeltadelta close to the solution, where mumumu is the regularization parameter.
Most quantum algorithms offering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. In spite of the linear algebraic nature of the postulates of quantum mechanics, until recent work by Lloyd and coauthors cite{LMR13, LMR13a, LMR13b} no quantum algorithms achieving speedups for linear algebra or machine learning had been proposed. A quantum machine learning algorithm must address three issues: encoding of classical data into a succinct quantum representation, processing the quantum representation and extraction of classically useful information from the processed quantum state. In this dissertation, we make progress on all three aspects of the quantum machine learning problem and obtain quantum algorithms for low rank approximation and regularized least squares. The oracle QRAMQRAMQRAM, the standard model studied in quantum query complexity, requires time O(sqrtn)O(sqrt{n})O(sqrtn) to encode vectors vinRnv in R^{n}vinRn into quantum states. We propose simple hardware augmentations to the oracle QRAMQRAMQRAM, that enable vectors vinRnv in R^{n}vinRn to be encoded in time O(logn)O(log n)O(logn), with pre-processing. The augmented QRAMQRAMQRAM incurs minimal hardware overheads, the pre-processing can be parallelized and is a flexible model that allows storage of multiple vectors and matrices. It provides a framework for designing quantum algorithms for linear algebra and machine learning. Using the augmented QRAMQRAMQRAM for vector state preparation, we present two different algorithms for singular value estimation where given singular vector ketvket{v}ketv for AinRmtimesnA in R^{mtimes n}AinRmtimesn, the singular value sigmaisigma_{i}sigmai​ is estimated within additive error epsilonnormAFepsilon norm{A}_{F}epsilonnormAF​. The first algorithm requires time wt1/epsilon3wt{1/epsilon^{3}}wt1/epsilon3 and uses the approach for simulating e−irhoe^{-i rho}e−irho in cite{LMR13}. However, the analysis cite{LMR13} does not establish the coherence of outputs, we provide a qualitatively different analysis that uses the quantum Zeno effect to establish coherence and reveals the probabilistic nature of the simulation technique. The second algorithm has a running time wt1/epsilonwt{1/epsilon}wt1/epsilon and uses Jordan's lemma from linear algebra and the augmented QRAMQRAMQRAM to implement reflections. We use quantum singular value estimation to obtain algorithms for low rank approximation by column selection, the algorithms are based on importance sampling from the leverage score distribution. We obtain quadratic speedups for a large class of linear algebra algorithms that rely on importance sampling from the leverage score distribution including approximate least squares and CXCXCX and CURCURCUR decompositions. Classical algorithms for these problems require time O(mnlogn+poly(1/epsilon))O(mn log n + poly(1/epsilon))O(mnlogn+poly(1/epsilon)), the quantum algorithms have running time O(sqrtmpoly(1/epsilon,k,Delta))O(sqrt{m}poly(1/epsilon, k, Delta))O(sqrtmpoly(1/epsilon,k,Delta)) where k,Deltak, Deltak,Delta are the rank and spectral gap. The running time of the quantum CXCXCX decomposition algorithm does not depend on mmm, it is polynomial in problem parameters. We also provide quantum algorithms for ell2ell_{2}ell2​ regularized regression problems, the quantum ridge regression algorithm requires time wt1/mu2deltawt{1/mu^{2} delta}wt1/mu2delta to output a quantum state that is deltadeltadelta close to the solution, where mumumu is the regularization parameter.
This chapter reviews the principal concepts and computational structures of classical machine learning that provide the foundation for quantum machine learning (QML). It examines linear models, support vector machines, principal component analysis, neural networks, probabilistic learning, generative modeling, and optimization, with emphasis on the linear algebraic, statistical, and variational principles that reappear in quantum algorithms. Particular attention is given to computational bottlenecks such as matrix inversion, eigenvalue decomposition, kernel evaluation, sampling, and high-dimensional optimization since these are among the areas where quantum methods may offer advantages under suitable assumptions. This chapter also discusses classes of learning problems that are most plausibly positioned to benefit from quantum speedups and highlights the importance of data access models, encoding assumptions, and realistic complexity analysis in assessing the promise of QML.
This chapter develops the theory and practice of quantum feature maps and quantum kernels as a central framework in quantum machine learning (QML). Building on the data-encoding methods introduced in the previous chapter, it explains how classical inputs are embedded into quantum states and how similarity between data points is then defined through overlaps in Hilbert space. This chapter introduces the quantum kernel as the squared overlap between encoded states, establishes its fundamental algebraic properties, and shows how it satisfies the conditions required for direct use in classical kernel-learning methods such as support vector machines, kernel ridge regression, and Gaussian process models. A major focus is placed on the circuit-level realization and estimation of quantum kernels. This chapter presents kernel evaluation through encoding unitaries and examines three principal estimation methods: the SWAP test, the Hadamard test, and the direct return-probability method, with attention to their trade-offs in qubit overhead, circuit depth, and suitability for NISQ hardware. It also develops the geometric interpretation of quantum kernels in Hilbert space, relating kernel values to angular separation, decision boundaries, expressivity, and generalization. Together, these ideas provide a unified foundation for understanding how quantum circuits, feature maps, and classical learning algorithms interact in hybrid quantum-classical kernel models.