Abstract As restricted quantum computers become available, research focuses on finding meaningful applications. For example, in quantum machine learning, a special type of quantum circuit called a quantum neural network is one of the most investigated approaches. However, we know little about suitable circuit architectures or important model hyperparameters for a given task. In this work, we apply the functional ANOVA framework to the quantum neural network architectures to analyze which of the quantum machine learning hyperparameters are most influential for their predictive performance. We restrict our study to 7 open-source datasets from the OpenML-CC18 classification benchmark, which are small enough for simulations on quantum hardware with fewer than 20 qubits. Using this framework, three main levels of importance were identified, confirming expected patterns and revealing new insights. For instance, the learning rate is identified as the most important hyperparameter on all datasets, whereas the particular choice of entangling gates used is found to be the least important on all except for one dataset. In addition to identifying the relevant hyperparameters, for each of them, we also learned data-driven priors based on values that perform well on previously seen datasets, which can then be used to steer hyperparameter optimization processes. We utilize these priors in the hyperparameter optimization method hyperband and show that these improve performance against uniform sampling across all datasets by, on average,
0.53% 0.53 % , up to
6.11% 6.11 % , in cross-validation accuracy. We also demonstrate that such improvements hold on average regardless of the configuration hyperband is run with. Our work introduces new methodologies for studying quantum machine learning models toward quantum model selection in practice. All research code is made publicly available.
The convergence of machine learning (ML) and quantum computing is quickly becoming a game-changing new frontier in computational science. The most intriguing area of overlap may be quantum optimization, which seeks to reduce the exponentially increasing complexity and computational cost of ML algorithms. As they grow larger and more complicated, traditional optimization methods, particularly in high-dimensional and non-convex space, will inevitably hit performance limits. Quantum optimization is the whole new paradigm that leverages quantum mechanical effects like superposition, entanglement, and tunnelling to explore solution spaces more efficiently. The chapter outlines the theory, practice, and potential of hybrid quantum-classical optimization algorithms for machine learning. The most brilliant research direction in the current NISQ (noisy intermediate-scale quantum) era is mixed systems leveraging classical computing capability augmented by quantum capacity. In such designs, computationally intensive subroutines like optimization for global optima or escape from local minima are left to quantum processors and data preprocessing, iterative refinement, and model testing are performed by classical processors. We start by defining the fundamental principles of quantum computing and acquainting readers with the fundamentals like qubits, quantum gates, and quantum circuits. These are the building blocks for quantum algorithms, which hopefully can be utilized for optimization, for example, the QAOA (quantum approximate optimization algorithm), the VQE (variational quantum Eigensolver), and quantum annealing. They all build to optimize the problems in some way differently: QAOA through parameterized quantum circuits, VQE through approximations of ground states of the Hamiltonians, and quantum annealing through quantum evolution to low-energy solutions. We then form a loop between these quantum algorithms and regular machine learning tasks. Optimization impacts virtually all ML phases, from training neural networks, where one reduces loss functions; feature selection, where one aims to discover the most relevant subset of input variables; and hyperparameter optimization, which often involves expensive search procedures over extremely large combinatorial spaces. Quantum optimization techniques can therefore, in principle, accelerate such procedures through computational complexity reduction and enhancing solutions over challenging landscapes. Realistic simulations and case studies emphasize the degree to which hybrid quantum-classical algorithms have been employed in solving ML problems, that is, QSVMs, quantum-enhanced clustering, and quantum reinforcement learning. Despite their experimental nature, these applications already show encouraging results in terms of improved performance metrics as well as reduction in time-to-solution compared to classical-only solutions. Notwithstanding that, this new field is not bereft of real challenges. Quantum hardware suffers from hard limits on qubit fidelity, decoherence, and scalability in the current times. Furthermore, quantum circuit integration into broad ML pipelines requires novel architectures and software platforms. As a counter to those issues, this chapter surveys recent advances in quantum programming languages, simulators, and hybrid platforms like PennyLane, Qiskit, and TensorFlow Quantum. We conclude by looking to the future possibility that quantum optimization will revolutionize the machine learning landscape. As hardware improves and algorithmic complexity increases, the possibility of quantum-powered learning systems solving hard problems becomes increasingly accessible. From speeding up deep network training to enabling new types of unsupervised learning, quantum optimization holds the potential to recast computational tractability in machine learning. This chapter is an all-encompassing introduction and state-of-the-art work for students, researchers, and practitioners interested in using quantum optimization for practical ML applications. It sets the foundation for understanding how to pursue hybrid approaches (how they may be applied today in the real world and bracing oneself for disruption in the future).
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
QRAM, the standard model studied in quantum query complexity, requires time
O(sqrtn) to encode vectors
vinRn into quantum states. We propose simple hardware augmentations to the oracle
QRAM, that enable vectors
vinRn to be encoded in time
O(logn), with pre-processing. The augmented
QRAM 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
QRAM for vector state preparation, we present two different algorithms for singular value estimation where given singular vector
ketv for
AinRmtimesn, the singular value
sigmai is estimated within additive error
epsilonnormAF. The first algorithm requires time
wt1/epsilon3 and uses the approach for simulating
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/epsilon and uses Jordan's lemma from linear algebra and the augmented
QRAM 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
CX and
CUR decompositions. Classical algorithms for these problems require time
O(mnlogn+poly(1/epsilon)), the quantum algorithms have running time
O(sqrtmpoly(1/epsilon,k,Delta)) where
k,Delta are the rank and spectral gap. The running time of the quantum
CX decomposition algorithm does not depend on
m, it is polynomial in problem parameters. We also provide quantum algorithms for
ell2 regularized regression problems, the quantum ridge regression algorithm requires time
wt1/mu2delta to output a quantum state that is
delta close to the solution, where
mu is the regularization parameter.
In this chapter and the next one, we will explore the exciting areas of quantum machine learning and quantum deep learning. Machine learning and deep learning have seen great success in recent years because of the increase in the computational power at our disposal and because of the high-end research in these fields. Quantum machine learning presents an exciting opportunity to increase the computational efficiency of the existing machine learning algorithms as well as presents a way to tackle some of the more computationally intractable problems. In this chapter, we start with the Harrow-Hassidim-Lloyd algorithm, popularly known as HHL, which acts as the matrix inversion routine in the quantum computing domain. Hence, HHL will be the default choice for algorithms such as linear regression and least square support vector machines. Subsequently, we touch upon quantum linear regression and support vector machines in detail in this chapter. We will then move on to implementing quantum routines such as quantum dot product and quantum Euclidean distances since they are integral to several machine learning algorithms such as the k-means clustering and nearest neighbor methods. In this regard, we will implement the k-means clustering method in detail. Also, we will discuss how Grover’s algorithm can be used to optimize quantum objectives by illustrating its usage in the cluster assignment in the k-means algorithm. Principal component analysis is an important machine learning technique, and we will walk through its quantum implementation in great detail as well.
Ridge regression (RR), also called regularized linear regression, is an important machine learning technique which introduces a regularization hyperparameter to ordinary multiple linear regression for analyzing data suffering from multicollinearity. Here we provide an efficient quantum algorithm for RR. Specifically, by giving the technique of parallel Hamiltonian simulation that can simulate a number of Hermitian matrices in parallel, we develop a quantum version of
K-fold cross-validation approach that can efficiently estimate the predictive performance of RR. Our algorithm involves two phases: (1) we first use the quantum
K-fold cross-validation to efficiently determine an approximately optimal regularization hyperparameter for RR with which RR can achieve very good predictive performance, and (2) then generate a quantum state encoding the optimal fitting parameters of RR with such hyperparameter, which can be further utilized to predict new data. Our algorithm can handle nonsparse data matrices, and is exponentially faster than the classical algorithm for (low-rank) design matrices with relatively small elements and low condition numbers.
Quantum Machine Learning (QML) has emerged as a promising field, offering potential advantages over classical machine learning in terms of speed, scalability, and the ability to model complex systems. One significant challenge lies in optimizing model parameters—a crucial step in achieving optimal performance. Traditional parameter tuning methods often rely on grid search, random search, or gradient-based optimization, which can be computationally expensive and time-consuming. This paper introduces a novel quantum machine learning parameter adjustment algorithm, leveraging quantum computing's unique properties to automate the parameter optimization process. The proposed algorithm utilizes a quantum circuit to efficiently explore the parameter space, leading to improved model accuracy and generalization capabilities. This work explores the potential of quantum computing to revolutionize the parameter tuning process, offering a more efficient and robust approach compared to classical methods. The core mechanism involves quantum state manipulation to represent and update model parameters, achieving a significant improvement in performance.