We give an algorithm that solves the problem of pattern classification on a quantum computer, effectively performing linear regression with least squares optimisation. Opposed to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on prediction for machine learning tasks and the result can be accessed through a single qubit measurement. The quantum algorithm is logarithmic in the dimension of the input space and independent of the size of the dataset if the data is given as quantum information.
We give an algorithm that solves the problem of pattern classification on a quantum computer, effectively performing linear regression with least squares optimisation. Opposed to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on prediction for machine learning tasks and the result can be accessed through a single qubit measurement. The quantum algorithm is logarithmic in the dimension of the input space and independent of the size of the dataset if the data is given as quantum information.
We give an algorithm for prediction on a quantum computer which is based on a linear regression model with least-squares optimization. In contrast to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on the machine-learning task of guessing the output corresponding to a new input given examples of data points. Furthermore, we adapt the algorithm to process nonsparse data matrices that can be represented by low-rank approximations, and significantly improve the dependency on its condition number. The prediction result can be accessed through a single-qubit measurement or used for further quantum information processing routines. The algorithm's runtime is logarithmic in the dimension of the input space provided the data is given as quantum information as an input to the routine.
We present a quantum algorithm for fitting a linear regression model to a given data set using the least-squares approach. Differently from previous algorithms which yield a quantum state encoding the optimal parameters, our algorithm outputs these numbers in the classical form. So by running it once, one completely determines the fitted model and then can use it to make predictions on new data at little cost. Moreover, our algorithm works in the standard oracle model, and can handle data sets with nonsparse design matrices. It runs in time $\text{poly}({log}_{2}(N),d,\ensuremath{\kappa},1/\ensuremath{\epsilon})$, where N is the size of the data set, d is the number of adjustable parameters, $\ensuremath{\kappa}$ is the condition number of the design matrix, and $\ensuremath{\epsilon}$ is the desired precision in the output. We also show that the polynomial dependence on d and $\ensuremath{\kappa}$ is necessary. Thus, our algorithm cannot be significantly improved. Furthermore, we also give a quantum algorithm that estimates the quality of the least-squares fit (without computing its parameters explicitly). This algorithm runs faster than the one for finding this fit, and can be used to check whether the given data set qualifies for linear regression in the first place.
Abstract Linear regression is one of the most important and common analytical methods in mathematical statistics. The letter studies a general model of linear regression problem based on least squares method, and investigates the impact of quantum algorithms on the time complexity of solving linear regression problem when quantum algorithms can be implemented on quantum computers. With the help of the general model of linear regression problem in terms of least squares method, we propose a novel simplified quantum scheme for solving linear regression equation based on the sparsity-independent quantum singular value estimation algorithm. For a linear regression equation with dimension n, our scheme can reduce the time complexity from O(Nn 2) to O ( N log N ) when both the condition number κ of related matrices and the reciprocal of precision ∈ are small in size O(play logN), where m is the dimension of the input u and N is the number of samples.
The literature reflects a mutually beneficial relationship between machine learning and quantum computing, where progress in one field frequently drives improvements in the other. Motivated by the rich connection between these areas, we use a resource-efficient and scalable Single-Qubit Quantum Neural Network (SQQNN) for both regression and classification tasks using a new data uploading technique. The SQQNN leverages parameterized single-qubit unitary operators and quantum measurements to achieve efficient learning. To train the model, we use gradient descent for regression tasks. For classification, we introduce a novel training method inspired by polynomial regression, which can efficiently find a global minimizer of the transformed least-squares objective in a single step. This approach significantly accelerates training compared to iterative methods. Evaluated across various applications, the SQQNN exhibits virtually error-free and strong performance in regression and classification tasks, including Wisconsin Breast Cancer and MNIST datasets. These results demonstrate the versatility, scalability, and suitability of the SQQNN for deployment on near-term quantum devices.
Logistic regression (LR) is an important machine learning model for classification, with wide applications in text classification, image analysis and medicine diagnosis, etc. However, training LR generally entails an iterative gradient descent method, and is quite time consuming when processing big data sets. To solve this problem, we present a quantum algorithm for LR to implement the key task of the gradient descent method, obtaining the classical gradients in each iteration. It is shown that our algorithm achieves exponential speedup over its classical counterpart in each iteration when the dimension of each data point M grows polylogarithmically with the number of data points N, i.e.,M=O(polylog N). It is worth noting that the optimal model parameters are finally derived by performing simple calculations on the obtained gradients. So once the optimal model parameters are determined, one can use them to classify new data at little cost.
Quantum computing is revolutionizing artificial intelligence (AI) by offering exponential speed-ups for complex computations. Quantum computing differs from classical computing, which uses bits leverages qubits that enable parallelism and superposition. This advancement enhances AI activities including pattern detection, optimization, and deep learning. By integrating quantum computing into AI, researchers aim to achieve superior computational power, driving breakthroughs in fields like healthcare, finance, and cryptography. In AI applications, machine learning techniques are essential., and their efficiency can be significantly improved using quantum computing. Linear Regression predicts continuous values, Random Forest Regression enhances prediction robustness by averaging multiple decision trees, and Support Vector Machines classify data by finding optimal hyperplanes. Quantum computing accelerates these algorithms by reducing computational complexity and improving model accuracy. Key input parameters for evaluating quantum AI performance include Quantum Bit Processing Speed (GHz), which determines data processing rate, Qubit Coherence Time (µs), which affects stability, and Quantum Gate Fidelity (%), which measures the accuracy of quantum operations. These parameters directly influence AI model efficiency and reliability. The AI Performance Score serves as a comprehensive evaluation metric, assessing computational efficiency, prediction accuracy, and overall robustness of quantum AI models. Keywords: Quantum Computing, Machine Learning, Qubit Coherence Time, AI Performance Score