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arXiv:2305.04287 · Quantum Machine Intelligence

A quantum k-nearest neighbors algorithm based on the Euclidean distance estimation

Enrico Zardini, Enrico Blanzieri, Davide Pastorello

Published Apr 23, 2024• 41 citations
arXivHTMLPDF

Abstract

Abstract The k-nearest neighbors (k-NN) is a basic machine learning (ML) algorithm, and several quantum versions of it, employing different distance metrics, have been presented in the last few years. Although the Euclidean distance is one of the most widely used distance metrics in ML, it has not received much consideration in the development of these quantum variants. In this article, a novel quantum k-NN algorithm based on the Euclidean distance is introduced. Specifically, the algorithm is characterized by a quantum encoding requiring a low number of qubits and a simple quantum circuit not involving oracles, aspects that favor its realization. In addition to the mathematical formulation and some complexity observations, a detailed empirical evaluation with simulations is presented. In particular, the results have shown the correctness of the formulation, a drop in the performance of the algorithm when the number of measurements is limited, the competitiveness with respect to some classical baseline methods in the ideal case, and the possibility of improving the performance by increasing the number of measurements.

DOI
10.1007/s42484-024-00155-2

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Much attention has recently been devoted to the k-nearest neighbors (KNN) algorithm as a case study for quantum computing algorithms. Several algorithms have devised methods to compute a similarity metric to solve Quantum KNN (QKNN). This paper presents a practical quantum encoding technique that enables Euclidean distance calculation between vectors with different L2 norms by appending a single component to each vector, then uses it in the QKNN problem. It is arguably simpler to understand and implement than comparable methods and uses fewer qubits. Further, an implementation of this algorithm is provided and is used to benchmark it against a recent Euclidean-distance QKNN method on standard labeled datasets. In addition to demonstrating proper formulation, the results show marked improvement over similar implementations during simulation with realistic noise due to the use of fewer qubits and thus more coherent circuits.
We present several quantum algorithms for performing nearest-neighbor learning. At the core of our algorithms are fast and coherent quantum methods for computing distance metrics such as the inner product and Euclidean distance. We prove upper bounds on the number of queries to the input data required to compute these metrics. In the worst case, our quantum algorithms lead to polynomial reductions in query complexity relative to the corresponding classical algorithm. In certain cases, we show exponential or even super-exponential reductions over the classical analog. We study the performance of our quantum nearest-neighbor algorithms on several real-world binary classification tasks and find that the classification accuracy is competitive with classical methods.
This paper presents a novel quantum K-nearest neighbors (QKNN) algorithm, which offers improved performance over the classical k-NN technique by incorporating quantum computing (QC) techniques to enhance classification accuracy, scalability, and robustness. The proposed modifications focus on optimizing quantum data encoding using Hadamard and rotation gates, ensuring more effective rendering of classical data in quantum states. In addition, the quantum feature extraction process is significantly enhanced by the use of entangled gates such as IsingXY and CNOT, which enables better feature interactions and class separability. A novel quantum distance metric, based on the swap test, is introduced to calculate similarity measures between various quantum states, offering superior accuracy and computational efficiency compared to traditional Euclidean distance metrics. We assess the achievement of the proposed QKNN algorithm on three benchmark datasets: Wisconsin Breast Cancer, Iris, and Bank Note Authentication, and have noted its superior performance relative to both classical k-NN (CKNN) and Quantum Neural Network (QNN). The proposed QKNN algorithm is found to achieve prediction accuracies of 98.25%, 100%, and 99.27% ,respectively, for the three datasets, while the customized QNN shows prediction accuracies of only 97.17%, 83.33%, and 86.18%, respectively. Furthermore, we address the challenges of quantum noise by incorporating a repetition encoding-based error mitigation strategy, which ensures the stability and resilience of the algorithm in noisy quantum environments. The results highlight the potential of the proposed QKNN as a scalable, efficient and robust quantum-enhanced machine learning algorithm, especially in high-dimensional and complex datasets, when traditional approaches frequently fail.
We present quantum algorithms for performing nearest-neighbor learning and kkk--means clustering. At the core of our algorithms are fast and coherent quantum methods for computing the Euclidean distance both directly and via the inner product which we couple with methods for performing amplitude estimation that do not require measurement. We prove upper bounds on the number of queries to the input data required to compute such distances and find the nearest vector to a given test example. In the worst case, our quantum algorithms lead to polynomial reductions in query complexity relative to Monte Carlo algorithms. We also study the performance of our quantum nearest-neighbor algorithms on several real-world binary classification tasks and find that the classification accuracy is competitive with classical methods.
Abstract Over the past few years, we observed a rethinking of classical artificial intelligence algorithms from a quantum computing perspective. This trend is driven by the peculiar properties of quantum mechanics, which offer the potential to enhance artificial intelligence capabilities, enabling it to surpass the constraints of classical computing. However, redesigning classical algorithms into their quantum equivalents is not straightforward and poses numerous challenges. In this study, we analyze in-depth two orthogonal designs of the Quantum K-Nearest Neighbor classifier. In particular, we show two solutions based on amplitude encoding and basis encoding of data, respectively. These two types of encoding impact the overall structure of the respective algorithms, which employ different distance metrics and show different performances. By breaking down each quantum algorithm, we clarify implementation aspects ranging from data preparation to classification. Eventually, we discuss the difficulties associated with data preparation, the theoretical advantage of quantum algorithms, and their impact on performance with respect to the classical counterpart.