Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension

We construct an efficient classical analogue of the quantum matrix inversion algorithm (HHL) for low-rank matrices. Inspired by recent work of Tang, assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix and sample from the solution to the problem $Ax=b$ using fast sampling techniques. We implement the pseudo-inverse by finding an approximate singular value decomposition of $A$ via subsampling, then inverting the singular values. In principle, the approach can also be used to apply any desired "smooth" function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem, our result suggests that more low-rank quantum algorithms can be effectively "dequantised" into classical length-square sampling algorithms.Comment: 10 page

Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension

We construct an efficient classical analogue of the quantum matrix inversion algorithm [HHL09] for low-rank matrices. Inspired by recent work of Tang [Tan18a], assuming length-square sampling access to input data, we implement the pseudo-inverse of a low-rank matrix and sample from the solution to the problem Ax = b using fast sampling techniques. We implement th

Quantum-Inspired Support Vector Machine

Support vector machine (SVM) is a particularly powerful and flexible supervised learning model that analyzes data for both classification and regression, whose usual algorithm complexity scales polynomially with the dimension of data space and the number of data points. To tackle the big data challenge, a quantum SVM algorithm was proposed, which is claimed to achieve exponential speedup for least squares SVM (LS-SVM). Here, inspired by the quantum SVM algorithm, we present a quantum-inspired classical algorithm for LS-SVM. In our approach, a improved fast sampling technique, namely indirect sampling, is proposed for sampling the kernel matrix and classifying. We first consider the LS-SVM with a linear kernel, and then discuss the generalization of our method to non-linear kernels. Theoretical analysis shows our algorithm can make classification with arbitrary success probability in logarithmic runtime of both the dimension of data space and the number of data points for low rank, low condition number and high dimensional data matrix, matching the runtime of the quantum SVM

Quantum-inspired algorithm for direct multi-class classification

Over the last few decades, quantum machine learning has emerged as a groundbreaking discipline. Harnessing the peculiarities of quantum computation for machine learning tasks offers promising advantages. Quantum-inspired machine learning has revealed how relevant benefits for machine learning problems can be obtained using the quantum information theory even without employing quantum computers. In the recent past, experiments have demonstrated how to design an algorithm for binary classification inspired by the method of quantum state discrimination, which exhibits high performance with respect to several standard classifiers. However, a generalization of this quantuminspired binary classifier to a multi-class scenario remains nontrivial. Typically, a simple solution in machine learning decomposes multi-class classification into a combinatorial number of binary classifications, with a concomitant increase in computational resources. In this study, we introduce a quantum-inspired classifier that avoids this problem. Inspired by quantum state discrimination, our classifier performs multi-class classification directly without using binary classifiers. We first compared the performance of the quantum-inspired multi-class classifier with eleven standard classifiers. The comparison revealed an excellent performance of the quantum-inspired classifier. Comparing these results with those obtained using the decomposition in binary classifiers shows that our method improves the accuracy and reduces the time complexity. Therefore, the quantum-inspired machine learning algorithm proposed in this work is an effective and efficient framework for multi-class classification. Finally, although these advantages can be attained without employing any quantum component in the hardware, we discuss how it is possible to implement the model in quantum hardware

超高次元データ解析のための量子インスパイア主成分分析の開発

主成分分析は多変量データから重要な低次元成分を抽出する統計手法である．しかし，主成分分析のアルゴリズムは特異値分解に基づくため，次元数（変量の数）が数百万を超えるデータには，計算時間の問題からしばしば適用が困難となる．我々は近年発案された「量子インスパイアアルゴリズム」を用い，計算時間を次元数の対数オーダーに抑えつつ，主成分分析を近似するアルゴリズムを計算機実装した．本報告において，複数の人工データ・実データを用いてその計算時間と性能を評価した結果を紹介する．第44回量子情報技術研究会（QIT44