ISSN 1000-1239 CN 11-1777/TP

计算机研究与发展 ›› 2021, Vol. 58 ›› Issue (9): 1835-1842.doi: 10.7544/issn1000-1239.2021.20210366

所属专题: 2021量子计算专题

• 基础理论 • 上一篇    下一篇



  1. 1(网络与交换技术国家重点实验室(北京邮电大学) 北京 100876);2(密码科学技术国家重点实验室 北京 100878) (
  • 出版日期: 2021-09-01
  • 基金资助: 

Quantum Algorithm for Spectral Regression

Pan Shijie, Gao Fei, Wan Linchun, Qin Sujuan, Wen Qiaoyan   

  1. 1(State Key Laboratory of Networking and Switching Technology (Beijing University of Posts and Telecommunications), Beijing 100876);2(State Key Laboratory of Cryptology, Beijing 100878)
  • Online: 2021-09-01
  • Supported by: 
    This work was supported by the Fundamental Research Funds for the Central Universities (2019XD-A01) and the National Natural Science Foundation of China (61976024, 61972048).

摘要: 子空间学习是机器学习领域的重要研究方向.为了降低子空间学习的复杂度,Cai等人提出了谱回归降维框架,并针对结合标签构造对应图的子空间学习提出了高效谱回归.近年来,量子计算的发展使进一步降低子空间学习算法的复杂度成为了可能.Meng等人率先提出了量子谱回归算法(MYXZ算法).MYXZ算法用了稀疏哈密顿量模拟技术来处理由权重矩阵生成的矩阵,但这个矩阵在较多的情况下是稠密矩阵.针对这种情况,指出了MYXZ算法的局限性,提出了一个改进的量子谱回归算法.改进算法采用了量子奇异值估计技术,在处理稠密矩阵时相对MYXZ算法有多项式加速.另外,提出了一个新的量子算法,对经典的高效谱回归进行加速.新算法能处理的这类问题是MYXZ算法无法处理的.新算法利用了量子岭回归和量子矩阵向量乘技术,在相同的参数条件下相对经典算法具有多项式加速效果.

关键词: 量子算法, 量子机器学习, 谱回归, 子空间学习, 稠密矩阵

Abstract: Subspace learning is an important research direction in the field of machine learning. In order to reduce the complexity of subspace learning, Cai et al. proposed a spectral regression dimensionality reduction framework, and proposed efficient spectral regression for subspace learning that constructed their graph by incorporating the label information. In recent years, the development of quantum computing has made it possible to further reduce the complexity of subspace learning algorithms. Meng et al. pioneered the quantum algorithm for spectral regression (MYXZ algorithm). In this article, the limitations of the MYXZ algorithm are pointed out. We point out that MYXZ algorithm uses sparse Hamiltonian simulation technology to process the matrix generated by the weight matrix, but this matrix is a dense matrix in some cases. In response to this situation, we propose an improved quantum spectral regression algorithm. Our improved algorithm uses quantum singular value estimation technology, which has a polynomial acceleration compared with MYXZ algorithm when dealing with dense matrices. In addition, we propose a new quantum algorithm to accelerate the classical efficient spectral regression. The problems that our new algorithm can handle cannot be handled by MYXZ algorithm. Our algorithm utilizes quantum ridge regression and quantum matrix vector multiplication technology. Under the same parameter conditions, our algorithm has polynomial acceleration effect compared with the classical algorithm.

Key words: quantum algorithm, quantum machine learning, spectral regression, subspace learning, dense matrix