General quantum matrix exponential dimensionality-reduction framework based on block encoding
Yongmei Li, Hai-Ling Liu, Shi‐Jie Pan, Su‐Juan Qin, Fei Gao, Qiaoyan Wen · Physical Review A · 2023
As a general framework, matrix exponential dimensionality reduction (MEDR) deals with the small-sample-size problem that appears in linear dimensionality-reduction (DR) algorithms. High complexity is the bottleneck in this type of DR algorithm because one has to solve a large-scale matrix exponential generalized eigenproblem. To address it, here we design a general quantum algorithm framework for MEDR based on the block-encoding technique. This framework is configurable, that is, by selecting suitable methods to design the block encodings of the data matrices, a series of efficient quantum algorithms can be derived from this framework. Specifically, by constructing the block encodings of the data matrix exponentials, we solve the generalized eigenproblem and then obtain the digital-encoded quantum state corresponding to the compressed low-dimensional data set, which can be directly utilized as the input state for other quantum machine learning tasks to overcome the curse of dimensionality. As applications, we apply this framework to four linear DR algorithms and design their quantum algorithms, which all achieve a polynomial speedup in the dimension of the sample over their classical counterparts.