Quantum machine learning with continuous variables

Abstract

Since Shor’s algorithm was proposed in 1994 which solves the factorization problem exponentially faster than classical algorithms, people realized that quantum computer has a great potential of solving hard problems which classical computers cannot. After the emergence of HHL algorithm which solves linear systems with an exponential reduction in computational time complexity in 2009, the interdisciplinary research field between quantum computation and machine learning, quantum machine learning started to develop. The number of literature has been growing rapidly since then, as HHL algorithm provides speedup on one of the key components in machine learning - linear algebra. Several quantum versions of machine learning algorithms have been developed afterward. Examples are quantum regression, quantum SVM, and quantum PCA. Recently, some new approach such as quantum machine learning using continuous-variable states, qumodes, and algorithms for near-term devices has been suggested. In this thesis, we investigated the possibility of solving different mathematical problems using continuous-variable states, including matrix inversion, principal component analysing, Fourier transform and using them as subroutines to build quantum machine learning algorithms such as linear regression and quantum-classical hybrid algorithms. Since a workable physical device for a quantum computer that can exhibit quantum supremacy has not been realized yet, some near-term solutions suggested outsourcing part of the computation to classical computers have been proposed recently. The proposal of building a new quantum kernel with different quantum states has been suggested. Examples are polynomial kernel and Gaussian kernel. Some new possible quantum kernels giving non-linear transformation on the data feature space based on Gaussian states are also proposed and investigated. The result of training these kernels in some quantum-classical hybrid algorithms with some standard datasets is analyzed as illustrative examples. The applications of kernels in different algorithms are then studied. Finally, the near-term quantum algorithms that have low quantum circuit depth and require fewer resources such as variational circuit and quantum kernel estimation are discussed. These algorithms usually consist of low-depth circuits which is executed repetitively. The optimization procedures can be left to classical computers or calculated with the near-term quantum computing devices.