The Research Of Analyzing And Applying Quantum Computing In The Field Of Probablistical Graph Model

Quantum computing is a probablistical computing model,which is based on the fundamental principles of quantum mechanics.Utilizing quantum superposition and quantum entanglement,quantum computing illustrates significant speed-up over classical computing models in solving certain specific problems,such as large integer decomposition,unstructured database searching and Hamiltonian simulation.With the rapid development of quantum supremacy,quantum computing has also been applied to the field of machine learning,and some efficient quantum algorithms on solving large scale and high dimensional machine learning problems have been proposed.However,the research of quantum machine learning algorithms is still in the initial stage,and many probablistical graph model problems have not been solved by efficient quantum algorithms.According to this phenomenon,this thesis aims at proposing quantum algorithms with significant speed-up advantages compared with classical algorithms for several important probablistical graph models.These quantum algorithms can also inspire the improvement of Shor algorithm.Specifically,the research in this thesis includes the following three aspects1.Aiming at the conditional random field model,which is an inference probablistical graph model for labeling and sequence analysis,we design the Hamiltonian corresponding to the realistic physical system and the measurement operator corresponding to the data set.Utilizing these physical quantities,the quantum condition random field model and its corresponding quantum training algorithm are proposed.Compared with the classical training algorithm,the quantum training algorithm achieves exponential speed-up when the condition number of the 0(n)scale Hamiltonian satisfies k=O(n).Furthermore,from the perspective of VC(Vapanik Chervonenkis)dimension,we also show the fact that the quantum conditional random field model has stronger data representation ability than the classical conditional random field model.2.Based on the well-known Bayesian learning theory,we propose a quantum Bayesian learning framework in the quantum-enhanced feature spaces and apply it on the restricted Boltzmann machine model to solve classification tasks.This quantum framework includes two phases:encoding phase and training phase.The encoding phase uses unitary feature operations and the Parallel hardware-efficient ansatz to encode real data and Boltzmann parameters into the quantum state space,respectively.The training phase provides two quantum algorithms to deliver the optimal inference function,in which one quantum algorithm is for calculating the maximum posterior probability distribution density matrix and another one is for computing the predictive probability distribution density matrix.We theoretically prove that these two quantum algorithms have exponential acceleration advantages over their classic counterparts.We also tested the proposed framework on an open source quantum computing cloud platform(HiQ),which can achieve almost the same classification performacne compared to the classic Bayesian learning algorithm.3.Aiming at the famous quantum integer decomposition algorithm—Shor algorithm,an improved Shor algorithm based on shallow quantum circuits is proposed.The proposed algorithm is inspired by the quantum probablistical graph model,which has the ability that adjusting the parameter ?.in the quantum model U(?)so that it can realize the mapping from the eigenvectors of the modular exponential operation Ux,N to the computational basis.The algorithm successfully reduced the quantum circuit depth required by the Shor algorithm,whose complexity decreases from O((logN)2)to O((logN))when decomposing the integerN.This performance allows us to use the proposed algorithm on the NISQ devices to implement integer decomposition tasks.