The Research Of Quantum Search And Quantum Optimization Algorithms

Due to its distinctive features,quantum computing has demonstrated so-called quantum advantage such as quantum speedup over its classical counterpart.This thesis focuses on two parts of quantum algorithms,quantum search algorithm and quantum optimization algorithm.The quantum computation model with continuous variables is an important model in quantum computation,besides the well-known qubit model.In order to verify the superiority of quantum computing,this thesis proposes three new results,a namely fixed-point quantum search algorithm based on continuous variables,a multilayer quantum search algorithm applied to both qubit models continuous-variable models and a quantum gradient algorithm based on continuous variables,with details as follows:1.Compared to the most established quantum search algorithms which mainly focus on discrete search problems,we propose a fixed-point quantum search algorithm based on a continuous-variable model to solve the continuous search problem.Specifically,we design the fixed-point quantum search algorithm by constructing the corresponding quantum state and unitary transformation and calculate the complexity of the quantum algorithm for the continuous search problem.In addition,based on previous work,this thesis proposes an optimal fixed-point quantum search algorithm,and gives the specific circuit constructed using oracle in the algorithm.In order to verify the effectiveness of the algorithm,we have applied it to a specific optimization problem for simulation,and have given an oracle circuit that can be physically realized.2.For the search algorithm,the known optimal quantum search algorithm has square speedup.In order to achieve better quantum search,we discuss a quantum multilayer search algorithm.It is assumed that there is a corresponding quantum oracle,which can invert the target state as a whole instead of acting on the base of the target state to invert.The search problem can be roughly divided into a multi-condition search problem and a single-condition search problem.Both problems can be implemented with a multi-layer quantum search algorithm.In addition,we use the fixed-point quantum search algorithm to propose a feasible alternative to achieve multi-layer quantum search,analyze the error and complexity of the algorithm,and prove that the algorithm is processing big data with known statistical laws there is exponential acceleration when searching problems,which reflects the superiority of quantum computing.3.In addition to the quantum search algorithm,this thesis also discusses a quantum optimization algorithm,proposes and analyzes the gradient descent algorithm based on continuous variables,and also gives specific quantum circuits.Under the framework of continuous-variable computation,there is a natural differentiator,and we use a special structure to apply this property to the algorithm design,thereby implementing the quantum gradient descent algorithm.Moreover,the full quantum circuit structure is given without using quantum measurement in the intermediate process,so there is no need to have enough copies of states to ensure the success of the algorithm.In conclusion,we have proposed quantum algorithms with quantum speedups over the classical counterparts,illustrating the quantum advantages.Our work in this thesis enriches the models of quantum search and quantum optimization algorithms.We also provide new methods and perspectives which can be further explored in future research.