Research On Adiabatic Quantum Search Algorithm And Its Performance

The quantum circuit algorithm has much lower time complexity than the corresponding classic algorithm in terms of large integer factorization,disordered search,and optimization.Its superior computing ability has attracted people's great attention.However,it is very difficult to use a quantum feature to design a quantum circuit algorithm for a specific type of problem.Because the quantum circuit algorithm must use the relevant properties of quantum to achieve quantum parallel computing,that is,a unitary transformation is used to correspond to each step in the classical computer.Until 2000,the adiabatic quantum computing model was proposed,which greatly reduced the design difficulty of quantum algorithms.In the model,people only need to consider to set the solution of a specific problem as the system final state and the superposed state of the problem solution and the non-solution as the system initial state.Then the quantum system is continuously adiabatically evolved from the initial state along the evolution path that is set previously.After the adiabatic evolution,we can obtain the solution to the required problem after measuring the final quantum state.The adiabatic quantum computing model is based on a physical adiabatic theorem,so its corresponding algorithm time complexity can be derived from the adiabatic theorem.Corresponding to the problem of search,the initial state design based on the adiabatic quantum search algorithm is generally set as a physically easy-to-prepare quantum state.The final state of the algorithm is generally set as the solution to the problem.In this way,after the adiabatic evolution,we can get the solution to the problem through the measurement system.The research content of this thesis mainly focuses on the comparison between adiabatic quantum algorithm and quantum circuit algorithm,performance optimization of adiabatic quantum algorithm,and algorithm failure.The main research innovations include the following aspects:Firstly,the Grover search algorithm(quantum line algorithm)and the adiabatic quantum search algorithm are compared and studied,and the flexibility of the adiabatic quantum search algorithm is further studied.First,we discretize the evolutionaryprocess of the adiabatic search algorithm to compare the iterative steps of the Grover search algorithm.We found that the adiabatic quantum search algorithm is not different from the Grover search algorithm in the implementation process.They both start from the initial state and reach the final target state through multiple unitary transformation.However,the success rate of the adiabatic quantum search algorithm does not decrease as the number of search targets increases,instead it will continue to increase.In contrary,the Grover search algorithm will quickly decline or even fail.Second,when additional physical energy is injected into the adiabatic quantum system,the evolutionary time complexity of the adiabatic quantum search algorithm will decrease.Finally,after appropriately increasing the physical energy of the adiabatic quantum system(adding extra drive Hamiltonian),even if the search target does not exist,the adiabatic search algorithm can complete the search within a constant time complexity,but the Grover algorithm will fail.Secondly,the performance of the adiabatic quantum search algorithm is studied.Firstly,from the adiabatic interpolation function and extra drive Hamiltonian,respectively study their effects on the performance of the adiabatic quantum search algorithm.The second is to discuss the factors affecting the performance of the algorithm from the form of matrix structure,and propose a general model for accelerating the adiabatic quantum algorithm.Finally,the real reason that affects the system evolution performance is obtained: once the quantum entanglement between the initial state and the final state of the system is increased to a constant level,the evolutionary time complexity of the system is accelerated to a constant as well.From the point of view of mathematics,it is to increase the value of the element of the diagonal line on the system evolution matrix.Thirdly,the failure problem of the adiabatic quantum search algorithm is analyzed and studied.The first is to prove that when the initial and the final Hamiltonian of the adiabatic system are orthogonal,the conventional type of adiabatic quantum search algorithm will inevitably fail.The second is to study the effects of different forms of extra drive Hamiltonian on the adiabatic algorithm.It is found that not all extra drive Hamiltonian can ensure the normal evolution of the system.The third is to analyze the possible correlation between the fidelity between the initial and final states of theadiabatic system and the extra drive Hamiltonian and work out the applicable range of various extra drive Hamiltonian.The fourth is to propose a kind of all-round adiabatic quantum search model to ensure that the adiabatic quantum search algorithm does not fail under any fidelity by constraining the system evolution path functions.Fourthly,taking the adiabatic quantum search algorithm as an example,the problem of conversion between the adiabatic quantum search algorithm and the corresponding quantum circuit model under three different types of evolution paths is studied in detail.We focused on the global adiabatic quantum search algorithm with extra drive Hamiltonian and analyzed the reason why the complexity of the algorithm evolution time is constant after increasing the driven Hamiltonian.The final research results show that the time slice number of the quantum circuit model corresponding to these adiabatic quantum search algorithms is consistent with the time complexity of the algorithm,further illustrating the equivalence of the adiabatic quantum search algorithm and the quantum circuit algorithm.