Quantum Computational Simulation Of Quantum Dynamical Systems And Its Characteristic Research

Quantum systems have rich and diverse dynamic properties,which directly affect the evolution of quantum states.Therefore,the study of the dynamic properties of quantum systems has become one of the important research contents in the fields of quantum regulation,quantum computing,and quantum sensing technology.But for many quantum systems,it is very difficult to directly solve the Schr(?)dinger equation satisfied by the wave function of the system.In order to study the evolution law of quantum dynamic systems,computational simulation has become an indispensable means,and it is more efficient and effective to use quantum computers to simulate quantum dynamic systems.General-purpose quantum computing hardware has developed rapidly in recent years,but quantum simulation based on this is still at the level of abstract algorithm design,lacking the support of corresponding quantum circuits.In this thesis,aiming at the HHL quantum algorithm,quantum baker’s map and period-driven quantum Harper(Quantum Kicked Harper,QKH)model,using IBM’s quantum computing hardware platform,we design the universal quantum circuit,quantum baker’s map and HHL quantum algorithm respectively.The quantum circuit of the QKH model was simulated on the qiskit platform.Mainly completed the following parts:(1)Quantum circuit design of HHL quantum algorithm.Combined with the principle of HHL quantum algorithm for solving linear equations,the key modules of the algorithm are designed top-down using universal quantum gates,including the universal quantum gate decomposition module of unitary matrix,quantum phase estimation module,quantum full adder and multiplier module,Quantum state conditional rotation transformation module,etc.,thus realizing the universal quantum circuit for solving linear equations.Quantum simulation experiments using the IBM qiskit quantum computing development platform show that the designed HHL quantum circuit can solve the general form of linear equations,and is easy to expand to medium and large-scale quantum circuits.(2)Quantum circuit design and characteristic analysis of quantum baker’s transformation model.Based on the quantum Fourier transform and its inverse transform,the quantum circuit of the quantum baker transform is designed.The statistical distribution of eigenvalues of the quantum baker’s transform Floquet operator under different qubits is studied and compared with the spectral statistics of highdimensional random matrices.Quantum simulations using 8 qubits show that there is dynamic localization of the quantum states transformed by the quantum baker.(3)Quantum circuit design and characteristic analysis of period-driven Haper model(Quantum Kicked Harper,QKH).Based on the quantum Fourier transform and a short-time slice decomposition algorithm,a quantum circuit that drives the quantum Harper model periodically is designed.Using the open source quantum computing cloud platform,the Floquet operator of the QKH model is obtained.When the QKH model corresponds to classical integrable and classical chaotic systems,the consistency between its quasi-energy statistical distribution and the distribution predicted by random matrix theory is studied.The statistical distribution of eigenvalues and the localization analysis of eigenstates show that the designed QKH quantum circuit can correctly describe the dynamics of the QKH model.In this thesis,the quantum circuit design of several typical quantum algorithms is carried out,and the simulation of quantum dynamic system is further transformed from the abstract algorithm level to the simulation and test that can be carried out on the quantum computing hardware platform,which will better promote the quantum Experimental study of dynamical systems and automatic design of medium-to-largescale quantum circuits.