Theory Research On Adiabatic Quantum Computation

Quantum computation, governed by the quantum mechanics, has a natural parallelism and therefor it’s more powerful at computing some hard problems compared with the traditional computers. Adiabatic quantum computation(AQC) is one of the models to realize quantum computer, which encoded the solution of the hard problems into the ground state of the target Hamiltonian, then adiabatically evolved from an initial Hamiltonian, whose ground state is easy to prepare, to the target Hamiltonian. The quantum adiabatic theorem guarantees that the final state of the system will be at the ground state of the target Hamiltonian with high probability, then a measurement on the final state will get the solution of the problem. Quantum simulator based on this model has made great progress in the past few years, for example, the D-Wave company announced that they constructed a 512-qubit quantum annealer and purchased by NASA and Google company.In this thesis, we invested on the AQC’s basis principle, the construction of the adiabatic algorithm, the algorithm’s efficiency, the physical realization of the AQC and so on. The main results and the creative points are as follows:1. A method which can reduce the 3-local Hamiltonian into 2-local with fewer physical qubits was proposed.The target Hamiltonian of AQC usually contains k-body interactions(k ≥ 3) whereas in the considered physical systems there are usually 2-body interactions. In previous, one can reduce a 3-local Hamiltonian with 3M auxiliary qubits where M is the number of 3-body terms in the Hamiltonian. Here we proposed a method which can reduce the 3-local Hamiltonian only with M auxiliary qubits, and in some special cases only need 3 auxiliary qubits.2. An approach on the combination of classical computation with the quantum computation was proposed.When analysing the relationship between the efficiency of the adiabatic algorithm and the choice of the initial Hamiltonians, we find that if we can guess the distribution of the possible solutions with classical computers and encode this information into the construction of the initial Hamiltonian, then the efficiency of the adiabatic algorithm can be enhanced with a good guess and vice versa. This result can be used as an approach to combine the classical computation(CC) with the quantum computation(QC). It is an unmature ideas, however, this can stimulate discussions of the combination between CC and QC.3. An adiabatic algorithm for solving linear systems of equations was proposed.Linear system of equations is ubiquitous in engineering and science, and it can be solved by classical computers efficiently. Here, we proposed an adiabatic algorithm for solving linear systems of equations and find that there may exist some "complementary relations" between the efficiency and the universality of the simulator in which the algorithm was implemented. In some situations, the algorithm has an exponential speedup compared with the classical algorithm.4. The possible adiabatic quantum simulations in the trapped ion system with the gradient magnetic field pulses are studied.In the trapped ion system, the coefficients of the interaction Hamiltonian between the laser and ions are determined by the laser amplitude, the trap frequency. When the laser and the trap are preset, then all the coupling strength between the ions can be computed.Here we studied the possible adiabatic quantum simulations that can be experimentally realized with the gradient magnetic field pulses and this can be as a foundation for further studies of adiabatic quantum simulation.