NMR realization of adiabatic quantum algorithms for the modified Simon problem

Having the advantages of inherent robustness and fault tolerance, adiabatic quantum algorithms have attracted much attention in recent years. In this paper, we report the NMR experimental realization of adiabatic algorithms for the modified Simon's problem in a NMR quantum information processor. We first realized an adiabatic quantum algorithm by Rao [Phys. Rev. APLRAAN1050-294710.1103/PhysRevA.67.052306 67, 052306 (2003)] and then proposed a quantum algorithm by using the method of enhanced symmetry of the Hamiltonian given in Schaller and Schtzhold [Quantum Inf. Comput. 10, 109 (2010)]. We demonstrated this symmetry-enhanced algorithm in experiment. We studied the effects of the number of Trotter evolution steps on the validity of adiabatic evolution and decoherence. In practical applications, a trade-off between pulse imperfection and adiabatic requirement must be made in the Trotter evolution step number. An optimal evolution step number and run time were indicated by our results. The experimental demonstration also showed that the symmetry-enhanced algorithm gave a better performance.