Quantum walk of two composite bosons

友之 星谷

Linda Sansoni

Xizhou Qin, Yongguan Ke, Xi‐Wen Guan +3 more

We investigate continuous-time quantum walks of two indistinguishable particles [two bosons, or two fermions, or two hard-core bosons (HCBs)] in one-dimensional lattices with nearest-neighbor interactions. The results for two HCBs are consistent with the recent experimental observation of two-magnon dynamics [Fukuhara et al., Nature (London) 502, 76 (2013)]. The two interacting particles can undergo independent walking and/or co-walking depending on both quantum statistics and interaction strength. Two strongly interacting particles may form a bound state and then co-walk like a single composite particle with a statistics-dependent walk speed. Analytical solutions for the scattering and bound states, which appear in the two-particle quantum walks, are obtained by solving the eigenvalue problem in the two-particle Hilbert space. In the context of degenerate perturbation theory, an effective single-particle model for the quantum co-walking is analytically derived and the walk speed of bosons is found to be exactly three times that of fermions and HCBs. Our result paves the way for experimentally exploring quantum statistics via two-particle quantum walks.

Xizhou Qin, Yongguan Ke, Xi-Wen Guan +3 more

We investigate continuous-time quantum walks of two indistinguishable particles (bosons, fermions or hard-core bosons) in one-dimensional lattices with nearest-neighbour interactions. The two interacting particles can undergo independent- and/or co-walking dependent on both quantum statistics and interaction strength. We find that two strongly interacting particles may form a bound state and then co-walk like a single composite particle with statistics-dependent propagation speed. Such an effective single-particle picture of co-walking is analytically derived in the context of degenerate perturbation and the analytical results are well consistent with direct numerical simulation. In addition to implementing universal quantum computation and observing bound states, two-particle quantum walks offer a novel route to detecting quantum statistics. Our theoretical results can be examined in experiments of light propagations in two-dimensional waveguide arrays or spin-impurity dynamics of ultracold atoms in one-dimensional optical lattices.

Eugene P. Gross

Xingliang Lou

It is well known that certain search problems are efficiently solved by quantum walk algorithms. Of particular interest are those problems whose efficient solutions involve nesting of search algorithms. The nesting of search algorithms generally incurs an extra logarithmic factor in query and time complexity, due to the use of majority voting as an error reduction technique. We study whether composition of search algorithms can be achieved without the said logarithmic factor in complexity. Two methods of composition have been proposed in this thesis. The first is a slight simplification of the quantum walk algorithm due to Magniez, Nayak, Roland, and Santha. The second is a method for composition of Markov chains. Neither approach appear to be generally applicable to all cases of quantum walk composition. Further work is required to determine the circumstances under which these approaches provide the logarithmic factor of increase in efficiency that we desire.