20200701
Complete classification of trapping coins for quantum walks on the twodimensional square lattice
Publication
Publication
Physical Review A , Volume 102  Issue 1
One of the unique features of discretetime quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A fourstate discretetime quantum walk on a square lattice is defined by its unitary coin operator, acting on the fourdimensional coin Hilbert space. The wellknown example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discretetime quantum walks on the twodimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet.
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doi.org/10.1103/PhysRevA.102.012207  
Physical Review A  
Kollár, B, Gilyén, A.P, Tkáčová, I, Kiss, T, Jex, I, & Štefaňák, M. (2020). Complete classification of trapping coins for quantum walks on the twodimensional square lattice. Physical Review A: Atomic, Molecular and Optical Physics, 102(1). doi:10.1103/PhysRevA.102.012207
