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One of the unique features of discrete-time quantum walks is called trapping,meaning the inability of the quantum walker to completely escape from itsinitial position, albeit the system is translationally invariant. The effect isdependent on the dimension and the explicit form of the local coin. A fourstate discrete-time quantum walk on a square lattice is defined by its unitarycoin operator, acting on the four dimensional coin Hilbert space. The wellknown example of the Grover coin leads to a partial trapping, i.e., thereexists some escaping initial state for which the probability of staying at theinitial position vanishes. On the other hand, some other coins are known toexhibit strong trapping, where such escaping state does not exist. We present asystematic study of coins leading to trapping, explicitly construct all suchcoins for discrete-time quantum walks on the 2D square lattice, and classifythem according to the structure of the operator and the manifestation of thetrapping effect. We distinguish three types of trapping coins exhibitingdistinct dynamical properties, as exemplified by the existence or non-existenceof the escaping state and the area covered by the spreading wave-packet.

Complete classification of trapping coins for quantum walks on the 2D square lattice