An unsharp logic from quantum computation
  Logical gates studied in quantum computation suggest a natural logical abstraction that gives rise to a new form of unsharp quantum logic. We study the logical connectives corresponding to the following gates: the Toffoli gate, the NOT and the squareroot of NOT (which admit of natural physical models). This leads to a semantic characterization of a logic that we call computational quantum logic CQL. 
G. Cattaneo, M.L. Dalla Chiara, R. Giuntini, R. Leporini