The existence of non-classical orthogonal quantum Latin squares

Quantum Latin squares are a generalization of classical Latin squares in quantum field and have wide applications in unitary error bases, mutually unbiased bases, $k$-uniform states and quantum error correcting codes. In this paper, we put forward some new quantum Latin squares with special properties, such as idempotent quantum Latin square, self-orthogonal quantum Latin square, holey quantum Latin square, and the notions of orthogonality on them. We present some forceful construction methods including PBD constructions and filling in holes constructions for non-classical quantum Latin squares. As consequences, we establish the existence of non-classical 2-idempotent MOQLS$(v)$, non-classical 2, 3-MOQLS$(v)$ and non-classical SOQLS$(v)$ except possibly for several definite values.