  In this paper, we present a polynomial-complexity algorithm to construct a
special orthogonal matrix for the deterministic remote state preparation (DRSP)
of an arbitrary n-qubit state, and prove that if n>3, such matrices do not
exist. Firstly, the construction problem is split into two sub-problems, i.e.,
finding a solution of a semi-orthogonal matrix and generating all
semi-orthogonal matrices. Through giving the definitions and properties of the
matching operators, it is proved that the orthogonality of a special matrix is
equivalent to the cooperation of multiple matching operators, and then the
construction problem is reduced to the problem of solving an XOR linear
equation system, which reduces the construction complexity from exponential to
polynomial level. Having proved that each semi-orthogonal matrix can be
simplified into a unique form, we use the proposed algorithm to confirm that
the unique form does not have any solution when n>3, which means it is
infeasible to construct such a special orthogonal matrix for the DRSP of an
arbitrary n-qubit state.
