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.

翻译：本文提出了一种多项式复杂度的算法，用于构造任意n量子比特态确定性远程态制备（DRSP）所需的特殊正交矩阵，并证明当n>3时此类矩阵不存在。首先，将构造问题分解为两个子问题：寻找半正交矩阵的解，以及生成所有半正交矩阵。通过给出匹配算子的定义与性质，证明了特殊矩阵的正交性等价于多个匹配算子的协同作用，进而将构造问题简化为求解XOR线性方程组，使构造复杂度从指数级降至多项式级。在证明每个半正交矩阵均可化为唯一规范形式后，利用所提算法验证了当n>3时该规范形式无解，这意味着无法为任意n量子比特态的DRSP构造此类特殊正交矩阵。