Realization of allowable qeneralized quantum gates

Abstract

The most general duality gates were introduced by Long, Liu and Wang and named allowable generalized quantum gates (AGQGs, for short). By definition, an allowable generalized quantum gate has the form of $\mathcal{U}$ = ∑ ^d−1_k=0 c _k U _k , where U _k ’s are unitary operators on a Hilbert space H and the coefficients c _k ’s are complex numbers with |∑ ^d−1_k=0 c _k | ⩽ 1 and |c _k | ⩽ 1 for all k = 0, 1, ..., d − 1. In this paper, we prove that an AGQG $\mathcal{U}$ = ∑ ^d−1_k=0 c _k U _k is realizable, i.e. there are two d by d unitary matrices W and V such that c _k = W _0k V _k0 (0 ⩽ k ⩽ d − 1) if and only if ∑ ^d−1_k=0 |c _k | ⩽ 1, in that case, the matrices W and V are constructed.