Quantum Np Is Hard for Ph

In analogy with NP, a language L is in the class NQP if the strings in L are exactly those accepted by a polynomial-time quantum machine with non-zero probability. Previously, Adleman, Demarrais and Huang showed that NQP PP. The sharper upper bound NQP coC=P is implicit in Adleman et al. and results of Fortnow and Rogers. Here we prove that in fact NQP and coC=P are the same class, by showing that determining whether a quantum computation has a non-zero probability of accepting is hard for coC=P. This implies that NQP is hard for the polynomial-time hierarchy. The hardness result also applies to determining whether a given quantum basis state appears with nonzero amplitude in a super-position, or whether a given quantum bit has positive expectation value at the end of a quantum computation.