Uniformity of quantum circuit families for error-free algorithms

In order to establish the computational equivalence between quantum Turing machines (QTMs) and quantum circuit families (QCFs) using Yao's quantum circuit simulation of QTMs, we have previously introduced the class of uniform QCFs based on an infinite set of elementary gates, which has been shown to be computationally equivalent to polynomial-time QTMs up to bounded error simulation. However, the complexity classes ZQP and EQP introduced by Bernstein and Vazirani for QTMs do not appear to equal their counterparts for uniform QCFs. Recently, we have introduced a subclass of uniform QCFs, the class of finitely generated uniform QCFs, and showed that they are perfectly equivalent to the class of polynomial-time QTMs in the sense that both classes can be exactly simulated with each other. Here, we further investigate the power of uniform QCFs comparing with that of finitely generated uniform QCFs in detail. We obtain the following results: (i) If a permutation Mf: |x〉↦|f(x)〉 can be implemented with zero error by a uniform QCF, then both f and f-1 can be exactly computed by uniform QCFs. (ii) The quantum Fourier transform (QFT) of any order cannot be implemented with zero error by any finitely generated uniform QCF, while it has been shown, in contrast, by Mosca and Zalka that the QFT of any order can be exactly implemented by a uniform QCF.