On quantum computation with some restricted amplitudes

Abstract

In this paper we explore the power of quantum computers with restricted amplitudes. Adleman, DeMarrais and Huang showed that quantum Turing machines (QTMs) with the amplitudes from A = (Formuls Present) are equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial time algorithms. We show that QTMs with the amplitudes from A is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms. Extending this result, it is shown that exact computers with rational biased ‘quantum coins’ are equivalent to classical computers. We also show that from the viewpoint of zero-error algorithms A is not more useful than B = (Formuls Present) but sufficient for the factoring problem as the set of amplitudes taken by QTMs.