On quantum versions of the yao principle

Abstract

The classical Yao principle states that the complexity R∈(f) of an optimal randomized algorithm for a function f with success probability1 - e equals the complexity max„ D∈μ(f) of an optimal deterministic algorithm for f that is correct on a fraction 1 - ∈ of the inputs, weighed according to the hardest distribution μ over the inputs. In this paper we investigate to what extent such a principle holds for quantum algorithms. We propose two natural candidate quantum Yao principles, a “weak” and a “strong” one. For both principles, we prove that the quantum bounded-error complexityis a lower bound on the quantum analogues of max„ D∈μ(f). We then prove that equalitycannot be obtained for the “strong” version, byexhibiting an exponential gap. On the other hand, as a positive result we prove that the “weak” version holds up to a constant factor for the querycomplexityof all symmetric Boolean functions.