nt - p h / 06 11 05 4 v 1 5 N ov 2 00 6 Negative weights makes adversaries stronger

The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations are information-theoretic and rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the existence of a measurement in a successful algorithm which gives the correct answer, with high probability. We show that this new method, which we call ADV, has all the advantages of the old: it is a lower bound on bounded-error quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV is always at least as large as the adversary method ADV, and we show an example of a monotone function for which ADV(f) = Ω(ADV(f)1.098). We also give examples showing that ADV does not face limitations of ADV such as the certificate complexity barrier and the property testing barrier.