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.

Consider the problem of evaluating an AND-OR formula on an N -bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time N 1/2+o(1) . In particular, approximately balanced formulas can be evaluated in O( √ N ) queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.

We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worst-case time O(n 5/3 ) and expected time O(n 5/3 / min(w,where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in time O(n 7/4 ). We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries.

A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function.Our direct product theorems imply a time-space tradeoff T 2 S = Ω N 3 for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication. * Supported by Canada's NSERC and MITACS, and by DFG grant KL 1470/1. † Supported in part by the EU fifth framework project RESQ, IST-2001-37559. expect a constant error on each instance and hence an exponentially small success probability for the k-vector as a whole. Such a statement is known as a weak direct product theorem:However, even if we give our algorithm roughly kt resources, on average it still has only t resources available per instance. So even here we expect a constant error per instance and an exponentially small success probability overall. Such a statement is known as a strong direct product theorem:Strong direct product theorems, though intuitively very plausible, are generally hard to prove and sometimes not even true. Shaltiel [Sha01] exhibits a general class of examples where strong direct product theorems fail. This applies for instance to query complexity, communication complexity, and circuit complexity. In his examples, success probability is taken under the uniform probability distribution on inputs. The function is chosen such that for most inputs, most of the k instances can be computed quickly and without any error probability. This leaves enough resources to solve the few hard instances with high success probability. Hence for his functions, with T ≈ tk, one can achieve average success probability close to 1.Accordingly, we can only establish direct product theorems in special cases. Examples are Nisan et al.'s [NRS94] strong direct product theorem for "decision forests", Parnafes et al.'s [PRW97] direct product theorem for "forests" of communication protocols, Shaltiel's strong direct product theorems for "fair" decision trees and his discrepancy bound for communication complexity [Sha01]. In the quantum case, Aaronson [Aar04, Theorem 10] established a result for the unordered search problem that lies in between the weak and the strong theorems: every T -query quantum algorithm for searching k marked items among N = kn input bits will have success probability σ ≤ O T 2 /N k .