The quantum adversary method is a powerful technique to prove lower bounds on quantum query complexity [BBBV97, Amb00, HNS01, BS04, Amb03, LM08]. The idea is to define a progress function varying from an initial value (before any query) to a final value (depending on the success probability of the algorithm) with one main property: the value of the progress function varies only when the oracle is queried. Then, a lower bound on the quantum query complexity of the problem can be obtained by bounding the amount of progress done by one query. Initially, different adversary methods were introduced, but they were later proved to be all equivalent [ ˇ SS06]. This unified method relied on optimizing an adversary matrix assigning weights to different pairs of inputs to the problem. While the original method only considered positive weights, it was later shown that negative weights also lead to a lower bound, which can actually be stronger in some cases [HL ˇ S07]. The relevance of this new adversary method with negative weights was made even clearer when it was shown that it is (almost) tight for Boolean functions [Rei09]. For non-Boolean functions, however, the situation is not so clear. For some problems
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