# The quantum query complexity of composition with a relation

The negative weight adversary method, ADV^±(g), is known to characterize the bounded-error quantum query complexity of any Boolean function g, and also obeys a perfect composition theorem ADV^±(f ∘ g^n) = ADV^±(f) ADV^±(g). Belovs gave a modified version of the negative weight adversary method, ADV_rel^±(f), that characterizes the bounded-error quantum query complexity of a relation f ⊆{0,1}^n × [K], provided the relation is efficiently verifiable. A relation is efficiently verifiable if ADV^±(f_a) = o(ADV_rel^±(f)) for every a ∈ [K], where f_a is the Boolean function defined as f_a(x) = 1 if and only if (x,a) ∈ f. In this note we show a perfect composition theorem for the composition of a relation f with a Boolean function g ADV_rel^±(f ∘ g^n) = ADV_rel^±(f) ADV^±(g) . For an efficiently verifiable relation f this means Q(f ∘ g^n) = Θ( ADV_rel^±(f) ADV^±(g) ).

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