Quantum Communication-Query Tradeoffs

For any function f : X × Y → Z, we prove that Q(f) ·Q(f) · (logQ(f) + log |Z|) ≥ Ω(log |X |). Here, Q(f) denotes the bounded-error communication complexity of f using an entanglementassisted two-way qubit channel, and Q(f) denotes the number of quantum queries needed to determine x with high probability given oracle access to the function fx(y) def = f(x, y). We show that this tradeoff is close to the best possible. We also give a generalization of this tradeoff for distributional query complexity. As an application, we prove an optimal Ω(log q) lower bound on the Q complexity of determining whether x + y is a perfect square, where Alice holds x ∈ Fq, Bob holds y ∈ Fq, and Fq is a finite field of odd characteristic. As another application, we give a new, simpler proof that searching an ordered size-N database requires Ω(logN/ log logN) quantum queries. (It was already known that Θ(logN) queries are required.)