Quantum Time-Space Tradeoffs by Recording Queries
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We use the recording queries technique of Zhandry [Zha19] to prove lower bounds in the exponentially small success probability regime, with applications to time-space tradeoffs. We first extend the recording technique to the case of non-uniform input distributions and we describe a new simple framework for using it. Then, as an application, we prove strong direct product theorems for K-Search under a natural product distribution not considered in previous works, and for finding K distinct collisions in a uniform random function. Finally, we use the latter result to obtain the first quantum time-space tradeoff that is not based on a reduction to K-Search. Namely, we demonstrate that any T-query algorithm using S qubits of memory must satisfy a tradeoff of T^3 S ≥Ω(N^4) for finding Θ(N) collisions in a random function. We conjecture that this result can be improved to T^2 S ≥Ω(N^3), and we show that it would imply a T^2 S ≥Ω̃(N^2) tradeoff for Element Distinctness.
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