We study tradeoffs between quantum and classical queries for hybrid
algorithms that have black-box access to a random oracle. Although there are
several established techniques for proving query lower bounds for both quantum
and classical algorithms, there is no such widely applicable technique for
hybrid algorithms and the optimal tradeoffs for many fundamental problems are
still unknown $unicode{x2013}$ an optimal tradeoff for the search problem was
only shown recently by Rosmanis, although not in the random oracle model. For
another fundamental problem, collision finding, the optimal tradeoff was not
known.

In this work, we develop a framework for recording a query transcript for
quantum-classical algorithms that represents the knowledge gained by the
algorithm. The main feature of this framework is to allow us to record queries
in two incompatible bases $unicode{x2013}$ classical queries in the standard
basis and quantum queries in the Fourier basis $unicode{x2013}$ in a
consistent way. We call the framework the hybrid compressed oracle as it
naturally interpolates between the classical way of recording queries and the
compressed oracle framework of Zhandry for recording quantum queries. We
demonstrate its applicability by giving a simpler proof of the optimal
quantum-classical tradeoff for search and by showing an optimal tradeoff for
collision finding.

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