Quantum Lazy Sampling and Path Recording for Any Group

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A central challenge in quantum algorithm analysis and cryptography is reasoning about algorithms with oracle access to a random group element (e.g. a random function, a random permutation, a random unitary). Can we efficiently simulate such algorithms? Can we determine what they know after $t$ queries? Classically, an important tool for this is lazy sampling, where the oracle does not commit to the full group element at the beginning, but rather samples partial information about it on the fly. W