Lower Bounds for Function Inversion with Quantum Advice
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Function inversion is that given a random function f: [M] → [N], we want to compute some auxiliary information of size S that we can find pre-image of any image with a few queries to the function given as a black box in time T. It is a well-studied problem in the classical settings, however, it is not clear how a quantum adversary can do better at this task besides invoking Grover's algorithm. Nayebi et al. proved a lower bound for adversaries inverting permutations leveraging only quantum queries to the black box. We give a matching lower bound for functions and permutations where M = O(N), and allowing adversaries to be fully quantum, and thus resolving the open question positively raised by Nayebi et al. of whether such lower bound is achievable for inverters with quantum advice. In order to prove these bounds, we also proved a lower bound for a generalized version of quantum random access code (originally introduced by Ambainis et al.), i.e. under the setting where the encoding length is variable and each element can be arbitrarily correlated, which may be of independent interest.
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