Robust quantum minimum finding with an application to hypothesis selection
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We consider the problem of finding the minimum element in a list of length N using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least α by some metric defined on the elements, then the comparison will be made correctly; if the values of the elements are closer than α, the outcome of the comparison is not subject to any guarantees. We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time Õ(√(N (1+Δ))), where Δ is an upper-bound on the number of elements within the interval α, and outputs a good approximation of the true minimum with high probability. Our noisy comparator model is motivated by the problem of hypothesis selection, where given a set of N known candidate probability distributions and samples from an unknown target distribution, one seeks to output some candidate distribution O(ε)-close to the unknown target. Much work on the classical front has been devoted to speeding up the run time of classical hypothesis selection from O(N^2) to O(N), in part by using statistical primitives such as the Scheffé test. Assuming a quantum oracle generalization of the classical data access and applying our noisy quantum minimum-finding algorithm, we take this run time into the sublinear regime. The final expected run time is Õ( √(N(1+Δ))), with the same O(log N) sample complexity from the unknown distribution as the classical algorithm. We expect robust quantum minimum-finding to be a useful building block for algorithms in situations where the comparator (which may be another quantum or classical algorithm) is resolution-limited or subject to some uncertainty.
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