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Consider the problem of evaluating an AND-OR formula on an N-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time N1/2+o(1). In particular, approximately balanced formulas can be evaluated in O(√N) queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.
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Ambainis et al. (2010) studied this question.
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