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Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer

Andrew M. Childs·Ben W. Reichardt·Robert Spalek·Shengyu Zhang
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Abstract

For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.

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Andrew M. Childs, Ben W. Reichardt, Robert Spalek, Shengyu Zhang. 2007-03-02. Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer. https://arxiv.org/abs/quant-ph/0703015

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