非线性估计的量子随机算法
Quantum and Randomised Algorithms for Non-linearity Estimation
摘要：
布尔函数的非线性表示它与任何线性函数的距离。尽管有几个强有力的结果，确定一个线性函数和区分一个充分的非线性函数，我们发现一个令人惊讶的缺乏工作计算的非线性函数。非线性与具有最大绝对值的沃尔什系数有关;然而，在构造沃尔什谱之后挑选最大值的天真尝试需要对$n$位函数的$Θ(2^n)$查询。我们通过设计高效的量子和随机化算法来近似非线性，允许添加剂误差，表示为$λ$，查询复杂性多项式依赖于$λ$的情况下，改善。我们证明了下界，表明这些都不是很远，从最佳的。我们的随机化算法所做的查询数量在$n$中是线性的，已经是指数级的改进，而我们的量子算法所做的查询数量令人惊讶地独立于$n$。我们的随机算法使用Goldreich-Levin风格的导航所有沃尔什系数和我们的量子算法使用Deutsch-Jozsa，振幅放大和振幅估计的巧妙组合，以改善现有的量子版本的Goldreich-Levin技术。
英文摘要：
Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires $Θ(2^n)$ queries to an $n$-bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted $λ$, with query complexities that depend polynomially on $λ$. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in $n$, already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of $n$. Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.
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