从量子上界导出经典下界
Classical lower bounds from quantum upper bounds
摘要：
我们证明下界的复杂性措施，如布尔函数的近似程度和布尔矩阵的近似秩，使用量子参数。我们证明了这些下界使用量子查询算法的组合组测试问题。我们证明了对于任何函数f，计算f的n个副本的OR的近似程度是Ω（sqrt{n}）乘以f的近似程度，这是最优的。在我们的工作之前，没有这样的一般结果，甚至AND函数的OR下限也是在2013年才解决的。然后，我们证明了一个类似的结果，在通信复杂性，显示的对数的近似秩（或更准确地说，近似gamma_2范数）的F：X x Y -> {0，1}增长的一个因素Ω ~（sqrt{n}）时，我们采取的或的n个副本的F，这也是本质上最优的。作为推论，我们给出了Razborov著名的Ω（sqrt{n}）下界的量子通信复杂性的不相交性问题的一个新的证明。最后，我们将这两个结果从与OR函数的组合推广到与任意对称函数的组合，在这种情况下也得到了接近最优的下界。
英文摘要：
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem. We show that for any function f, the approximate degree of computing the OR of n copies of f is Omega(sqrt{n}) times the approximate degree of f, which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013. We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate gamma_2 norm) of F: X x Y -> {0,1} grows by a factor of Omega~(sqrt{n}) when we take the OR of n copies of F, which is also essentially optimal. As a corollary, we give a new proof of Razborov's celebrated Omega(sqrt{n}) lower bound on the quantum communication complexity of the disjointness problem. Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.
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