近似中位数的量子查询复杂度及相关统计
The quantum query complexity of approximating the median and related statistics
摘要：
令X =（x_0，...，x_{n-1}）$是n个数的序列。当ε> 0时，如果严格小于xi的元素个数和严格大于xi的元素个数均小于（1+ε）n/2，则称xi是ε-近似中值。我们考虑计算ε-近似中值的量子查询复杂度，给定序列X作为预言机。我们证明了一个下界Ω（\min{{1/ε},n}）查询的任何量子算法，计算一个ε-近似中位数与任何常数概率大于1/2。我们还展示了如何使用O（{1/ε} \log（{1\/ε}）\log \log（{1/ε}））Oracle查询计算ε-近似中位数，这代表了对Grover早期算法的改进。因此，我们得到的下界基本上是最优的。在比较树模型中，上界和下界都成立。我们的下界结果是最近引入量子复杂性理论的多项式范式的应用Beals等人。在证明中的主要成分是一个多项式的真实多线性多项式的“近似”对称部分布尔函数的程度下界。度界扩展了Paturi的一个结果，并且也立即产生了近似第k个最小元素的问题的下界，近似一个数字序列的平均值，以及近似计算布尔函数的个数的下界。得到的所有界限都在最优的多对数因子内（正如我们通过提出算法所示，其中没有这样的最优或接近最优的算法是已知的），从而证明了多项式方法的力量。
英文摘要：
Let X = (x_0,...,x_{n-1})$ be a sequence of n numbers. For ε> 0, we say that x_i is an ε-approximate median if the number of elements strictly less than x_i, and the number of elements strictly greater than x_i are each less than (1+ε)n/2. We consider the quantum query complexity of computing an ε-approximate median, given the sequence X as an oracle. We prove a lower bound of Ω(\min{{1/ε},n}) queries for any quantum algorithm that computes an ε-approximate median with any constant probability greater than 1/2. We also show how an ε-approximate median may be computed with O({1/ε}\log({1\/ε}) \log\log({1/ε})) oracle queries, which represents an improvement over an earlier algorithm due to Grover. Thus, the lower bound we obtain is essentially optimal. The upper and the lower bound both hold in the comparison tree model as well. Our lower bound result is an application of the polynomial paradigm recently introduced to quantum complexity theory by Beals et al. The main ingredient in the proof is a polynomial degree lower bound for real multilinear polynomials that ``approximate'' symmetric partial boolean functions. The degree bound extends a result of Paturi and also immediately yields lower bounds for the problems of approximating the kth-smallest element, approximating the mean of a sequence of numbers, and that of approximately counting the number of ones of a boolean function. All bounds obtained come within polylogarithmic factors of the optimal (as we show by presenting algorithms where no such optimal or near optimal algorithms were known), thus demonstrating the power of the polynomial method.
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