The speedup of quantum algorithms over classical algorithms has been the main driving force for the current interests in quantum computing. Although dramatic speedups seem possible, as Shor's polynomial time quantum algorithms for factoring and for finding discrete logarithms suggest, provable speedups are only found in very restricted models, such as the decision trees.; In the decision tree model, the inputs x1, x2,…,xn are known to an oracle, and the only way that the algorithm can access the inputs is to ask the oracle questions of the type: “xi = ?”. The complexity measure is the number of queries used.; Many problems can be naturally formulated in this model. On the one hand many provable quantum speedups have been found. On the other hand, much progress has also been made in proving quantum lower bounds, which reveal the limitations of the quantum computational power. This dissertation is along the latter lines. We prove the following lower bound results. (a) The quantum decision tree complexity of any function is lower bounded by its average sensitivity , which measures how likely the function value will change if a random input is changed in a random coordinate. (b) With a natural quantification of the amount of information, each classical query can only obtain at most, 1 bit of information about the input, while a quantum query can obtain up to Θ(log n) bits. In contrast to an example in which one quantum query obtains log n bits of information, we show that in computing any total function, on average each quantum query can obtain at most some constant number of bits of information. (c) We prove that any comparison-based quantum algorithm for sorting n numbers must compare Ω(n log n) times. If no error is allowed, at least 0.110n log2 n−0.067n + O(1) comparisons are necessary. We also prove quantum lower bounds on some other problems related to sorting, such as the problems of element distinctness, matching nuts and bolts, and non-comparison-based sorting.
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机译:量子算法相对于经典算法的加速,已经成为当前量子计算兴趣的主要驱动力。尽管似乎可以实现极大的提速,但是正如Shor的多项式时间量子算法进行分解和查找离散对数所表明的那样,可证明的 italic>提速仅在非常有限的模型中才能找到,例如决策树。在决策树模型中,输入 x italic> 1 sub>, x italic> 2 sub>,…, x n sub> italic>,并且算法访问输入的唯一方法是询问oracle类型的问题:“ x i sub> italic> =?”。复杂度度量是所使用查询的数量。在此模型中自然可以提出许多问题。一方面,已经发现了许多可证明的量子加速。另一方面,在证明量子下界方面也取得了很大进展,这揭示了量子计算能力的局限性。本文的研究方向与前者相同。我们证明以下下界结果。 (a)任何函数的量子决策树复杂度都以其平均灵敏度 italic>为下限,该函数衡量如果随机坐标中的随机输入发生变化,函数值发生变化的可能性。 (b)通过自然量化信息量,每个经典查询最多只能获取有关输入的1位信息,而量子查询最多可以获取Θ(log n italic>)位。与一个量子查询获得log n italic>位信息的示例相反,我们表明,在计算任何 total italic>函数时,平均每个量子查询最多可以获取一些恒定的信息位数。 (c)我们证明,用于对 n italic>个数字进行排序的任何基于比较的量子算法都必须比较Ω( n italic> log n italic>)次。如果不允许错误,则至少需要进行0.110 n italic> log2 n italic> -0.067 n italic> + O(1)比较。我们还证明了与排序有关的其他一些问题的量子下界,例如元素区别,匹配的螺母和螺栓以及基于非比较的排序。
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