一维非线性结构的量子算法
Quantum Algorithms for One-Dimensional Infrastructures
摘要：
几何结构是一种类群对象，在研究有限域上的数域和函数域的计算问题时出现在算术几何中。基础结构最突出的计算任务是计算基础结构的周长和广义离散矩阵。这两个问题是不知道有效率的经典算法的任意基础设施。我们的主要贡献是多项式时间量子算法的一维基础设施，满足一定的条件。例如，对于从单位秩为1的数域和函数域获得的基础结构，这些条件总是满足的。由于二次数域给出了这样的基础结构，因此该算法可用于求解Pell方程和主理想问题。在这个意义上，我们推广Hallgren的量子算法的二次数域，同时也提供了一个多项式加速。我们更一般的方法表明，这些量子算法也可以应用于从复杂的立方和完全复杂的四次数域获得的基础设施。我们改进的分析性能的方法，可以表明，这些算法成功的恒定概率独立的问题的大小。相比之下，Hallgren的成功概率下限随着圆周对数的四次幂而减小。我们的分析还表明，需要更少的量子位。我们还有助于基础设施的研究，并展示如何在基础设施内有效地计算。
英文摘要：
Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.
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