多项式插值的最优量子算法
Optimal quantum algorithm for polynomial interpolation
摘要：
我们考虑确定GF（q）上的d次多项式的系数所需的量子查询的数量。Kane和Kendall以及Meyer和Pommersheim独立给出的下界表明，需要d/2+1/2量子查询才能以有界误差解决这个问题，而Boneh和Zhandry的算法表明d量子查询是足够的。我们表明，下界是可实现的：d/2+1/2量子查询足以确定有界误差的多项式。此外，我们表明，d/2+1查询足以实现概率接近1大q。这些上界改进了Boneh和Zhandry关于密码协议抵抗量子攻击的不安全性的结果。我们还表明，我们的算法的成功概率作为查询的数量的函数是完全最优的。此外，该算法可以实现与门复杂度poly（log q）的成功概率可以忽略不计的减少。最后，我们对多元多项式插值的量子查询复杂度进行了一个猜想。
英文摘要：
We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm's success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation.
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