用于指数和及黎曼ζ函数求值的量子算法
Quantum algorithms for exponential sums and the evaluation of the Riemann zeta function
AI总结：
研究给出估计加权指数和的量子算法，在特定预言机假设下，通过振幅估计可得到该和，将其应用于黎曼 - 西格尔公式可改进估计精度，还表明相关技术无法得出\(\text{polylog}(t)\)算法等，给出了对特定状态振幅和的估计器及相关子程序。
摘要：
我们给出了用于估计加权指数和\(S(f,w,N)=\sum_{k=0}^{N - 1} w_k e^{2 \pi i f(k)}\)的量子算法，其中\(w_k \geq 0\)，\(\sum_k w_k = 1\)且\(N = 2^n\)为指数级大。在两个明确的预言机假设下，通过振幅估计可得到\(S\)，其加法误差为\(\varepsilon\)，预言机使用次数为\(O(\varepsilon^{-1}\log(1/\gamma))\)，每次使用\(\text{polylog}(N)\)个门。将此应用于黎曼 - 西格尔公式，证明了在临界带中\(\zeta(\sigma + it)\)可被估计到精度\(\delta\)，所需门数为\(\widetilde O(t^{(1 - \sigma)/2}\, \delta^{-1})\)，在临界线上为\(\widetilde O(t^{1/4}\, \delta^{-1})\)；在固定精度下，这改进了黎曼 - 西格尔的\(t^{1/2}\)成本以及最佳严格经典算法的\(t^{4/13 + o(1)}\)。还表明这些技术无法得出\(\text{polylog}(t)\)算法，且Hiary型块分解不能改善量子查询复杂度。此外，给出了对任何可有效制备状态的振幅和\(\lvert \sum_k a_k \rvert\)的估计器，并回顾了所需的振幅和相位估计子程序。
英文摘要：
We give quantum algorithms for estimating weighted exponential sums $S(f,w,N)= \sum_{k=0}^{N-1} w_k e^{2 πi f(k)}$, with $w_k \ge 0$, $\sum_k w_k =1$, and $N=2^{n}$ exponentially large. Under two explicit oracle assumptions -- efficiently computable prefix sums for the weights, enabling Grover--Rudolph state preparation, and a fixed-point circuit for $f$ -- amplitude estimation yields $S$ to additive error $\varepsilon$ with $O(\varepsilon^{-1}\log(1/γ))$ oracle uses and $\operatorname{polylog}(N)$ gates per use; all bounds are full gate complexities, and the saving over classical sampling is quadratic in $1/\varepsilon$. Applying this to the Riemann--Siegel formula, we prove that $ζ(σ+it)$ in the critical strip can be estimated to accuracy $δ$ with $\widetilde O(t^{(1-σ)/2}\, δ^{-1})$ gates, hence $\widetilde O(t^{1/4}\, δ^{-1})$ on the critical line, where $\widetilde O$ suppresses factors polylogarithmic in $t/δ$. At fixed accuracy this improves on the $t^{1/2}$ Riemann--Siegel cost and on the best rigorous classical algorithm's $t^{4/13+o(1)}$; the quantum algorithm is advantageous precisely when $δ\gg t^{-3/52}$, which covers the accuracy needed to locate and count zeros. We show that a $\operatorname{polylog}(t)$ algorithm does not follow from these techniques -- undoing the normalization costs the $\ell^{1}$ mass $\sum_{k\le N} k^{-σ} = Θ(t^{(1-σ)/2})$ of the main sum -- and that Hiary-type block decompositions cannot improve the quantum query complexity. We also give an estimator for the magnitude of the amplitude sum $\lvert \sum_k a_k \rvert$ of any efficiently preparable state, and review the required amplitude- and phase-estimation subroutines.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
