Quantum algorithms for exponential sums and the evaluation of the Riemann zeta function

Abstract

We give quantum algorithms for estimating weighted exponential sums $S(f,w,N)= \sum_{k=0}^{N-1} w_k e^{2 \pi i f(k)}$, 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 $O(\varepsilon^{-1}\log(1/\gamma))$ 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 $\zeta(\sigma+it)$ in the critical strip can be estimated to accuracy $\delta$ with $\widetilde O(t^{(1-\sigma)/2}\, \delta^{-1})$ gates, hence $\widetilde O(t^{1/4}\, \delta^{-1})$ on the critical line, where $\widetilde O$ suppresses factors polylogarithmic in $t/\delta$. 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 $\delta \gg t^{-3/52}$, 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 $\sum_{k\le N} k^{-\sigma} = \Theta(t^{(1-\sigma)/2})$ 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.