This work introduces a quantum algorithm for computing the function arcsine, with arbitrary accuracy. We leverage a technique from embedded computing and Field-Programmable Gate Arrays, called COordinate Rotation DIgital Computer (CORDIC). CORDIC is a family of iterative algorithms that, in a classical context, can approximate various trigonometric, hyperbolic, and elementary functions using only bit shifts and additions. Adapting CORDIC to the quantum context is non-trivial, as the algorithm traditionally uses several non-reversible operations. We detail a method for CORDIC that avoids such non-reversible operations. We propose multiple approaches to calculate the arcsine function reversibly with CORDIC. For n bits of precision, our method has space complexity of order n qubits, a layer count in the order of n times log n, and a CNOT count in the order of n squared. This primitive function is a required step for the Harrow-Hassidim-Lloyd (HHL) algorithm, is necessary for quantum digital-to-analog conversion, can simplify a quantum speed-up for Monte-Carlo methods, and has direct applications in the quantum estimation of Shapley values.

翻译：本文提出了一种用于计算反正弦函数的量子算法，该算法可实现任意精度。我们借鉴了嵌入式计算和现场可编程门阵列中的坐标旋转数字计算机技术。CORDIC是一类迭代算法族，在经典计算中仅通过位移和加法操作即可逼近多种三角函数、双曲函数和基本初等函数。将CORDIC适配至量子计算环境并非易事，因为该算法传统上依赖多项不可逆操作。我们详细阐述了一种避免此类不可逆操作的CORDIC实现方法，并提出多种基于CORDIC的可逆反正弦函数计算方案。针对n位精度，我们的方法的空间复杂度为n量子比特量级，电路层数为n乘以log n量级，CNOT门数量为n平方量级。该基础函数是Harrow-Hassidim-Lloyd算法的必要步骤，对量子数模转换至关重要，可简化蒙特卡洛方法的量子加速，并在Shapley值的量子估计中具有直接应用。