Quantum CORDIC -- Arcsine on a Budget

TL;DR

This paper presents a quantum algorithm for efficiently computing the arcsine function with arbitrary precision, adapting classical CORDIC techniques to the quantum realm while minimizing non-reversible operations.

Contribution

It introduces a reversible quantum CORDIC method for arcsine calculation, detailing space complexity, layer count, and CNOT count for quantum implementation.

Findings

Achieves space complexity of order n qubits for n-bit precision

Layer count scales as n log n, CNOT count as n squared

Enables quantum applications like HHL, digital-to-analog conversion, and Shapley value estimation

Abstract

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.