A spectral quantum algorithm for numerical differentiation and integration

TL;DR

This paper introduces quantum algorithms that use spectral methods and quantum Fourier transforms to efficiently compute derivatives and integrals from sampled data, enabling advanced quantum data processing tasks.

Contribution

The paper develops novel quantum algorithms for numerical differentiation and integration based on spectral methods, suitable for data given as samples rather than closed-form functions.

Findings

Algorithms produce quantum states proportional to derivatives and integrals

Utilizes quantum Fourier transform for computational efficiency

Extends to gradient estimation and sign recovery procedures

Abstract

Numerical calculus algorithms which estimate derivatives and integrals from data series acquired either via measurements or by sampling functions are essential in scientific computing. To date, a few quantum algorithms have been developed to perform calculus operations based on closed form functional inputs; yet, in many practical applications, field variables are numerically described via series of samples rather than closed form expressions. This paper presents the theoretical development and the gate-level circuit implementation of novel quantum algorithms for numerical differentiation and indefinite integration with a prescribed integration constant. The methodology relies on a spectral approach that leverages the computational efficiency of the quantum Fourier transform and the parallel computing capability afforded by quantum superposition to evaluate outputs at all domain points simultaneously. The differentiation approach is also extended to enable gradient estimation, and post-processing procedures are presented to recover sign information. The primary output of the proposed algorithms are quantum state vectors directly proportional to the numerical derivative or integral of the given data; therefore, the correctly signed results are made available to proceeding quantum computations. This result lays the foundation for the proposed algorithms to serve as core subroutines in applied quantum computing operations such as image processing, data analysis, and machine learning.