# Quantum spectral methods for differential equations

## TL;DR

This paper introduces a quantum spectral method algorithm for solving linear ordinary differential equations with time-dependent coefficients, achieving exponential speedup in problem size and accuracy compared to classical methods.

## Contribution

It develops the first quantum algorithm for time-dependent linear differential equations using spectral methods, expanding quantum numerical analysis capabilities.

## Key findings

- Quantum algorithm solves time-dependent ODEs with poly(log d, log 1/ε) complexity.
- Spectral methods provide a global approximation approach in quantum algorithms.
- Algorithm extends quantum linear algebra techniques to differential equations with time-dependent coefficients.

## Abstract

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system of linear equations or linear differential equations with complexity $\mathrm{poly}(\log d)$. While several of these algorithms approximate the solution to within $\epsilon$ with complexity $\mathrm{poly}(\log(1/\epsilon))$, no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity $\mathrm{poly}(\log d, \log(1/\epsilon))$.