# Quantum algorithm for linear differential equations with exponentially improved dependence on precision

## TL;DR

This paper introduces a quantum algorithm for solving linear differential equations that significantly reduces complexity dependence on precision, leveraging recent quantum linear systems techniques.

## Contribution

It presents a novel quantum algorithm with exponential improvement in precision dependence for solving linear differential equations.

## Key findings

- Complexity is polynomial in log(1/error)
- No additional stability hypotheses needed
- Builds on recent quantum linear systems advances

## Abstract

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.