The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts

Understand

How well can quantum computers simulate classical dynamical systems? There is increasing effort in developing quantum algorithms to efficiently simulate dynamics beyond Hamiltonian simulation, but so far exact resource estimates are not known.

• In this work, we provide two significant contributions.
• First, we give the first non-asymptotic computation of the cost of encoding the solution to general linear ordinary differential equations into quantum states -- either the solution at a final time, or an encoding of the whole history within a time interval.
• Second, we show that the stability properties of a large class of classical dynamics allow their fast-forwarding, making their quantum simulation much more time-efficient.