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

Simulating complex dynamical systems—like plasma behavior, fluid flow, or coupled oscillators—is one of the biggest computational challenges in science and engineering. Classical computers struggle with these problems due to the curse of dimensionality: as systems grow larger and more complex, the computational resources required scale exponentially.

Our paper, The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts, tackles this challenge head-on. We provide the first rigorous, non-asymptotic resource estimates for solving general linear ordinary differential equations (ODEs) on a quantum computer. This means we calculated, in concrete terms, what it would cost in qubits and gates—not just scaling theory—to simulate such systems.

Even more significantly, the study shows that a wide class of stable linear systems can be “fast-forwarded” on a quantum computer. In practice, this means their dynamics can be simulated in a time sublinear in the simulated time itself. The work builds on Lyapunov stability theory to show how the inherent stability of certain physical systems improves computational efficiency when simulated on a quantum device.

Key takeaways from the paper:

- Provides the first detailed resource counts for solving arbitrary linear ODEs on a quantum computer.

- Demonstrates that stable classical systems can be fast-forwarded, leading to quadratic improvements in efficiency. Other groups have extended our work showing that even greater fast-forwarding are possible [https://arxiv.org/abs/2410.13189].

- Applies results to real-world examples, including linearized collisional plasma problems, damped oscillators, and dissipative nonlinear systems.

This work contributes to a recent research program that looks into how fault-tolerant quantum computing could make it feasible to simulate classical dynamical systems that are currently beyond reach, with major implications for aerospace, energy, materials science, and plasma physics.