Practical Quantum Computing: solving the wave equation using a quantum approach

TL;DR

This paper experimentally analyzes the costs of a direct quantum algorithm for solving the wave equation, demonstrating its feasibility on idealized hardware and discussing potential improvements and future requirements.

Contribution

It provides an experimental cost analysis of a direct quantum PDE solver and details implementation steps, highlighting its potential on future error-corrected quantum hardware.

Findings

Quantum wave equation solver aligns with theoretical complexity.

Implementation details and potential improvements are discussed.

PDEs can be solved on quantum computers with future error correction.

Abstract

In the last years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised. On one side, "direct" quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit. On the other side, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the "direct" quantum algorithm: the wave equation solver devised in [PCS. Costa, S. Jordan, A. Ostrander, Phys. Rev. A 99, 012323, 2019]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great details the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.