Quantum Realization of the Finite Element Method

Matthias Deiml, Daniel Peterseim, 2024

Supplementary material to the paper

https://arxiv.org/abs/2403.19512

This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional domain. An essential step in the construction is a BPX preconditioner, which transforms the linear system into a sufficiently well-conditioned one, making it amenable to quantum computation. We provide a constructive proof demonstrating that, for any fixed dimension, our quantum algorithm can compute suitable functionals of the solution to a given tolerance $\mathtt{tol}$ with an optimal complexity of order $\mathtt{tol}^{-1}$ up to logarithmic terms, significantly improving over existing approaches. Notably, this approach does not rely on regularity of the solution and achieves quantum advantage over classical solvers in two dimensions, whereas prior quantum methods required at least four dimensions for asymptotic benefits. We further detail the design and implementation of a quantum circuit capable of executing our algorithm, present simulator results, and report numerical experiments on current quantum hardware, confirming the feasibility of preconditioned finite element methods for near-term quantum computing.

Structure of the code

• quantum_bpx.py contains the implementation of the circuits described in Section 6 of the paper and the numerical experiment of Section 7.1.
• qsp.py contains code for implementing quantum signal processing (QSP).
• block_encoding.py contains code for managing block encodings as described in Section 3 of the paper, as well as the operations of Proposition 3.5.
• compare_condition.py contains the experiment of Section 7.2 which compares the performance of methods with and without preconditioning.

Requirements

This code requires the python packages numpy, scipy, qiskit, and qiskit_aer. To install them run

pip3 install numpy scipy qiskit qiskit_aer

or

pip3 install -r requirements.txt

The code was tested with the versions qiskit 0.43.1 and qiskit_aer 0.12.0.

Citation

If you use this code please cite our paper

@article{DP2024quantum,
      title={Quantum Realization of the Finite Element Method}, 
      author={Matthias Deiml and Daniel Peterseim},
      year={2024},
      eprint={2403.19512},
      archivePrefix={arXiv},
      primaryClass={quant-ph}
}