Quantum Domain Decomposition for Preconditioning the Finite Element Method
Abstract
Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be inverted is a decisive parameter. A well known classical, and now quantum, remedy is to precondition the linear system A x = b by premultiplying it by a matrix H in such a way that the condition number of HA is significantly smaller than the condition number of A. In this work, we focus on a family of preconditioners called domain decomposition. First, we prove that it is feasible to apply quantum domain decomposition. We provide upper bounds for the block-encoding parameters of the Poisson problem discretized by the finite element method and preconditioned by the two-level Additive Schwarz preconditioner (one of the most fundamental domain decomposition techniques). From these bounds, we deduce the complexity of the quantum linear system solver. Second, we focus on a particular choice of local solver within the domain decomposition preconditioner by applying recent work by [Deiml and Peterseim, Math. Comput., 2025] on the Bramble--Pasciak--Xu (BPX) preconditioner. Finally, we provide details on how the operators are implemented.
Related
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BibTeX
@misc{fressart2026quantum,
title = {{Quantum Domain Decomposition for Preconditioning the Finite Element Method}},
author = {Elise Fressart and Michel Nowak and Nicole Spillane},
howpublished = {HAL},
year = {2026},
month = may,
eprint = {2605.26090},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05851534v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Domain Decomposition for Preconditioning the Finite Element Method
AU - Elise Fressart
AU - Michel Nowak
AU - Nicole Spillane
PY - 2026
DA - 2026/05/25
PB - HAL
UR - https://hal.science/hal-05851534v1
AB - Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be inverted is a decisive parameter. A well known classical, and now quantum, remedy is to precondition the linear system A x = b by premultiplying it by a matrix H in such a way that the condition number of HA is significantly smaller than the condition number of A. In this work, we focus on a family of preconditioners called domain decomposition. First, we prove that it is feasible to apply quantum domain decomposition. We provide upper bounds for the block-encoding parameters of the Poisson problem discretized by the finite element method and preconditioned by the two-level Additive Schwarz preconditioner (one of the most fundamental domain decomposition techniques). From these bounds, we deduce the complexity of the quantum linear system solver. Second, we focus on a particular choice of local solver within the domain decomposition preconditioner by applying recent work by [Deiml and Peterseim, Math. Comput., 2025] on the Bramble--Pasciak--Xu (BPX) preconditioner. Finally, we provide details on how the operators are implemented.
ER -