Optimal kernel functions for linear combination of Hamiltonian simulation
Abstract
Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error ε, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion 2e/π(log1/ε−loglog1/ε+o(1)) as ε↓0. The provably optimal leading coefficient 2e/π improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.
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BibTeX
@misc{apel2026optimal,
title = {{Optimal kernel functions for linear combination of Hamiltonian simulation}},
author = {Harriet Apel and Bjorn K. Berntson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03669},
eprint = {2610.03669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03669},
}
RIS
TY - UNPB
TI - Optimal kernel functions for linear combination of Hamiltonian simulation
AU - Harriet Apel
AU - Bjorn K. Berntson
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03669
UR - https://arxiv.org/abs/2610.03669
AB - Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error ε, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion 2e/π(log1/ε−loglog1/ε+o(1)) as ε↓0. The provably optimal leading coefficient 2e/π improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.
ER -