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Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling
Abstract
On the span of the first N oscillator modes, Gauss--Hermite quadrature gives an exact change of basis between mode coefficients and N weighted position space samples. We implement this transform with O(Npolylog(N,1/ε)) logical gates and polylogarithmic quantum width. The operator-error bound ε holds on arbitrary superpositions and includes all auxiliary registers. The construction uses signed averages on adaptive windows to convert uniform-grid samples into weighted Hermite-root samples. Their varying widths control the amplification cost, giving the near-linear bound.
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BibTeX
@misc{mayo2026efficient,
title = {{Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling}},
author = {Nitay Mayo and Aryeh Lev Zabokritskiy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20739},
eprint = {2609.20739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20739},
}
RIS
TY - UNPB
TI - Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling
AU - Nitay Mayo
AU - Aryeh Lev Zabokritskiy
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.20739
UR - https://arxiv.org/abs/2609.20739
AB - On the span of the first N oscillator modes, Gauss--Hermite quadrature gives an exact change of basis between mode coefficients and N weighted position space samples. We implement this transform with O(Npolylog(N,1/ε)) logical gates and polylogarithmic quantum width. The operator-error bound ε holds on arbitrary superpositions and includes all auxiliary registers. The construction uses signed averages on adaptive windows to convert uniform-grid samples into weighted Hermite-root samples. Their varying widths control the amplification cost, giving the near-linear bound.
ER -