Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling

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 $O(N\operatorname{polylog}(N,1/\varepsilon))$ logical gates and polylogarithmic quantum width. The operator-error bound $\varepsilon$ 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.

Source: arXiv:2609.20739v1 - http://arxiv.org/abs/2609.20739v1 PDF: https://arxiv.org/pdf/2609.20739v1 Original Link: http://arxiv.org/abs/2609.20739v1