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