{"ID":23507422,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20739","arxiv_id":"2609.20739","title":"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 $O(N\\operatorname{polylog}(N,1/\\varepsilon))$ 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.","short_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 $O(N\\operatorname{polylog}(N,1/\\varepsilon))$ logical gates and polylogarithmic quantum width. The operator-error bo...","url_abs":"https://arxiv.org/abs/2609.20739","url_pdf":"https://arxiv.org/pdf/2609.20739v1","authors":"[\"Nitay Mayo\",\"Aryeh Lev Zabokritskiy\"]","published":"2026-09-17T17:26:53Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.DC\",\"cs.ET\"]","methods":"[]","has_code":false}