发表机构
Global Technology Applied Research, JPMorganChase（摩根大通全球应用技术研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出 Askey 方案多项式族的离散与连续量子正交多项式变换算法，实现 polylog 时间，并给出 Hahn 变换二次加速及全阶 Laguerre 变换新框架。
AI 中文摘要
量子正交多项式变换（QOPT）是一种算法原语，它将标准基态叠加 $\sum_k \alpha_{k} \ket{k}$ 相干映射到关于概率测度 $\mu(x)$ 正交的归一化单变量多项式基上。我们为 Askey 方案中的多项式族提供了新的离散和连续 QOPT，扩展了量子 Hermite 变换的结果（Jain 等人，STOC'26）。我们高效的 QOPT 算法对于离散 Charlier、Meixner 和 Krawtchouk 变换以及连续情形下的整数阶 Laguerre 变换，需要时间 $O(\text{polylog}(N, 1/\epsilon))$，其中 $N$ 是网格大小，$\epsilon$ 是误差。这些高效的 QOPT 是通过揭示 Askey 方案多项式与高斯量子光学门之间的联系，并开发一个在笛卡尔网格上编译 $SU(2)$ 和 $SU(1,1)$ 光学门的框架而获得的，该框架扩展了 Iyer 等人（arXiv:2602.15180）开发的框架。此外，我们将连续 Jacobi 变换归约为离散 Hahn 变换，并提供 $O\\!\left(N\operatorname{polylog}((N+\alpha+\beta+1)/\epsilon)\right)$ 时间的 Hahn 变换，这是相对于朴素实现的二次加速。这基于对耦合自旋 $(j_{1}, j_{2})$ 的 $\mathrm{SU}(2)$ 表示的 Clebsch--Gordan 变换的更高效编译。最后，我们通过使用三项啁啾分解和在对数网格上高效的量子 Hankel 变换算法来快速推进相应的径向振荡器，为所有阶数 $\nu>0$ 的 Laguerre 变换开发了一个新框架。
英文摘要
A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states $\sum_k α_{k} \ket{k}$ coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure $μ(x)$. We provide new discrete and continuous QOPTs for polynomial families in the Askey scheme extending the results for the quantum Hermite transform (Jain et al., STOC'26). Our efficient QOPT algorithms require time $O(\text{polylog}(N, 1/ε))$, where $N$ is the grid size and $ε$ is the error, for the discrete Charlier, Meixner and Krawtchouk transforms and for the integer-order Laguerre transform in the continuous setting. The efficient QOPTs are obtained by uncovering the links between Askey scheme polynomials and Gaussian quantum optical gates and developing a compilation framework for $SU(2)$ and $SU(1,1)$ optical gates on Cartesian grids extending the framework developed by Iyer et al. (arXiv:2602.15180). Further, we reduce the continuous Jacobi transform to the discrete Hahn transform and provide an $O\!\left(N\operatorname{polylog}((N+α+β+1)/ε)\right)$-time Hahn transform, a quadratic speedup over the naive implementation. This is based on a more efficient compilation of the Clebsch--Gordan transform for coupling $\mathrm{SU}(2)$ representations with spins $(j_{1}, j_{2})$. Finally, we develop a new framework for Laguerre transforms for all orders $ν>0$ by fast-forwarding the corresponding radial oscillator using a 3-term chirp decomposition and an efficient algorithm for the Quantum Hankel Transform on a logarithmic grid.
