Quantum algorithms for orthogonal polynomial transforms

2026-10-02

AI Summary

This study addresses the acceleration of orthogonal polynomial transforms in scientific computing by proposing discrete and continuous quantum orthogonal polynomial transform algorithms encompassing the entire polynomial families of the Askey scheme. Methodologically, it reveals intrinsic connections between polynomials and Gaussian optical gates, constructs a compilation framework based on SU(2)/SU(1,1) groups, and integrates quantum optical gates, group representation theory, Chirp decomposition, and quantum Hankel transforms to achieve efficient circuit design. The core contribution lies in attaining near-logarithmic time complexity for multiple transform classes, with the Hahn transform specifically achieving a quadratic speedup. These advances establish a novel paradigm for quantum-enhanced scientific computing.

Abstract

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.

Problem

• Quantum orthogonal polynomial transform
• Askey scheme
• Quantum algorithms
• Polynomial families
• Quantum optical gates

Innovation

• Quantum orthogonal polynomial transform
• Askey scheme
• Gaussian quantum optical gates
• Clebsch-Gordan transform
• Quantum Hankel transform