Abstract We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner–Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials is $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$ , where $\ell_{\mathcal{D}}$ is an even positive integer ( $\mathcal{D}$ : a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.Abstract We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner–Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials is $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

arXiv:1907.12218Progress of Theoretical and Experimental PhysicsOct 11, 20198 citations

Quantum algorithms for orthogonal polynomial transforms

Anupam Prakash, Shree Hari Sureshbabu, Dylan Herman +1 more

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.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.

quant-pharXiv:2610.03581Oct 5, 20260 citations

Orthogonal polynomials and operator orderings

Adel Hamdi, Jiang Zeng B

An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.

Jan 1, 20106 citations

Quantum systems with finite Hilbert space and Chebyshev polynomials

Apostol Vourdas

Journal of Computational and Applied MathematicsAug 1, 20012 citations

Quantum Systems in Finite-Dimensional Hilbert Spaces

Gianluca Stefanucci

Springer eBooksJan 1, 20240 citations

\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}

, where

\ell_{\mathcal{D}}

is an even positive integer (

\mathcal{D}

: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

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.