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Quantum algorithms for orthogonal polynomial transforms

Anupam Prakash, Shree Hari Sureshbabu, Dylan Herman, Shouvanik Chakrabarti

arXiv:2610.03581v1quant-phcs.DS

TL;DR

Orthogonality and a three-term recurrence alone do not guarantee a fast basis change. This paper links Askey-scheme polynomials to quantum optical gates, realizing efficient transforms for several families and a quadratic-speedup Hahn transform.

  • Problem

    A three-term recurrence completely characterizes orthogonality, yet a generic resulting basis change can be dense and cost Θ(N^2).

    Source: §1 · p. 3

  • Method

    The paper maps Askey-scheme polynomial transforms to quantum optical gates and extends digital compilation to SU(2) and SU(1, 1).

    Source: §1 · p. 6, §1.3 · p. 11

  • Results

    Efficient transforms are realized for Charlier, Meixner, Krawtchouk, and Laguerre; the Hahn transform runs in O(N polylog N), a quadratic speedup over naive O(N^2).

    Source: §1 · pp. 4, 6

  • Takeaways & Limitations

    Applications of the designed transforms to PDEs and quadrature remain future directions.

    Source: §1.3 · p. 12

    The Jacobi transform remains incomplete because the radial label l must be erased efficiently, a bottleneck left for future work.

    Source: §4.2 · p. 43

Abstract

from arXiv · show

A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states $\sum_k \alpha_{k} \ket{k}$ coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure $\mu(x)$. 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 $O(\text{polylog}(N, 1/\epsilon))$, where $N$ is the grid size and $\epsilon$ 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 $O\!\left(N\operatorname{polylog}((N+\alpha+\beta+1)/\epsilon)\right)$-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 $\nu>0$ 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.

1 Introduction

The section presents quantum orthogonal polynomial transforms by relating Askey-scheme polynomials to rank-one groups and Gaussian optical operations, and situates them among Gaussian-gate simulation and prior QOPTs. It outlines constructions for Meixner, Hahn, and Laguerre transforms, reports their stated algorithmic results and Jacobi bottleneck, and discusses hardware fit and future applications.

1 Introduction

  • Fast quantum transforms require structure beyond a three-term recurrence; the paper identifies Askey families’ second-order operators and rank-one group representations as exploitable structure.The relevant groups are the Heisenberg–Weyl group, SU(2), and SU(1, 1), whose operations connect to Gaussian optical gates.Source: §1 · pp. 3–4
  • Continuous circuit implementations use a discrete grid and energy cutoff, so they do not preserve exact orthogonality; the paper leaves several higher Askey families and general efficient Jacobi transforms unresolved.The stated efficient Jacobi route is restricted, and the Wilson, Racah, continuous Hahn, and continuous dual Hahn families are not addressed by these techniques.Source: §1 · pp. 4, 7

1.1 Discrete QOPT Results

  • Charlier, Meixner, and Krawtchouk transforms have polylogarithmic dependence on grid size through Gaussian optical gates identified with their polynomial families.Displacements, squeezers, and beam splitters correspond respectively to Charlier, Meixner, and Krawtchouk; their measures are Poisson, negative binomial, and binomial.Source: §1 · p. 6, §1.1 · pp. 7–8
  • The paper generalizes Gaussian-gate compilation to compositions including squeezing, with an error analysis that permits a nonconstant number of modes.The revised analysis suppresses a doubly exponential error term in the mode count.Source: §1.1 · pp. 7–8
  • The Hahn transform runs in O(N polylog N) time, giving a quadratic speedup over the naive O(N^2) algorithm via SU(2) Clebsch–Gordan coefficients and Racah recoupling.For allowed integer parameters α and β, the algorithm has polylogarithmic dependence on their magnitudes and no additional polynomial dependence on |α−β|.Source: §1.1 · pp. 8–9

1.2 Continuous QOPT Results

  • For integer m, the Cartesian-grid Laguerre transform uses a fast-forwarded oscillator, with gate count and live workspace polynomial in log(NF + 1) and log(1/ϵ).The paper also presents a logarithmic-grid Laguerre framework and identifies efficient Hahn transformation and angular-quantum-number measurement as bottlenecks for Jacobi.Source: §1 · p. 7, §1.2 · pp. 9–10

1.3 Summary and related work

  • The summary maps Charlier polynomials to displacement operators and Krawtchouk polynomials to beam splitters, with Krawtchouk operations also implementable as arithmetic rotations or qubit rotations.The corresponding Charlier measure is Poisson, while Krawtchouk beam splitters act at fixed total photon number.Source: §1.3 · pp. 10–11
  • Single-mode squeezing on the SU(1,1) number basis realizes the Meixner transform with a negative-binomial thermal measure, implementable via hyperbolic rotations and Gaussian-gate composition.Source: §1.3 · p. 11
  • An SU(2) Clebsch–Gordan coupling in a fixed spin-j sector realizes the Hahn transform on {0, ..., N}, while the described cascade-style implementation leaves polylogarithmic implementation open.The construction identifies Hahn polynomials with Clebsch–Gordan coefficients and resembles the cascade implementation of the Schur transform.Source: §1.3 · p. 11
  • The Laguerre transform uses Cartesian-to-polar grid conversion and a circular beam splitter for a single integer parameter, while a logarithmic grid supports non-integer parameters.Source: §1.3 · p. 11
  • The quantum Hermite transform [JIS+25] was the first QOPT in a quantum circuit, based on digital fast-forwarding of the harmonic oscillator derived from Lie-algebraic methods and Wei–Norman transformations.Its exponentially efficient circuit implementation on a discrete grid required the error analysis in [JIS+25].Source: §1.3 · p. 11
  • The digital QOPTs extend simulation from SU(2) gates to SU(1,1) gates and sharpen analysis in parameter regimes treated as constant in.All digital implementations in this work use the extended framework; the earlier quantum Hermite transform [JIS+25] used digital fast-forwarding of the quantum harmonic oscillator, with error analysis for efficient circuits.Source: §1.3 · p. 11
  • The work unifies Krawtchouk-transform appearances in, [MFGH26], and [AA11] and provides a conceptual framework and family of new QOPTs.Source: §1.3 · p. 11
  • Applying these QOPTs to PDE solution and quadrature remains future work; [CLO21] uses a quantum Chebyshev transform for PDEs, with Chebyshev as Jacobi α = β = −1/2.Source: §1.3 · p. 12
  • The optical gates are native to continuous-variable and hybrid processors, but this work digitally compiles them on qubit-encoded truncated Fock spaces for standard gate-based systems.On hybrid hardware, a displacement, beam splitter, or squeeze gate can be a single physical instruction; the digital route uses the quantum Hermite transform [JIS+25] and does not assume native bosonic hardware.Source: §1.3 · p. 12

2 Preliminaries

The preliminaries establish the oscillator and Schwinger representations used in the Charlier, Meixner, and Krawtchouk QOPTs and Laguerre constructions, then describe optical gates and their digital implementation.

2.0.1 Quantum harmonic oscillator

  • A single bosonic mode has position and momentum operators, with Fock states as number and energy eigenstates represented by Hermite functions in position.Source: §2.0.1 · pp. 12–13

2.0.2 Schwinger SU(2) representations

  • Fixing total photon number in two oscillator modes realizes a spin-j SU(2) representation, where number-preserving optical operations such as beam splitters act as rotations.Source: §2.0.2 · pp. 13–14

2.0.3 Schwinger SU(1, 1) representations

  • Two-mode quadratic generators preserve number difference and yield SU(1,1) sectors; Laguerre and Meixner constructions use this representation theory in different oscillator bases.Source: §2.0.3 · pp. 14–15

2.1 Quantum optical gates and their digital implementations

  • Optical gates are unitaries generated by linear or quadratic oscillator expressions, including displacements, beam splitters, single-mode squeezers, and phase rotations.Source: §2.1 · p. 15

2.2 Digital implementation of optical operations

  • Adding the Fock-diagonal Kerr gate to Gaussian gates gives universal optical computation, with an n-gate, K-mode circuit simulated using O(poly(n, K, log(1/epsilon))) one- and two-qubit gates and O(poly(log(K), log(n))) qubits.Source: §2.2 · pp. 16, 18
  • Digital Gaussian-gate simulation embeds the first N Fock levels in a larger grid, applies discretized quadrature evolution, and maps back, choosing N and L to control leakage and discretization error.Source: §2.2 · pp. 16–17

3 Discrete orthogonal quantum polynomial transforms

This section develops discrete quantum orthogonal polynomial transforms for Charlier, Meixner, Krawtchouk, and Hahn polynomials using coherent-state, generalized-displacement, and SU(1,1)/SU(2) optical-operation approaches. For Hahn, it relates the transform to Clebsch–Gordan coupling and Racah recoupling, and gives a linear-operation compilation with O(N polylog((N+α+β+1)/ε)) gate complexity and clean ancillary registers.

3 Discrete orthogonal quantum polynomial transforms

  • Charlier, Meixner, and Krawtchouk transforms use generalized displacement operators with polylogarithmic grid-dimension resources, while the Hahn transform uses SU(2) Clebsch–Gordan coefficients with resources linear in lattice dimension.Source: §3 · p. 19

3.1 Unified framework for Charlier, Meixner and Krawtchouk Transforms

  • For Charlier, Meixner, and Krawtchouk transforms, generalized displacements are ordinary displacements, SU(1,1) squeezes, or SU(2) beam splitters, whose number-basis measurements reproduce the corresponding orthogonality measures.Applying the displacement to arbitrary number states yields the polynomial transform up to a diagonal coherent-state envelope and phase.Source: §3.1 · pp. 19–20

3.1.1 Charlier transform

  • The displacement operator maps each number state |k⟩ coherently to the degree-k Charlier polynomial state.Its number-basis matrix elements factor into a coherent-state amplitude and the Charlier polynomial.Source: §3.1.1 · pp. 21, 23

3.1.2 Meixner transform

  • In a fixed number-difference sector q, the SU(1,1) squeeze-operator matrix elements give Meixner polynomials with β=2k and c=|z|², orthogonal under the negative binomial distribution.The construction realizes integer β=|q|+1 families.Source: §3.1.2 · pp. 24–25
  • The Meixner transform uses a fixed-q SU(1,1) squeeze with rMx = O(1 + rho_z) bounded Gaussian factors, assuming Nq = N + |q| = Omega(log(1/epsilon)).The squeeze is decomposed into bounded Gaussian pieces, with Cartesian-to-radial basis changes and reversible arithmetic embedding and unembedding.Source: §3.1.2 · pp. 26–27

3.1.3 Krawtchouk transform

  • Krawtchouk coefficients are beam-splitter matrix elements in the fixed-total-photon-number SU(2) sector, with p = sin^2(theta) and O(1) bounded Gaussian factors uniformly for 0 < p < 1.The beam-splitter kernel yields the Krawtchouk polynomial representation; a fixed-number sector embedding and unembedding complete the transform.Source: §3.1.3 · pp. 28–29

3.2 Hahn Transform

  • The Hahn transform is a fixed-M Clebsch–Gordan block implemented with O(N) coupling and recoupling operations rather than O((N+1)^2) Givens operations.The procedure uses O(N) qubits plus polylogarithmic arithmetic workspace; the same circuit implements the normalized dual Hahn transform when the roles of x and l are exchanged.Source: §3.2 · pp. 30–32
  • Racah recoupling maps the symmetric coupling-path state to a fixed path, disentangling it from the total-spin register so the path can be uncomputed.The sequential coupling-path state is generally a superposition of paths, so it remains entangled with the total-spin output until recoupling.Source: §3.2 · p. 34
  • In the reversible arithmetic and controlled-rotation model, Algorithm 4 implements the quantum Hahn transform for α, β, N ∈ Z≥0 and 0 < ε < 1, returning all ancillary registers to their initial states.Source: §3.2 · pp. 35–36
  • The circuit uses O(N) controlled-rotation and arithmetic layers, O(N) qubits, and rotation synthesis precision Θ(ε/(N+1)) to bound total error by ε.Spin labels use O(log(N+α+β+1)) bits, while arithmetic workspace is reused.Source: §3.2 · p. 37

4 Continuous quantum polynomial transforms

The continuous transforms are framed through Schrödinger operators obtained from classical orthogonal-polynomial equations. The section develops two Laguerre-transform routes and identifies the remaining bottlenecks for a Jacobi transform.

4 Continuous quantum polynomial transforms

  • The Hermite, Laguerre, and Jacobi transforms correspond to the quantum harmonic oscillator, radial oscillator, and trigonometric Pöschl–Teller operator, respectively.Fast forwardability of these Schrödinger operators is equivalent to an efficient quantum orthogonal polynomial transform.Source: §4 · p. 38

4.1 Quantum Laguerre transform

  • The Laguerre transform has a Cartesian-grid route for integer orders and a logarithmic-grid route for all orders ν > 0.The Cartesian approach uses planar-oscillator separation; the logarithmic approach fast-forwards the radial oscillator.Source: §4.1 · p. 39
  • The Cartesian construction loads circular-mode occupations, converts to Cartesian modes, applies two quantum Hermite transforms, and corrects the angular phase.The phase correction may be omitted when the full angular wavefunction is required.Source: §4.1 · pp. 40–41
  • For integer ν = |m|, the Cartesian-grid Laguerre circuit has gate count and live workspace polynomial in log(NF + 1) and log(1/ϵ).The stated cutoff is NF = Θ(NCart + log(1/ϵ) + 1), with NCart = 2N + |m|.Source: §4.1 · pp. 41–42

4.2 Quantum Jacobi transform

  • The Laguerre method does not generalize to Jacobi because the relevant operators HSW, A, and B do not close into a Lie algebra.A and B belong to distinct maximal commuting sets and do not commute with each other.Source: §4.2 · p. 43
  • For the Smorodinsky–Winternitz oscillator, the Hahn transform is the overlap between uncoupled and coupled bases, but an efficient Jacobi transform also requires erasing the radial label l.The required efficient measurement of l remains a bottleneck and is left to future work.Source: §4.2 · p. 43

5 Quantum Laguerre transform on a logarithmic grid

This section develops a fixed-order quantum Laguerre transform on a logarithmic radial grid, combining radial-oscillator evolution via Hankel transforms and chirps with phase estimation to prepare the requested Laguerre degree. It approximates the target map coherently, with gate and qubit bounds for geometrically spaced radii.

5.1 Radial oscillators and Laguerre functions

  • For every fixed ν ≥ 0, the Friedrichs radial oscillator has Laguerre functions as an orthonormal eigenbasis, allowing phase estimation to recover the polynomial degree.After subtracting the fixed eigenvalue offset ν + 1, the eigenvalue is 2n; choosing a phase-estimation modulus above the retained cutoff recovers n on that subspace.Source: §5.1 · p. 45

5.2 Radial factorization

  • Radial-oscillator evolution factors exactly into seven operations: four Hankel transforms and three radial chirps, with the potential diagonal on the logarithmic grid.The factorization follows from the oscillator operators’ closed su(1, 1) commutators, and the kinetic evolution is a Hankel-conjugated chirp.Source: §5.2 · p. 45, Then the Friedrichs radial oscillator… · p. 47

5.3 Hankel transform in logarithmic coordinates

  • In logarithmic coordinates, the Hankel transform becomes a Fourier multiplier followed by frequency reflection, with a pure-phase multiplier whose order dependence is confined to its phase table.The multiplier has unit magnitude for real frequencies, so the transcendental-kernel transform reduces to a Fourier transform, phase application, and reflection.Source: §5.3 · pp. 47–48
  • A WKB trial state with positive overlap is filtered by phase estimation and fixed-point amplification to prepare the requested degree without measuring its degree register.The algorithm evaluates trial amplitudes and diagonal phases reversibly rather than loading an L-entry table.Source: §5.3 · pp. 49–50
  • For fixed ν ≥ 0, the transform approximates normalized Laguerre samples on a logarithmic grid with Euclidean-norm error at most ϵ and grid size L = polyν(N, ϵ^-1).The grid represents geometrically spaced radii; the construction does not resample onto a grid uniform in r.Source: §5.3 · pp. 50–51
  • Phase estimation writes the exact degree k, and each controlled power uses one finite propagator rather than repeating a short-time simulation 2j times.The exact-degree phase encoding distinguishes k from its residue modulo M.Source: §5.3 · p. 51
  • The circuit output is within ε of map (157) for arbitrary coherent superpositions in the retained sector.Operator-norm evolution bounds and degree-label-preserving preparation and filtering give the stated error guarantee.Source: §5.3 · p. 52
  • The circuit uses Oν(B^15) gates and Oν(B^12) qubits.Reusable arithmetic workspace yields the qubit bound.Source: §5.3 · p. 52
  • The algorithm represents amplitudes on a logarithmic grid at radii r_j = e^x_j; it does not include resampling to uniform-r or Cartesian grids.A coherent conversion to either representation would require a separate resampling algorithm.Source: §5.3 · p. 52

6 Additional Details on Digital Implementation

This section details digital implementations of Gaussian optical gates, including discrete oscillator-state approximations to continuous Hermite states and extensions to larger gate sets such as non-compact SU(1,1). It also describes position-basis diagonal grid evolutions with logarithmic gate and qubit costs.

6.1 Phase space representations of Gaussian optical gates

  • An R-gate optical circuit compiles to an O~(R^3) one- and two-qubit gate circuit by decomposing Gaussian gates into simpler evolutions and implementing them digitally.The compilation uses the quantum Hermite transform to move between oscillator and position representations.Source: §6.1 · pp. 52–53

6.2 The discrete oscillator and discrete quantum Hermite transform

  • For low-energy states, the discrete oscillator states are approximately orthonormal and approximate eigenfunctions of the discrete harmonic oscillator, supporting the digital use of continuous Hermite states.The stated regime is m ≤ cL for a constant c.Source: §6.2 · p. 54

6.3 Efficient digital implementation of Gaussian optical gates

  • The framework generalizes [JIS+25] to larger Gaussian gate sets, handles non-compact SU(1,1), improves digital-resource bounds, and gives circuits on O(K log L) qubits.The digital implementation theorem concerns Gaussian optical gates and specifies one- and two-qubit gates.Source: §6.3 · pp. 55, 59, 60
  • Each of at most RM grid evolutions is position-basis diagonal up to a constant number of L-point Fourier transforms, uses O(log_2 L) gates, and K grid registers require O(K log L) qubits.The gate count is per grid evolution.Source: §6.3 · p. 60

Disclaimer

The disclaimer states that the document is not investment research or advice and disclaims warranties and liability.

Disclaimer

  • The document disclaims warranties and liability and says it is not investment research, investment advice, or a recommendation to transact.It also says the paper is for informational purposes and is not a product of JPMorgan Chase’s Research Department or its affiliates.Source: Disclaimer · p. 61

A Log-grid discretization and error control

This appendix establishes that one logarithmic grid can approximate the Hankel transforms used during bounded-time radial evolution and preserve inner products among retained Laguerre eigenfunctions. Its error analysis confines wavefunctions before treating periodization and sampling.

A Log-grid discretization and error control

  • One logarithmic grid approximates each Hankel transform in bounded-time radial evolution and approximately preserves inner products among retained Laguerre eigenfunctions.The proof controls tails before periodization and sampling, because Fourier accuracy requires confinement to a finite log interval.Source: A Log-grid discretization and error… · p. 63
  • The radial eigenfunctions selected for accurate phase estimation are represented on the finite logarithmic grid by a sampled-column map.The operator PE selects the eigenfunctions, while AE records their sampled representations.Source: A Log-grid discretization and error… · p. 63

A.1 Intermediate Hankel-chirp states and endpoint control

  • Intermediate states are kept in finite-dimensional families stable under exact Hankel and chirp operations, so the same finite Hankel block applies at each stage.The factorization contains four Hankel transforms and three chirps.Source: §A.1 · p. 63
  • The endpoint analysis controls the singular origin uniformly over the stated compact width region.Removing the phase reduces the estimate to a Laguerre-Gaussian space with real Gaussian parameter β in [1/2, 1].Source: §A.1 · p. 64

A.2 Periodization and sampling aliases

The appendix analyzes how finite log-grid periodization and sampling approximate radial Hankel evolution. It bounds the resulting errors and establishes a polynomial grid-size requirement for the approximation and sampled-family inner products.

A.2 Periodization and sampling aliases

  • The analysis first confines radial tails with a smooth log-coordinate window, then chooses interval length and grid size to keep periodization, spectral, and reconstruction errors within δ budgets.The window’s derivatives are supported only on its transition region.Source: §A.2 · pp. 64, 66
  • Periodization folds translated log-coordinate intervals of the Hankel output back into the finite interval, while sampling errors include aliasing and a separate multiplier-phase error.Aliasing combines Fourier coefficients whose indices differ by multiples of L; the multiplier error assigns one finite-grid phase to their sum.Source: §A.2 · p. 65
  • The low-ringing shift preserves conjugate phases across paired frequencies and assigns a dedicated phase to the Nyquist frequency.The stated choices make the finite diagonal phase relation exact; rounding the shift changes the output by a bounded amount.Source: §A.2 · p. 66
  • L = O(poly_ν(E, δ^-1)) suffices for spectral aliasing and sampling estimates, and the same grid bounds inner products among retained Laguerre states.The appendix also uses the grid reconstruction estimate to bound the sampled Gram matrix.Source: §A.2 · p. 66
  • The finite FFTLog block approximates the exact Hankel transform at every intermediate stage of the radial factorization.The theorem states this for all |t| ≤ π/2 and unit inputs in ran PE.Source: §A.2 · p. 67
  • Only the four Hankel replacements contribute approximation error in the finite radial evolution block, because chirps are diagonal on the same log grid.The passage says sampling commutes exactly with each radial chirp and the Hankel-stage errors sum to the displayed 4δ bound.Source: §A.2 · p. 67

A.3 Sampled states and spectral filtering

The section prepares sampled Laguerre trial states and uses spectral filtering to obtain degree-labeled states coherently, despite finite-grid nonorthogonality. Its resource estimate gives an Oν(B^15)-gate circuit using Oν(B^12) logical qubits.

A.3 Sampled states and spectral filtering

  • A clean unitary controlled by degree n prepares a sampled WKB state, while an orthonormal comparison family addresses finite-grid loss of exact Laguerre orthogonality.The comparison family’s columns differ from the requested normalized samples by a Gram-error bound.Source: §A.3 · pp. 68–69
  • When λ+d+2g ≤ c_trial(ν)/4, equality testing and one common fixed-point amplification sequence prepare every |v_n⟩ coherently.The same amplification polynomial acts in every degree block, so the superposition error is the largest block error rather than the sum of block errors.Source: §A.3 · pp. 69–70
  • The degree and phase-estimation labels can be erased coherently by relabeling |v_n⟩, subtracting n modulo M, and reversing the labeler.The subtraction is an exact reversible permutation; the two approximate-labeler uses add at most 2λ error.Source: §A.3 · pp. 69–70

A.4 Resource estimates

  • The full circuit uses Oν(B^15) gates and Oν(B^12) logical qubits.The resource estimate chooses local operator errors within the total error budget and reuses work registers across calls.Source: §A.4 · p. 70

B Circular beam splitters

The section identifies radial oscillator states from circular-mode occupation numbers and describes a Cartesian-gate phase correction within the circular-to-radial basis change.

B Circular beam splitters

  • The phase correction is also expressible using Cartesian gates as part of the Cartesian-to-radial mode basis change.The passage identifies the gates Rj as phase/rotation gates on mode j.Source: B Circular beam splitters · p. 71
  • A Cartesian-gate product P is given as exp(−iπN−/2) for the phase correction.The product is specified using two mode rotations and a beam splitter, each with angle −π/4.Source: B Circular beam splitters · p. 72
  • For |n1,n2⟩+ states, the radial quantum number is min(n1,n2) and angular momentum is n1−n2.This identification establishes the stated radial-mode correspondence.Source: B Circular beam splitters · p. 72

C Displacement operator and Charlier transform

The section connects displacement-operator matrix elements to Charlier polynomials and derives a backward-shift representation of the family. The representation is tied to the Poisson weight’s generating-function structure.

C Displacement operator and Charlier transform

D Squeeze operator and the Meixner transform

The section relates SU(1,1) squeeze-operator matrix elements to Meixner polynomials. The resulting Meixner family is orthogonal with respect to a negative-binomial distribution.

D Squeeze operator and the Meixner transform

E Beamsplitter operator and the Krawtchouk transform

The section relates the beam-splitter action on two bosonic modes to Krawtchouk polynomials. Its operator uses p = sin^2 θ and q = cos^2 θ.

E Beamsplitter operator and the Krawtchouk transform

  • The Krawtchouk polynomials are stated to satisfy an orthogonality relation with binomial-distribution weights.The supplied fragment identifies the weights as p^n q^(N−n).Source: E Beamsplitter operator and the… · p. 75
  • The beam-splitter operator is defined on two bosonic modes and intertwines their creation operators with sine- and cosine-weighted combinations.The identities specify the transformed creation operators under B(θ).Source: E Beamsplitter operator and the… · p. 76
  • For p = sin^2 θ and q = cos^2 θ, the Krawtchouk polynomial K_k(n; p, N) is identified with a 2F1 expression.The supplied passage gives K_k(n; p, N) = 2F1(−k, −n; −N; 1/p).Source: E Beamsplitter operator and the… · p. 76

F Hahn polynomials and Clebsch-Gordan coefficients

The section identifies Hahn polynomials with Clebsch–Gordan coefficients for specified SU(2) representations. It establishes the identification by matching their difference equation to the total-spin eigenvalue problem and fixing the sign by normalization.

F Hahn polynomials and Clebsch-Gordan coefficients

G Computation of the Racah Rotation Amplitudes

The section derives the Racah rotation amplitudes for the relevant intermediate labels and gives their role in a layered circuit. The resulting matrix is orthogonal, with a one-channel transformation at triangle boundaries.

G Computation of the Racah Rotation Amplitudes

  • The computed Racah matrix is orthogonal, with its row indexed by e and column by d.The two numerators in the displayed entries add to the denominator.Source: G Computation of the Racah Rotation… · pp. 83–84
  • For each layer, the circuit computes ϑt from j1, St−1, and Kt, applies a phase-consistent single-qubit rotation, then uncomputes its arithmetic workspace.At a triangle boundary, it uses the corresponding one-dimensional transformation because one channel is absent.Source: G Computation of the Racah Rotation… · p. 84
  • The layer's rotation coefficient is obtained by substituting the layer labels into the Racah matrix entry formulas.The supplied passages identify e = Kt−1, f = Kt, and d = St.Source: G Computation of the Racah Rotation… · p. 84

H WKB analysis for the radial Laguerre trial state

The section constructs localized WKB trial states for fixed-order Laguerre functions and bounds their spectral leakage. The cutoff stays inside the oscillatory region, while a separate low-label construction handles the finite initial branch.

H WKB analysis for the radial Laguerre trial state

  • The oscillatory approximation is localized by a C2 cutoff supported on [1/4, 3/4] and equal to one on [1/3, 2/3].The section scales position by Λn,ν = 4n + 2ν + 2 and uses zn,ν(x) = e^(2x).Source: H WKB analysis for the radial Laguerre… · pp. 84–85
  • For each fixed ν ≥ 0, the WKB trial-state lemma provides normalized trial states with a spectral-leakage bound for K ≥ n + 1.The bound is stated as ||...||^2 ≤ Ctrial(ν)(n + 1)/(K + 1), with constants depending on ν.Source: H WKB analysis for the radial Laguerre… · p. 85
  • A fixed low branch is constructed from a nonnegative compactly supported function, with positivity and overlap certified for the finitely many lower labels.The construction uses interval arithmetic and interval quadrature to certify positivity and lower bounds.Source: H WKB analysis for the radial Laguerre… · p. 86
  • The trial states lie in the Friedrichs form domain, and their energy estimate bounds the spectral weight at energies at least 2K + ν + 1.The proof bounds the quadratic form on the compact support and combines it with the energy threshold.Source: H WKB analysis for the radial Laguerre… · pp. 86–87
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