I’m happy to share QRecoupling.jl, a new Julia package for the exact and numerically stable evaluation of quantum recoupling coefficients ( quantum 3j- and 6j-symbols), topological fusion data (F-symbols, R-matrices), and general q-hypergeometric series.
It supports evaluations at roots of unity (e.g., \mathrm{SU}(2)_k model) as well as for generic q \in \mathbb{C}^\times.
The Problem
Direct evaluation of q-deformed symbols is highly unstable. Because these amplitudes are alternating sums of massive quantum factorials, standard approaches suffer from these bottlenecks:
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Standard Numerics (
Float64): Catastrophic cancellation destroys all significant digits at moderate spins. -
Standard Symbolic (CAS): Expanding q-factorials as rational polynomials triggers massive intermediate expression swell, leading to GB-scale memory limits.
The Solution: Deferred Cyclotomic Representation (DCR)
QRecoupling.jl avoids both issues by fundamentally changing the order of operations. Instead of the standard expand → cancel → evaluate pipeline, we do:
factor → exact cancellation → evaluate
The package compiles each amplitude once into a Deferred Cyclotomic Representation—a sparse integer-exponent vector over the basis of irreducible cyclotomic polynomials \Phi_d(q^2). All cancellations happen symbolically and exactly at the integer-exponent level before any field evaluation occurs.
Why this matters (The Performance):
For a symmetric 6j-symbol at j=120:
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Standard exact polynomial methods exceed 50 GB of memory usage.
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DCR construction takes ~50 KB, and exact cyclotomic evaluation completes using < 400 MB.
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For numerical evaluation, the DCR compresses the dynamic range of intermediate terms by thousands of orders of magnitude, vastly extending the accuracy of
Float64.
Quick Example: Compile Once, Evaluate across many regimes
Because the DCR isolates the algebraic structure, the same compiled object can be seamlessly projected into different regimes:
using QRecoupling
julia> ex1 = q6j(1,1,1,1,1,1);
# Float64 evaluation (uses optimized Log-Sum-Exp)
julia> qeval(ex1,k=10)
0.1547005383792515
#Alternatively, get full evaluation directly
julia> q6j(1,1,1,1,1,1, k=10)
0.1547005383792515
# Exact algebraic evaluation in the cyclotomic field Q(ζ_n) via `Nemo.jl`
julia> qeval(ex1,k=10, exact=true)
Exact Algebraic Result in ℚ(ζ₂₄):
Value: (-2//3*ζ^6 + 4//3*ζ^2 - 1)
# The Classical limit (q → 1)
julia> qeval(ex1,q=1)
0.16666666666666663
# The Classical limit (q → 1) compatible with `WignerSymbols.jl`
julia> q6j(2,2,2,2,2,2, q=1.0, exact=true)
-3//70
# F-symbol
julia> fsymbol(5,5,5,5,4,4, k=50)
0.40931278202373267
# deferred fractional phase
julia> rmatrix(1/2, 1/2, 1, exact=true)
q^(1//2)
Who might find this useful?
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Mathematical Physics & Quantum Gravity: Turaev-Viro invariants and state sum models require summing 6j-symbols over all admissible colorings. The zero-allocation, thread-safe architecture of
QRecoupling.jlmakes massive, large-k state-sum contractions computationally tractable. -
Topological Quantum Computation: F-symbols and R-matrices for non-abelian anyon models are directly available with exact algebraic outputs, perfect for verifying coherence identities.
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Special Functions & Combinatorics: The underlying DCR engine can handle arbitrary q-hypergeometric series (e.g. {}_{r}\Phi_s), offering a new way to precondition combinatorial sums.
Links
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Documentation: https://sethkasante.github.io/QRecoupling.jl/stable/
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Preprint: arXiv:2604.13196 (Details on the DCR mathematics)
If you work with q-series, q-deformed quantum gravity, spin foams, or tensor categories, I’d love your feedback. Issues, ideas, and PRs are all very welcome!