Deferred Cyclotomic Representation for Stable and Exact Evaluation of q-Hypergeometric Series

1. Problem Statement

Finite $q$-hypergeometric series are fundamental to quantum groups, topological quantum field theories (TQFT), and quantum gravity (e.g., state-sum models like Turaev-Viro). However, their evaluation faces two critical, intertwined computational bottlenecks:

• Numerical Instability (Catastrophic Cancellation): In floating-point arithmetic, these series often involve alternating sums of exponentially large terms. The condition number ($\kappa$) grows super-polynomially with parameters (e.g., spin $j$), leading to massive loss of precision. Standard remedies like Log-Sum-Exp (LSE) mitigate overflow but fail to resolve the cancellation of large, nearly equal numbers.
• Symbolic Expression Swell: In exact symbolic computation (Computer Algebra Systems), expanding $q$-factorials into dense polynomials in $q$ causes intermediate expressions to grow explosively in size. This "expression swell" makes exact evaluation computationally intractable for moderate parameters due to memory exhaustion and the high cost of polynomial GCD reductions.

The root cause identified is a representation mismatch: Conventional methods treat $q$-hypergeometric series as dense rational functions, obscuring the underlying multiplicative structure of quantum factorials and deferring cancellations until after expansion.

2. Methodology: Deferred Cyclotomic Representation (DCR)

The author introduces the Deferred Cyclotomic Representation (DCR), a framework that separates the algebraic structure of the series from its numerical evaluation.

Core Concepts

• Cyclotomic Factorization: Instead of treating quantum integers $[n]_q$ as polynomials, they are factorized into irreducible cyclotomic polynomials $\Phi_d(q^2)$.
  $$[n]_q = q^{1-n} \prod_{d|n, d>1} \Phi_d(q^2)$$
• Sparse Exponent Vectors: The series is represented not as a function of $q$, but as a tuple of integer exponent vectors corresponding to the basis $\{q, \Phi_2(q^2), \Phi_3(q^2), \dots\}$.
  • Multiplication and division of quantum factorials become component-wise integer addition and subtraction of these exponent vectors.
  • This converts multiplicative algebra into sparse integer arithmetic.
• Deferred Evaluation: The representation is constructed once as a parameter-independent combinatorial object (the DCR). Evaluation in any target field (e.g., $\mathbb{R}$, $\mathbb{C}$, or cyclotomic fields $\mathbb{Q}(\zeta)$) is realized later via a projection map $\Pi_T$.

The Two-Stage Process

1. Compilation Stage:
   • The series is converted into a DCR object: $S_{DCR} = (M_{base}, \{R_z\}, M_{root}, M_{rad})$.
   • $M_{base}$ is the initial term; $\{R_z\}$ are local update ratios; $M_{root}$ and $M_{rad}$ handle square roots of geometric prefactors (e.g., triangle coefficients) by separating even and odd exponents.
   • Key Feature: All algebraic cancellations (GCDs) are performed exactly at the integer exponent level before any value of $q$ is assigned.
2. Projection Stage:
   • The fixed DCR object is mapped to a target field $T$ by assigning values to $q$ and $\Phi_d(q^2)$.
   • This allows the same compiled object to be reused for exact symbolic evaluation, floating-point arithmetic, or classical limits ($q \to 1$).

3. Key Contributions

1. Cyclotomic Representation: Proved that finite $q$-hypergeometric amplitudes admit a sparse representation in a cyclotomic exponent basis, revealing the intrinsic multiplicative structure of quantum factorials.
2. Deferred Evaluation Framework: Introduced the DCR, which decouples combinatorial compilation from numerical projection, acting as a universal algebraic preconditioner.
3. Complexity Reduction: Demonstrated that DCR construction scales near-linearly with summation length, avoiding the exponential memory growth of polynomial representations.
4. Numerical Stability: Showed that by performing exact cancellations at the exponent level, the dynamic range of intermediate quantities is drastically reduced, mitigating catastrophic cancellation in floating-point arithmetic.
5. Structural Unification: Provided a unified perspective where admissibility at roots of unity, $q$-deformation, and classical limits emerge as intrinsic properties of a single combinatorial object.

4. Results and Empirical Validation

The framework was benchmarked using the Quantum 6j-symbol (a canonical test case for $U_q(\mathfrak{sl}_2)$ recoupling) in the Julia package QRecoupling.jl.

• Memory Scaling (Exact Arithmetic):
  • Eager CAS (Standard): Memory usage exploded, exceeding 50 GB for spin $j=120$ due to expression swell.
  • DCR Construction: Scales linearly ($O(j)$), using only ~51 KB for $j=120$.
  • DCR Projection: Scales roughly cubically ($O(j^3)$), remaining under 380 MB for $j=120$.
• Numerical Stability (Floating Point):
  • Dynamic Range Compression: The DCR reduced the intermediate dynamic range by over 8,400 orders of magnitude compared to eager evaluation (e.g., at $j=500$, $\gamma_{DCR} \approx 1079$ vs. $\gamma_{eager} \approx 9518$).
  • Precision: Standard double-precision (Float64) evaluation using DCR maintained correct signs and accuracy up to $j \approx 200$, whereas the eager LSE method failed (100% error) around $j \approx 90$.
• Latency and Amortization:
  • While the initial DCR construction has a cost, subsequent evaluations across different $q$ values are extremely fast (microseconds) because the combinatorial structure is pre-compiled.
  • For repeated parameter sweeps (common in TQFT partition functions), the DCR offers massive amortized speedups by avoiding redundant recomputation of factorial structures.

5. Significance and Implications

• Computational Physics: The DCR makes large-scale computations in topological quantum field theories (e.g., Turaev-Viro invariants) and spin foam models tractable. It allows for the efficient evaluation of state sums over triangulated manifolds by compiling local geometric weights once and reusing them.
• Theoretical Insight: The framework reframes $q$-deformation not as a change in the algebraic object, but as a change in the evaluation map. This suggests that topological invariance and coherence identities (like the Pentagon identity) may be rooted in purely combinatorial properties of the exponent data, independent of the deformation parameter.
• General Applicability: The principle of aligning data structures with the intrinsic multiplicative structure of a problem (rather than forcing an additive polynomial representation) offers a blueprint for improving stability and efficiency in other areas of special functions and quantum amplitude calculations.

In conclusion, the Deferred Cyclotomic Representation fundamentally alters the computational profile of $q$-hypergeometric series by shifting the complexity from algebraic manipulation to controlled projection, simultaneously solving the problems of expression swell and numerical instability.