Topological quantum gate compiler for SU(2)_k WZW anyonic models.
Runs entirely in the browser — one HTML file, zero dependencies. Open fiblib_v5-4.html and compile.
Intended as a reference implementation and verification layer for Fibonacci anyon gate synthesis — the compilation step that no standard quantum compiler (pytket, quilc) currently supports.
fiblib synthesises quantum gates from braid sequences of non-abelian anyons using the Solovay-Kitaev algorithm, with exact algebraic data for SU(2)_k Wess-Zumino-Witten models at levels k = 1–10.
The full pipeline:
- Braid synthesis — iterative-deepening DFS (IDDFS) over the generator set {σ₁, σ₂, σ₁⁻¹, σ₂⁻¹}, guaranteed shortest braid word achieving target ε
- SK refinement — true iterative Solovay-Kitaev: εₙ = c · εₙ₋₁^(3/2) via group commutator construction, gate count O(log^3.97(1/ε))
- 10 error checks — pentagon coherence, braid relation, unitarity, basis completeness, determinant, SK approximation distance, and more
- 57-test benchmark suite — algebraic correctness, metric properties, matrix square roots, SK convergence, cross-level consistency, numerical precision
| k | Anyons | Universal | d_{1/2} |
|---|---|---|---|
| 1 | SU(2)₁ | No | 1 |
| 2 | Ising (σ, ψ) | No | √2 |
| 3 | Fibonacci (τ) | Yes | φ = 1.6180… |
| 4 | — | No | √3 |
| 5 | — | Yes | 2cos(π/7) |
| 6 | — | No | 2cos(π/8) |
| 7 | — | Yes | 2cos(π/9) |
| 8 | — | No | 2cos(π/10) |
| 9 | — | Yes | 2cos(π/11) |
| 10 | — | No | 2cos(π/12) |
Universal levels (odd k ≥ 3) support dense approximation of any SU(2) gate by braiding alone.
R¹_{ττ} = e^{−4πi/5} (fusion to vacuum)
R^τ_{ττ} = e^{+3πi/5} (fusion to τ)
F^τ_{ττ} = [[φ⁻¹, φ⁻¹/²],
[φ⁻¹/², −φ⁻¹]] φ = (1+√5)/2
σ₁ = diag(R¹_{ττ}, R^τ_{ττ})
σ₂ = F · σ₁ · F (F self-inverse)
Braid relation: σ₁σ₂σ₁ = σ₂σ₁σ₂, error < 2.4×10⁻¹⁶
For k ≠ 3, F-matrix entries are computed via exact quantum 6j-symbols:
F^{j₁j₂}[m][n] = (−1)^{2(j₁+j₂+cₘ+cₙ)} · √(dₘ dₙ) · {j₁ j₂ cₙ; j₁ j₂ cₘ}_q
[n]_q = sin(nπ/(k+2)) / sin(π/(k+2)) q-deformed integer
The k = 3 diagonal entry F[1,1] = −φ⁻¹ is hardcoded because the q6j formula returns the wrong sign for this entry (phase convention mismatch on the diagonal). The benchmark explicitly guards against this regression.
The FIB_DATA/twist-formula phases for k = 2 (−135°, +45°) are the eigenvalues of σ₁² — the squared braid generator — not σ₁ itself. The actual matrix eigenvalues are their principal square roots:
R^1_{σσ} = e^{−3iπ/8} (−67.5°) σ₁²: e^{−3iπ/4} = −135° ✓
R^ψ_{σσ} = e^{+iπ/8} (+22.5°) σ₁²: e^{+iπ/4} = +45° ✓
The ratio β/α = e^{iπ/2} = i is the condition for braid closure with a Hadamard F-matrix. The fusion table R-phase display values (−135°, +45°) are unchanged — they are the correct topological spin ratios θ_c/(θ_a θ_b).
| Group | Tests | What is checked |
|---|---|---|
| 1 — Algebraic Correctness | 13 | R-symbols, F-matrix, pentagon, braid relation (k=3 exact) |
| 1b — Ising Braid Relation | 7 | k=2 braid relation, F-matrix, σ₁² ↔ twist consistency |
| 2 — phaseDist Metric | 5 | Phase-invariant distance: identity, symmetry, triangle inequality |
| 3 — Matrix Square Root | 10 | U(2) eigendecomposition sqrt for σ₁, σ₂, products, edge cases |
| 4 — SK Convergence | 6 | Rz(144°), Rz(72°), Z exact; S 99.9%, T 99.7%, X 99.4% |
| 5 — Verlinde / WZW | 11 | D² formula, golden ratio, palindrome symmetry, fusion positivity |
| 6 — Numerical Precision | 5 | Complex arithmetic, identity multiplication, machine-epsilon bounds |
Gate fidelities use the phase-invariant metric: fidelity = 1 − ε²/4.
| # | Name | Statement |
|---|---|---|
| T1 | Primary Quantum Dimension | d_{1/2}(k) = 2cos(π/(k+2)) |
| T2 | Palindrome Symmetry | d_j = d_{k/2−j} |
| T3 | Tetrahedral Admissible Count | ` |
| T4 | Total Quantum Dimension | D² = (k+2) / (2sin²(π/(k+2))) |
| T5 | Braiding Phase Quantisation | R ∈ exp(iπ·ℤ/(k+2)) |
| T6 | WZW ↔ LQG Equivalence | N^k_{ij} > 0 ⟺ (j₁,j₂,j₃) LQG-admissible |
| T7 | Lossless Gate Guarantee | ` |
- F[1,1] hardcode (k=3): The q6j path returns the wrong sign for the F[1,1] diagonal entry due to a phase convention mismatch. The value −φ⁻¹ is hardcoded directly. The benchmark includes a regression guard that will catch any future removal of this hardcode.
- SK constant c = 0.97 is empirical. The true Dawson-Nielsen bound is gate-set dependent; convergence is not guaranteed for all targets at k ≠ 3. The constant was chosen to avoid divergence (c = 4 diverges for ε₀ > 1/16).
- Single-qubit only. Two-qubit gates (CNOT, Toffoli) are not implemented. The compiler handles single-qubit SU(2) synthesis only. Two-qubit encoding requires a 6-anyon fusion space and is future work.
- No hardware interface. fiblib outputs braid words and matrices. Translating these to physical control pulses for Majorana or photonic anyon hardware is outside the scope of this implementation.
Open the HTML file in any modern browser. No build step, no server, no install.
Compiler tab — select gate, anyon level, precision ε, and hit Compile. Shows fusion tree, F/R matrices, composed unitary, braid word, SK refinement table, and 10 error checks.
Level Overview — quantum dimensions, admissible triple counts, universality for k = 1–10.
Fusion Tables — all admissible triples with R-phases and lossless certification.
Validity Certificate — check any spin sequence for pairwise Verlinde channel consistency.
7 Theorems — proofs and k = 1–10 verification matrix.
SK Convergence — interactive εₙ table for any k and starting error.
Benchmark Suite — runs all 57 tests in-browser, ~4–6 seconds on first run (epsilon-net is cached for the session).
- Kitaev, A. (2006). Anyons in an exactly solved model and beyond. Annals of Physics, 321(1), 2–111.
- Preskill, J. (2004). Lecture Notes on Quantum Computation, Chapter 9: Topological Quantum Computation.
- Dawson, C. M., & Nielsen, M. A. (2006). The Solovay-Kitaev algorithm. Quantum Information & Computation, 6(1), 81–95.
- Kauffman, L. H., & Lins, S. L. (1994). Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Princeton University Press.
- Finkelberg, M. (1996). An equivalence of fusion categories. Geometric and Functional Analysis, 6(2), 249–267.
- Bonderson, P. (2007). Non-Abelian Anyons and Interferometry (PhD thesis). Caltech.
If you use fiblib in research, please cite this repository:
@software{fiblib,
title = {fiblib v5.4: Topological Quantum Gate Compiler},
author = {Alfred Kimani},
year = {2026},
url = {https://github.com/eifla-a/fiblib}
}
MIT
Questions and discussion welcome — open an issue.