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
- Dual-certificate validity — algebraic (WZW fusion) and geometric (Penrose inflation) certificates, both derived from the same matrix M
| 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 central structural result underlying the dual certificate is the exact equality:
M_Penrose = M_WZW = [[0, 1],
[1, 1]]
The Penrose quasicrystal inflation matrix and the WZW Fibonacci anyon fusion matrix are the same 2×2 integer matrix. This is not an analogy — it is an equality. Both have characteristic polynomial λ²−λ−1=0 and Perron-Frobenius eigenvalue φ = (1+√5)/2. M² = M + I in both representations.
Consequence for fiblib: Any theorem proven about M in one domain transfers automatically to the other. Every valid gate sequence has a geometric realisation as a Penrose inflation path, and every closed Penrose inflation orbit corresponds to a candidate high-fidelity gate. The two certificates check the same underlying M-consistency from different physical representations.
Question: Does the anyon braiding sequence close consistently in the k=3 Hilbert space?
Method: For each consecutive spin pair (j₁, j₂), the Verlinde formula computes N^k_{j₁j₂}^{j₃} — the number of open fusion channels. The sequence is algebraically valid if every pair has at least one open channel (N > 0).
Physical meaning: The standard topological protection check. An invalid sequence would require anyons to fuse into a channel that the WZW theory forbids.
Available: k = 1–10.
Question: Does the gate sequence correspond to a valid inflation path in a Penrose tiling?
Tile encoding:
τ-anyon (j=1, d=φ) → fat tile (F)
vacuum (j=0, d=1) → thin tile (T)
Inflation rules (rows of M = [[0,1],[1,1]]):
fat → fat + thin (F → F T)
thin → fat (T → F)
Method: Each consecutive tile pair (A, B) is checked against the Penrose inflation rule: B must appear in inflate(A). A sequence passes if all edges are valid inflation edges and the path closes — the final tile type matches the initial tile type.
Closure condition: A gate corresponding to a closed Penrose inflation orbit is topologically stable in both representations. The geometry of the tiling encodes the topology of the gate space exactly, because M is the same operator.
Available: k = 3 only. At k≠3 the fusion matrices are distinct from M_Penrose.
| Algebraic | Geometric | Verdict |
|---|---|---|
| ✓ Pass | ✓ Closed orbit | M-consistent from both angles. Gate is valid. Strong indicator of high fidelity. |
| ✓ Pass | ⚠ Open path | Valid gate; Penrose path is non-returning. Extend the sequence until closure for geometric stability. |
| ✗ Fail | ✓ Pass | Impossible in a correct implementation — signals a compiler bug or edge case. |
| ✗ Fail | ✗ Fail | Sequence is invalid in both Hilbert space and tile space. Reject. |
Two independent error-detection mechanisms means a subtle compiler bug or numerical edge case that slips past one certificate will be caught by the other. Disagreement is not a probabilistic warning — it is a logical contradiction derived from the same underlying M, and the source must be identified.
Gate sequences can be designed as inflation paths in a Penrose tiling. A sequence that starts and ends on the same tile type (closed orbit) is topologically stable in both representations. This is exact.
Example closed orbits (k=3, τ-anyons):
1, 1, 1 → F, F, F closes: F→F ✓
1, 0, 1 → F, T, F closes: F→F ✓
1, 1, 0, 1 → F, F, T, F closes: F→F ✓
| 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 99.3% |
| 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.
Achieved fidelities at benchmark thresholds:
| Gate | Threshold | Fidelity |
|---|---|---|
| Rz(144°) | ε = 0 (exact) | 100% |
| Rz(72°) | ε = 0 (exact) | 100% |
| Z | ε = 0 (exact) | 100% |
| S | ε < 0.070 | > 99.9% |
| T | ε < 0.115 | > 99.7% |
| X | ε < 0.170 | > 99.3% |
| # | 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.
- 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.
- Single-qubit only. Two-qubit gates (CNOT, Toffoli) are not implemented. Two-qubit encoding requires a 6-anyon fusion space and is future work.
- No hardware interface. fiblib outputs braid words and matrices. Translating to physical control pulses is outside scope.
- Penrose certificate (k=3 only). The geometric certificate uses M_Penrose = M_WZW, which holds exactly only at k=3.
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.
Dual-Certificate — check any spin sequence with two independent certificates: algebraic (WZW Verlinde) and geometric (Penrose inflation). For k=3 both run simultaneously; for k≠3, algebraic only.
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, results in ~1 second.
- 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.
- Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bulletin of the Institute of Mathematics and its Applications, 10, 266–271.
If you use fiblib in research, please cite this repository:
@software{fiblib,
title = {fiblib v5.4: Topological Quantum Gate Compiler with Dual-Certificate Validity},
author = {Alfred Kimani},
year = {2026},
url = {https://github.com/eifla-a/fiblib}
}
MIT
Questions and discussion welcome —— open an issue.