The Fibonacci model has only two topological charges: the vacuum \(1\) and the nontrivial charge \(\tau\). Its defining fusion rule is short,
\[\tau \times \tau = 1 + \tau,
\]
but the plus sign changes everything. Two \(\tau\) anyons do not have a single deterministic fusion outcome. They span two allowed channels, and those channels form a small quantum state space on which braids act as matrices rather than ordinary phases.
In this project “anyon” refers to a digitally encoded mathematical model. The physical device contains superconducting qubits, not topologically protected Fibonacci quasiparticles.
A logical qubit made from fusion channels
Choose a fusion tree for several \(\tau\) charges and fix the total charge. In the first nontrivial sector, the intermediate fusion result can be either \(1\) or \(\tau\). We label those alternatives \(|0_L\rangle\) and \(|1_L\rangle\). They are not two spatial paths; they are two consistent ways to associate the same overall fusion process.
The golden ratio \(\phi=(1+\sqrt{5})/2\) fixes the relative weights. In the convention used throughout the project, the nontrivial F-matrix is
\[F = \begin{pmatrix}
1/\phi & 1/\sqrt{\phi} \\
1/\sqrt{\phi} & -1/\phi
\end{pmatrix}.
\]
Because \(1/\phi^2+1/\phi=1\), this matrix is unitary and involutory: \(F^\dagger F=I\) and \(F^2=I\). Acting on \(|0_L\rangle\) gives the frozen input used for the braid-order test,
\[F|0_L\rangle = \frac{1}{\phi}|0_L\rangle + \frac{1}{\sqrt{\phi}}|1_L\rangle.
\]
F changes the fusion basis; R performs an exchange
An F-move does not physically braid anything. It changes the association of a fusion tree — for example, from “fuse the first pair, then the third” to “fuse the second pair, then the first.” An R-move exchanges two adjacent anyons in a basis where their shared fusion channel is definite.
For two \(\tau\) charges, the counterclockwise exchange phases in the locked convention are
\[R^{\tau\tau}_{1}=e^{-4\pi i/5}, \qquad
R^{\tau\tau}_{\tau}=e^{3\pi i/5}.
\]
The first braid generator is diagonal in the chosen basis. To braid the next adjacent pair, we first change basis, apply the same channel-dependent R phases, and change back. Schematically, \(B_1=R\) and \(B_2=FRF\).
Non-Abelian means order can become observable
The braid generators obey the braid-group relation \(B_1B_2B_1=B_2B_1B_2\), but they do not commute. The project finds
| Exact check | Numerical value |
|---|---|
| Braid-relation residual | \(2.00\times10^{-16}\) |
| Frobenius norm of \([B_1,B_2]\) | \(1.7989\) |
| Pentagon maximum residual | \(3.80\times10^{-138}\) |
| Hexagon maximum residual | \(0\) |
A nonzero matrix commutator is necessary, but it is not yet an experimental witness. Two statevectors might differ only by an overall phase, which no measurement can detect. We therefore freeze both an input and an observable.
For \(P_0=|0_L\rangle\langle0_L|\), the two braid orders give
| Sequence | \(P_0\) |
|---|---|
| \(B_1\) then \(B_2\) | \(1\) |
| \(B_2\) then \(B_1\) | \(1/\phi^2=0.381966\ldots\) |
The signed contrast is exactly \(1/\phi=0.618034\ldots\). It survives optimal global-phase alignment and appears in a directly measurable probability. This is the central observable carried through every later circuit and noise test.
Coherence before compilation
The algebra was not validated in only the familiar two-dimensional sector. The implementation enumerates all 12 nonempty F-unitarity sectors, 27 nonempty pentagon sectors, and 12 hexagon sectors in both orientations. Independent SymPy, NumPy, and Qiskit paths agree to numerical precision. That matters because a correct-looking \(2\times2\) matrix does not by itself establish a coherent fusion category.
The outcome of this first layer is therefore precise: the digitally encoded representation is coherent, noncommuting, and measurement-sensitive. It says nothing yet about whether a nine-qubit circuit is shallow enough for hardware, and it says nothing about computational scaling or quantum advantage.