[1, 2, 3], identity
The Physical Seam | paths with memory
The Path That Does Not Unhappen
Exchange three points, watch their endpoints forget a winding that the braid group retains, then compile a chosen qubit rotation with Fibonacci and Ising braid matrices to see universality separate from noncommutativity and from physical protection.
In a plane, identical particles can pass around one another without colliding. The final arrangement records only where they ended. Their worldlines record how they got there. Those are different ledgers.
Layer one | exchange something
Make the endpoints lie
Press sigma1 twice. The labels return to their starting slots, so the permutation is the identity. The purple certificate still changes because the two strands wound. Then press the inverse once and watch one crossing cancel.
Three-strand braid bench
no exchanges yet
Ready. Apply an exchange.
Endpoint permutation, the S3 quotient
Artin action on the free group F3
x1 maps to x1; x2 maps to x2; x3 maps to x3
The certificate uses Artin's faithful action. For a positive adjacent exchange, it substitutes
sigma_i: x_i -> x_i x_(i+1) x_i^-1, x_(i+1) -> x_i
and cancels adjacent inverse letters after every substitution. The permutation makes the extra relation sigma_i squared = identity; the braid group does not. In two spatial dimensions the collision-free configuration space has braid group B_n as its fundamental group. In three or more dimensions the corresponding exchange paths reduce to S_n. This topological permission does not make every arbitrary matrix assignment into a physical anyon theory.
Layer two | the further result
Noncommuting is not the same as universal
The easy dismissal is fair: a pretty noncommuting braid proves little about computation. So choose a rotation axis and angle, set a maximum word length, and exhaust the braid words. The same search runs against two consistent non-Abelian representations. Fibonacci keeps finding new projective gates. This Ising representation closes into a finite projective set.
That contrast concerns what each image can reach in the limit, not which model wins at every chosen target or finite cutoff. When the target lies in the finite Ising Clifford image, Ising reaches it exactly within a phase-insensitive error tolerance of 1e-7. Fibonacci ties at gates shared by both images, but at other Clifford targets Ising can win outright. On an audit grid of 720 round-number slider settings, 216 settings, exactly 30 percent, land in the Ising image. Repeated axis-angle descriptions of the same gate count separately. Away from those points, either model can still have the smaller finite-cutoff error. The default 57, 31, 73 degree target is a generic example where Fibonacci has the lower error through length ten. The readout flags whether the current target is a Clifford point.
Exhaustive single-qubit braid compiler
ready
representationFibonacci
candidates testednot run
best wordnot run
word lengthnot run
phase-insensitive errornot run
distinct projective matrices seennot run
projective image closed under generatorsnot run
target in finite Ising imagenot run
Run the compiler to print the fusion-basis action.
Fibonacci generator convention
sigma1 = R, sigma2 = F R F, in the basis where the first pair fuses to vacuum or tau.
F = [[phi^-1, phi^-1/2], [phi^-1/2, -phi^-1]]
R = diag(exp(-4 pi i/5), exp(3 pi i/5))
R = diag(exp(-4 pi i/5), exp(3 pi i/5))
Ising control convention
The analogous vacuum or psi fusion basis, with the standard Ising F and R. Braiding alone gives projective Clifford gates.
F = [[1, 1], [1, -1]] / sqrt(2)
R = diag(exp(-pi i/8), exp(3 pi i/8))
R = diag(exp(-pi i/8), exp(3 pi i/8))
The error is sqrt(1 - |Tr(U_target dagger U_word)| / 2). Absolute phase is discarded, as it must be for a single isolated gate. Immediate generator-inverse pairs are omitted, but braid-relation duplicates remain in the search. Both models are projectively deduplicated to show growth of the image, while every freely reduced word is still tested as a candidate. A lower error found at a longer cutoff is an explicit approximation, not a proof of the density theorem. The growing Fibonacci count and the finite, closed Ising image are the live facts this search can show.
The check
These values are rebuilt now by the same equations that drive both instruments. The repository verifier separately derives them, extracts this page's own engine, and mutation-tests the checks.
Fusion root d from d squared = 1 + d
Ideal Fibonacci monodromy, -1 / d squared
Fibonacci Yang-Baxter residual
Ising Yang-Baxter residual
Words through selected length
Ising projective image found through length ten
Every fixed choice and uncertainty
- Orientation: positive generators use the displayed Artin action. Reversing every crossing conjugates the convention, not the topology.
- Fusion basis: the first two anyons fuse first. Fibonacci basis order is vacuum, tau. Ising basis order is vacuum, psi.
- Gauge: the displayed real symmetric F matrices and diagonal R matrices are one gauge. Basis rephasing changes entries but not the projective approximation error.
- Global phase: deliberately removed by the trace metric and by projective deduplication.
- Truncation: at most generators and at most freely reduced words including the empty word. This is exhaustive only inside that finite ball.
- Free reduction: only adjacent inverse pairs are forbidden during enumeration. Algebraically equal words related by sigma1 sigma2 sigma1 = sigma2 sigma1 sigma2 may both be tested.
- Numerics: ordinary double-precision complex arithmetic. Projective matrices are counted equal after phase normalization and rounding to decimal places. Residuals near machine precision are not exact proofs.
- Target: an SU(2) axis-angle rotation. The default angles are free interface choices, not physical constants.
- Physical scope: the matrices are ideal theory. The browser includes no device noise, leakage, gap, separation, adiabatic path, decoder or error correction.
Reported measurements, kept separate from computed ideals
These are source transcriptions, not browser experiments. The graph-defect paper used a -qubit square grid. It reported postselected fermion-detection signs for the indistinguishable braid and for its distinguishable control, using CZ layers and single-qubit layers. Its GHZ-like overlap was , with reported purity . These were programmed projective Ising-type graph defects, without active error correction during braiding.
A different fractional-quantum-Hall experiment in reported time-domain correlations consistent, without fitted parameters, with the one-third-state phase . This was an Abelian correlation measurement, not tracked Fibonacci transport.
The superconducting-processor paper specifically reporting Fibonacci braiding appeared in . It extracted for the quantum dimension and for monodromy, compared with the computed ideals above. Its braid was implemented by physical-qubit gates in a digital simulation, without intrinsic topological protection.
Layer three | the open edge
The algebra has outrun the protection
What is closed here is the mathematical contrast: this Fibonacci representation is braid-universal for a qubit, while standard Ising braiding is not. What remains open is the full physical conjunction: create, store, braid, fuse and read non-Abelian anyons at scale while preserving a gapped topological regime and correcting realistic faults.
A trapped-ion result prepared a -qubit quantum-double state and demonstrated a universal gate set by combining braiding with fusion. That advances the operational boundary beyond the two processor studies above. It still does not turn this browser's ideal matrices, or a digitally prepared wavefunction, into evidence of passive material protection. The dated open edge is therefore not "can anyon algebra express universal gates?" It is "under measured noise, can the whole protected protocol scale?" This page has no verified threshold number for that system and does not invent one.
honest apparatus Anyons here mean collective excitations or engineered defects whose low-energy motion is effectively confined to two spatial dimensions. Bosons and fermions remain allowed there. A changed phase, fusion outcome or programmed unitary must be interpreted through the particular model and controls that produced it.