Topological Braid Gates, Encodings, and Universality — Examples

Under the stated interface [F3], the Ising conclusion is [F1]: signed permutations of the Majorana generators normalize the finite logical Pauli group, so the projective image is finite.

By [F2], the subgroup generated by the two Fibonacci braid generators is dense in $PU(2)$. In contrast, [F1] gives a finite Ising projective image, which cannot be dense in the infinite projective unitary group. Thus these models have different projective gate power under their stated hypotheses.

In the pairwise tree $T_p=((({\tau }_1\otimes {\tau }_2)\otimes ({\tau }_3\otimes {\tau }_4))\otimes ({\tau }_5\otimes {\tau }_6))$, let $q_i\in \{0,1\}$ label the charge $1$ or $\tau$ of pair $(2i-1,2i)$, respectively. Fusion to total charge $1$ allows exactly the strings $000,011,101,110,111.$ For $000$, the intermediate charge after the first two pair outputs is $0$. For $011,101,110,111$, it is uniquely $1,1,0,1$, respectively; each then fuses with the last pair output to total charge $1$. A string with exactly one $1$ has total charge $\tau$ and is inadmissible. Thus the four even strings span a four-dimensional pair-charge code ${\mathcal{D}}_1$, and the all-$\tau$ string $111$ spans its one-dimensional complement. Fix a unitary associator composite from $T_p$ to the left-associated B6 tree. The triple code ${\mathcal{C}}_1$ in [F6] is transported to that same tree by its own fixed associator composite. Both are four-dimensional subspaces of the five-dimensional total-charge-$1$ fusion space. The argument below treats them as independent code choices and makes no projector-equality claim.

Let $\mathcal{K}$ be any four-dimensional subspace of the five-dimensional total-charge-$1$ fusion space, including either ${\mathcal{D}}_1$ or ${\mathcal{C}}_1$, and let $U\in U(\mathcal{K})$. Its orthogonal complement is one-dimensional. Define $T=U\oplus (\det U{)}^{-1}$ on $\mathcal{K}\oplus {\mathcal{K}}^{\perp }$; this is unitary with determinant one, so $T\in SU(5)$. By [F1], every operator-norm neighborhood of $T$ contains a normalized B6 word. Applying [F8] to a word within $\delta$ of $T$ gives logical block error and leakage at most $\delta$ for the selected code. Therefore the pair-charge and triple codes each admit the same density guarantee without identifying their subspaces.

The five-dimensional certificate runs research/scaffold-audit-2026-10-16/topological-quantum-computing/b6-lie-rank-certificate.py and returns the exact minor determinant $17$ in ${\mathbb{F}}_{31}$. The eight-dimensional deterministic selection runs research/scaffold-audit-2026-10-16/topological-quantum-computing/b6-su8-lie-rank-certificate.py and returns determinant $22$. Both scripts use integer arithmetic only; [F4] proves that these nonzero reductions imply nonzero exact complex determinants.

Fix either selected charge-$1$ code ${\mathcal{K}}_1={\mathcal{D}}_1$ or ${\mathcal{K}}_1={\mathcal{C}}_1$. For ${\mathcal{D}}_1$ use the ordered transported even paths $(000,011,101,110)$ of step 1.1 as the computational basis; for ${\mathcal{C}}_1$ use the ordered triple paths in [F6]. In the charge-$\tau$ block use the triple code ${\mathcal{K}}_{\tau }={\mathcal{C}}_{\tau }$ with its ordered basis from [F6]. Each code is thereby identified with ${\mathbb{C}}^2\otimes {\mathbb{C}}^2$. Write each ideal circuit gate as $U_j\in U(4)$: a one-qubit gate A is $A\otimes I_2$ or $I_2\otimes A$, and an adjacent two-qubit gate is its full $U(4)$ matrix. In each sector take the orthonormal complementary paths in the corresponding grouped fusion tree and transport them by the same unitary associator. These complements have dimensions $1$ and $4$. With ${\lambda }_j=(\det U_j{)}^{-1}$, define ambient code-preserving targets, in these code/complement coordinates, by $T_{j,1}=U_j\oplus {\lambda }_j,T_{j,\tau }=U_j\oplus \mathrm{diag}({\lambda }_j,1,1,1)\mathrm{.}$ Each complement action is unitary because $\mathrm{\mid }{\lambda }_j\mathrm{\mid }=1$, and $\det T_{j,1}=\det T_{j,\tau }=\det (U_j){\lambda }_j=1$. Transporting these targets to the fixed left-associated ambient bases preserves determinant and norm. Thus they are actual elements of $SU(5)$ and $SU(8)$ and preserve the selected code in each sector. For the pair-charge choice, the charge-$\tau$ target just uses the separately specified triple code; no equality of pair and triple projectors is used.

Let $G>0$, take target accuracy $ϵ>0$, and set ${\delta }_j=ϵ\mathrm{/}G$ for $1\le j\le G$. The product of the open operator-norm ${\delta }_j$-balls around $(T_{j,1},T_{j,\tau })$ meets the normalized simultaneous B6 image by [F2]. Choose one word $b_j$ in this product, giving $\mathrm{\parallel }{\tilde{\rho }}_1(b_j)-T_{j,1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\delta }_j,\mathrm{\parallel }{\tilde{\rho }}_{\tau }(b_j)-T_{j,\tau }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\delta }_j\mathrm{.}$ These are ambient bounds on the entire five- and eight-dimensional sectors for every gate, including the one-qubit factors. The choices for the two sectors are the same braid word.

For either $a=1,\tau$, the normalized ambient matrices ${\tilde{\rho }}_a(b_j)$ and the code-preserving ideal matrices $T_{j,a}$ satisfy every hypothesis of [F3] by step 2.1, step 3.1. Its finite telescoping estimate therefore gives $\mathrm{\parallel }{\tilde{\rho }}_a(b_G)\cdots {\tilde{\rho }}_a(b_1)-T_{G,a}\cdots T_{1,a}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\sum }_{j=1}^G{\delta }_j=ϵ\mathrm{.}$ The ideal product restricts to $U_G\cdots U_1$ on the selected code. Apply [F8] to this ambient bound: total logical block error and leakage are both at most $ϵ$. For a concrete distinction between these measurements, take $V={\mathbb{C}}^2$, code $\mathbb{C}{e}_0$, target $U=1$, and $W_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right),0<\theta <\pi \mathrm{/}2.$ The compressed block is $\cos \theta$, so its logical error is $1-\cos \theta$, while the leakage amplitude is $\sin \theta$. It has the correct one-dimensional logical direction with reduced amplitude, not an exactly correct block. Conversely, exact block equality forces zero leakage by [F3].

The bounds in step 4.1 concern normalized braid matrices, or equivalently categorical matrices compared to targets with the common phase matched. By [F9], the concatenated categorical braid has the common factor $R_1^{{\sum }_{j}e(b_j)}$ in both sectors. Multiplication by that unit scalar leaves leakage unchanged and yields the corresponding single projective logical phase; it cannot be independently discarded in different sectors or under coherent control. This existence proof assigns no braid-length, search-time or physical-preparation bound.