Topological Braid Gates, Encodings, and Universality

Let $V$ be a nonzero finite-dimensional complex inner-product space and let $\rho :B_n\to U(V)$ be a fixed mathematical braid representation. A physical implementation record for a braid $b\in B_n$ consists of all of the following data and hypotheses:

1. a finite-dimensional physical Hilbert space ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$;
2. a $C^2$ path $H(s):{\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ of Hermitian endomorphisms, $0\le s\le 1$, and a norm-continuous constant-rank orthogonal spectral projection $P(s)$ of rank $m=\dim V$ that is a coherently followed band in the sense of Finite adiabatic path model, including its local-contour continuation, and is separated from its complement by a uniform positive full gap. The initial low-energy sector is $\mathrm{ran}P(0)$;
3. a unitary identification ${\iota }_0:V\to \mathrm{ran}P(0)$, an explicit preparation isometry $A_0:{\mathbb{C}}^m\to {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ with image exactly $\mathrm{ran}P(0)$, its declared preparation cost, and a stated slow-time schedule $T>0$;
4. a unitary final-band identification ${\iota }_1:V\to \mathrm{ran}P(1)$ and a separate identification of the band's transported action along the chosen exchange path, using ${\iota }_0$ and ${\iota }_1$, with $\rho (b)$ up to a stated scalar phase.

The first three clauses form an instance of the finite adiabatic path model Finite adiabatic path model on ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. Under those hypotheses, Finite JRS adiabatic projector bound bounds the deviation from ideal adiabatic band following. That estimate does not establish that a material realizes the fusion category, that the exchange path has the prescribed holonomy, or that the initial state can be prepared at no cost. Locality, temperature, quasiparticle separation, stray excitations, and tunnelling require their own physical error model.

For every integer $n\ge 1$, the matrix group $SU(n)=\{A\in M_n(\mathbb{C}):A^{\ast }A=I,\det A=1\}$ is path connected.

Fix a unitary fusion category $\mathcal{D}$ and the representative set $\mathrm{Irr}(\mathcal{D})$ of Multiplicity-aware fusion-tree basis. For an ordered list $x_1,\dots ,x_n\in \mathrm{Irr}(\mathcal{D})$, $n\ge 1$, a total charge $c\in \mathrm{Irr}(\mathcal{D})$, and a parenthesized input tensor $T$, let $V=\mathrm{Hom}(T,c)\mathrm{\ne }0$, equipped with that definition's orthonormal fusion-tree inner product. The identical-input case $x_1=\cdots =x_n=x$ retains the notation $V=V(x^n\to c)$. Let $\mathcal{C}={\mathbb{C}}^k$ with its ordered orthonormal computational basis, where $k\ge 1$. A fixed-charge encoding is an isometry $J:\mathcal{C}\to V$, so $J^{\ast }J=I_{\mathcal{C}}$. Its code subspace is $J\mathcal{C}$, and its code projector is $P:=J{J}^{\ast }:V\to V\mathrm{.}$ An encoding records a chosen subspace of a fusion space; it does not identify that subspace with an object of the fusion category.

Let $J:\mathcal{C}\to V$ be a fixed-charge encoding, $P=J{J}^{\ast }$ its code projector, $U\in U(\mathcal{C})$ a logical target, and $W\in U(V)$ an ambient unitary. Its zero-on-the-complement target extension is $\tilde{U}:=JU{J}^{\ast }:V\to V$. Define the logical block error and leakage amplitude by ${ϵ}_{\mathrm{l}\mathrm{o}\mathrm{g}}(W,U):=\mathrm{\parallel }PWP-\tilde{U}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}},{ϵ}_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}(W):=\mathrm{\parallel }(I-P)WP{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{.}$ Both are finite operator norms. The first measures the compressed block against the target; the second measures transfer from the code into its orthogonal complement. They diagnose different errors.

Fix a unitary braided fusion category $\mathcal{C}$ and the simple-object representative set $\mathrm{Irr}(\mathcal{C})$ used in Multiplicity-aware fusion-tree basis. Let $x,c\in \mathrm{Irr}(\mathcal{C})$, let $n\ge 2$, and suppose $V(x^n\to c)\mathrm{\ne }0$. Use the left-associated fusion-tree basis. For the Artin generator ${\sigma }_i$ that exchanges the $i$th and $(i+1)$st strands, let $T_i$ be the specified sequence of $F$-moves that changes from the fixed tree basis to a basis where those two charges fuse first, and let $R_i$ apply the corresponding $R$-symbol on each pair-charge channel. Define ${\rho }_c({\sigma }_i):=T_i^{-1}{R}_i{T}_i\mathrm{.}$ This defines the generator assignment. The next theorem proves that the assignment descends from words to a well-defined group homomorphism. Word products use matrix multiplication acting right-to-left on column vectors. The abstract group in this definition is the Artin group The braid group by Artin presentation; the positive crossing is the generator convention used for the displayed $R$-symbols. If a motion is described chronologically, its matrix product is written in reverse temporal order.

The diagram depicts the positive adjacent generator on two identical charges. Time runs from top to bottom; the continuous left-input strand passes over the broken right-input strand. The crossing represents $c_{x,x}$, while its pullback on a fusion space uses the inverse in the formula above. This fixes the $R$-matrix orientation; the proof of the Artin relations is categorical.

The geometric configuration convention in Geometric braid classes and the unordered configuration fundamental group uses first-loop-then-second multiplication, while raw slicing reverses stacking. This definition fixes the algebraic Artin action directly and makes no extra assertion that the library's geometric group is identified with the Artin presentation under a stronger axiom branch.

Let $J:\mathcal{C}\to V$ be an isometric encoding with $P=J{J}^{\ast }$, let $U$ be unitary on $\mathcal{C}$, and let $W$ be unitary on $V$. If the logical block error $\mathrm{\parallel }PWP-JU{J}^{\ast }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ is zero, then the leakage amplitude $\mathrm{\parallel }(I-P)WP{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ is zero.

Let $V=PV\oplus (I-P)V$ be an orthogonal decomposition of a finite-dimensional Hilbert space, let $U$ and $W$ be unitary operators on the two summands, and let $T:=U\oplus W$. If an ambient unitary $G$ satisfies $\mathrm{\parallel }G-T{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \delta$, then the logical block error of $G$ relative to $U$ and its leakage amplitude both are at most $\delta$.

Let $m\ge 2$ and let $\mathcal{H}=({\mathbb{C}}^2{)}^{\otimes m}$. A Majorana system of length $2m$ is a family of operators ${\gamma }_1,\dots ,{\gamma }_{2m}\in \mathrm{End}(\mathcal{H})$ satisfying ${\gamma }_j^{\ast }={\gamma }_j,{\gamma }_j^2=I,{\gamma }_j{\gamma }_k=-{\gamma }_k{\gamma }_j(j\mathrm{\ne }k)\mathrm{.}$ Its parity operator is $\mathrm{\Pi }=i^m{\gamma }_1\cdots {\gamma }_{2m}$. For each adjacent pair, put $Q_1(j)=\frac{I-i{\gamma }_j{\gamma }_{j+1}}{2},Q_{\psi }(j)=\frac{I+i{\gamma }_j{\gamma }_{j+1}}{2},U_j=\exp \text{\hspace{0.17em}⁣}(-\frac{\pi }{4}{\gamma }_j{\gamma }_{j+1})\mathrm{.}$

A Majorana–Ising exchange interface on a fixed parity sector consists of such a system, a choice $\eta \in \{1,-1\}$ and the sector ${\mathcal{H}}_{\eta }=\ker (\mathrm{\Pi }-\eta I)$, together with the following explicit model-identification hypotheses for every $1\le j<2m$: the Ising pair-charge projections are identified with $Q_1(j),Q_{\psi }(j)$, and the exchange is $R_1{Q}_1(j)+R_{\psi }{Q}_{\psi }(j)$ with the same relative phase $R_{\psi }\mathrm{/}{R}_1=i$. The last identity is required for all adjacent pairs as part of the interface; the three-anyon supplier below establishes it only in its stated sector and does not prove this all-$m$ identification.

Let $V\mathrm{\ne }0$ be a finite-dimensional complex inner-product space and let $U(V)$ be its unitary group. Two unitaries are projectively equivalent when $A=zB$ for some $z\in \mathbb{C}$ with $\mathrm{\mid }z\mathrm{\mid }=1$; write $[A]\in PU(V)$ for the equivalence class. Define $d_P([A],[B]):={\min }_{\mathrm{\mid }z\mathrm{\mid }=1}\mathrm{\parallel }A-zB{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\mathrm{.}$ This is the distance after quotienting by one common scalar phase. The phase is minimized over the unit circle; it is not chosen independently in different charge blocks.

For the fixed-charge braid representation ${\rho }_c$ in Braid action on a fixed-charge fusion space, the projective braid gate of a braid $b\in B_n$ is $[{\rho }_c(b)]\in PU(V(x^n\to c))$.

Assume the Fibonacci fusion rules $1\otimes x=x\otimes 1=x$ and $\tau \otimes \tau =1\oplus \tau$, with each listed summand of multiplicity one. For six ordered $\tau$ charges, let $W_a$ be the fixed-input, fixed-total fusion space of total charge $a\in \{1,\tau \}$ in the tree that first fuses leaves $1,2,3$ and $4,5,6$ separately, then fuses those triple outputs. Let $V_a$ be the same fixed-charge space in the left-associated tree.

To identify these spaces, write $A=(({\tau }_1\otimes {\tau }_2)\otimes {\tau }_3)$. Use the object associator composite $g=({\alpha }_{A\otimes {\tau }_4,{\tau }_5,{\tau }_6}^{-1})\circ {\alpha }_{A,{\tau }_4,{\tau }_5\otimes {\tau }_6}^{-1}\circ ({\mathrm{id}}_A\otimes {\alpha }_{{\tau }_4,{\tau }_5,{\tau }_6})$ from the triple-grouped input tree to the left-associated tree. On a fusion morphism $f$ into $a$, define $G_a(f)=f\circ g^{-1}:V_a$; this is a unitary change of fusion-tree coordinates.

For one triple of total charge $\tau$, its two orthonormal path vectors are indexed by the charge $x\in \{1,\tau \}$ after the first two anyons: both $x\otimes \tau \to \tau$ channels have multiplicity one. Write these vectors as $\mathrm{\mid }x⟩$. For $x,y\in \{1,\tau \}$ and $a\in \{1,\tau \}$, let $\mathrm{\mid }x,y;a⟩$ be the grouped-tree vector with first triple path $x$, first triple total $\tau$, second triple path $y$, second triple total $\tau$, and final vertex channel $\tau \otimes \tau \to a$. The last channel has multiplicity one. Define ${\mathcal{C}}_a:=G_a(\mathrm{span}\{\mathrm{\mid }x,y;a⟩:x,y\in \{1,\tau \}\})\subseteq V_a\mathrm{.}$ In the ordered basis $(G_a\mathrm{\mid }1,1;a⟩,G_a\mathrm{\mid }1,\tau ;a⟩,G_a\mathrm{\mid }\tau ,1;a⟩,G_a\mathrm{\mid }\tau ,\tau ;a⟩)$, the inclusion $J_a:{\mathbb{C}}^4\to V_a$ is an isometry; its code projector is $P_a=J_a{J}_a^{\ast }$. The same ordered local triple paths are used for $a=1$ and $a=\tau$; only the final pair-charge coordinate differs. The orthogonal complements are called leakage spaces.

For three $\tau$ anyons of total charge $\tau$, the fusion space is two-dimensional. Fix the chiral convention $F=\left(\begin{array}{cc}{\varphi }^{-1}& {\varphi }^{-1\mathrm{/}2}\\ {\varphi }^{-1\mathrm{/}2}& -{\varphi }^{-1}\end{array}\right),r=R_{\tau }\mathrm{/}{R}_1=e^{-3\pi i\mathrm{/}5}\mathrm{.}$ After removing the common scalar $R_1$, the adjacent generators are $B_1=\mathrm{diag}(1,r),B_2=F{B}_{1}F\mathrm{.}$

The verification checks these matrices directly as a unitary $B_3$ representation. It uses the listed sector data as input and does not infer global category coherence from this calculation.

In the three-$\sigma$, total-$\sigma$ Ising sector, use $F=H=2^{-1\mathrm{/}2}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$, $R_1=e^{-i\pi \mathrm{/}8}$, and $R_{\psi }=e^{3i\pi \mathrm{/}8}$. After removing the common phase $R_1$, the two projective braid generators are $B_1=\left(\begin{array}{cc}1& 0\\ 0& i\end{array}\right),B_2=H{B}_{1}H\mathrm{.}$

This is a projective statement for an isolated fixed-charge sector. The scalar phase cannot be discarded when the braid is coherently controlled. The supplier gives sector data only; this item makes no all-label Ising category claim.

Fix the Fibonacci convention with common exchange scalar $R_1$. Let $e:B_n\to \mathbb{Z}$ be exponent sum, sending each Artin generator ${\sigma }_i$ to $1$. If $\tilde{\rho }$ is the generator-normalized representation $\tilde{\rho }({\sigma }_i)=R_1^{-1}\rho ({\sigma }_i)$, then for every braid $b$ and every total-charge block, $\rho (b)=R_1^{e(b)}\tilde{\rho }(b)\mathrm{.}$ The scalar is the same on every total-charge block and every spectator multiplicity. It is a global phase for an uncontrolled fixed-sector computation; under coherent control it is a relative phase and must be tracked.

Use the Fibonacci fusion rule $\tau \otimes \tau =1\oplus \tau$ and the fixed-chirality data $F_{\tau }^{\tau \tau \tau }=\left(\begin{array}{cc}{\varphi }^{-1}& {\varphi }^{-1\mathrm{/}2}\\ {\varphi }^{-1\mathrm{/}2}& -{\varphi }^{-1}\end{array}\right)$ and $r=R_{\tau }\mathrm{/}{R}_1=e^{-3\pi i\mathrm{/}5}$. The fixed-total-charge B6 path bases are

${\mathcal{P}}_1=\{0101010,0101110,0110110,0111010,0111110\},$

${\mathcal{P}}_{\tau }=\{0101011,0101101,0101111,0110101,0110111,0111011,0111101,0111111\},$

where each path is $(x_0,\dots ,x_6)$, $x_0=0$ is the unit label, $x_1=1$ is $\tau$, and adjacent labels obey the fusion rule. For $d_0=1,d_1=\varphi$, define $E_i$ on either path basis by $(E_i{)}_{x,y}=\{\begin{array}{ll}\sqrt{d_{x_i}{d}_{y_i}}\mathrm{/}{d}_{x_{i-1}},& x_j=y_j(j\mathrm{\ne }i),x_{i-1}=x_{i+1}=y_{i-1}=y_{i+1},\\ 0,& \text{otherwise,}\end{array}{P}_i:={\varphi }^{-1}{E}_i\mathrm{.}$ Then $P_i$ are orthogonal projections, have ranks $2$ on ${\mathcal{P}}_1$ and $3$ on ${\mathcal{P}}_{\tau }$, satisfy $P_i{P}_{i+1}{P}_i={\varphi }^{-2}{P}_i$ and commute at separation at least two. Consequently $B_i=rI+(1-r)P_i$ are unitary matrices satisfying the Artin relations of $B_6$, with $\det B_i=r^3$ on $V({\tau }^6\to 1)$ and $\det B_i=r^5$ on $V({\tau }^6\to \tau )$.