[ultra thick,->,red] (5,4) – (5,4.3);
[ultra thick,->,teal] (4,4) – (4,4.3);
[ultra thick,->,blue] (6,4) – (6,4.3);
[ultra thick,blue] (5,3) – (6,4);
[ultra thick,knot=red] (6,3) – (5,4);
[ultra thick,knot=teal] (4,3) – (4,4);
[ultra thick,knot=blue] (4,2) – (5,3);
[ultra thick,knot=teal] (5,2) – (4,3);
[ultra thick,red] (6,2) – (6,3);
[ultra thick,knot=red] (5,1) – (6,2);
[ultra thick,knot=blue] (4,1) – (4,2);
[ultra thick,knot=teal] (6,1) – (5,2);
[above] at (4,4.2) $U$;
[above] at (5,4.2) $V$;
[above] at (6,4.2) $W$;
[below] at (4,1) $W$;
[below] at (5,1) $V$;
[below] at (6,1) $U$;

The braiding in a braided tensor category.

[->,node distance=3cm,thick,>=stealth,shorten >=1pt,scale=0.5]

[] (A) $(V_1\otimes \bm{1}) \otimes V_2$;
[below right of =A] (B) $V_1\otimes V_2$;
[above right of=B] (C) $V_1\otimes (\bm{1} \otimes V_2)$;

[blue,ultra thick] (A) edge node [above,blue] $\alpha$ (C);
(A) [blue,ultra thick] edge node [left,blue] $id\otimes \lambda$ (B);
(C) [blue,ultra thick] edge node [right,blue] $\rho \otimes id$ (B);

[->,node distance=3.5cm,thick,>=stealth,shorten >=1pt,scale=0.7]

[] (A) at (6,3) $((V_1\otimes V_2)\otimes V_3)\otimes V_4$;
[] (B) at (1.5,0) $(V_1\otimes (V_2\otimes V_3))\otimes V_4$;
[] (C) at (2.5,-4) $(V_1\otimes ((V_2\otimes V_3)\otimes V_4)$;
[] (D) at (10,-4) $V_1\otimes (V_2\otimes (V_3\otimes V_4))$;
[] (E) at (11,0) $((V_1\otimes V_2)\otimes V_3)\otimes V_4$;

[blue,ultra thick] (A) edge node [left=3,blue] $\alpha_{1,2,3}\otimes id_4$ (B);
[blue,ultra thick] (B) edge node [left,blue] $\alpha_{1,23,4}$ (C);
[blue,ultra thick] (C) edge node [below,blue] $id_1\otimes \alpha_{2,3,4}$ (D);
[blue,ultra thick] (D) edge node [right,blue] $\alpha_{1,2,34}$ (E);
[blue,ultra thick] (A) edge node [right,blue] $\alpha_{12,3,4}$ (E);

[->,node distance=2.5cm,thick,>=stealth,shorten >=1pt]

[] (A) $(V_1\otimes V_2) \otimes V_3$;
[above right of=A] (B) $V_1\otimes (V_2 \otimes V_2)$;
[right of =B,node distance=4cm] (C) $(V_2\otimes V_3)\otimes V_1$;
[below right of =C] (D) $V_2\otimes (V_3\otimes V_1)$;
[below right of =A] (E) $(V_2\otimes V_1)\otimes V_3$;
[right of =E,node distance=4cm] (F) $V_2\otimes (V_1\otimes V_3)$;

[blue,ultra thick] (A) edge node [left,blue] $\alpha_{1,2,3}$ (B);
[blue,ultra thick] (B) edge node [above,blue] $\sigma_{1,2}$ (C);
[blue,ultra thick] (C) edge node [right,blue] $\alpha_{2,3,1}$ (D);

[blue,ultra thick] (A) edge node [left,blue] $\sigma_{1,2}\otimes id_3$ (E);
[blue,ultra thick] (E) edge node [below,blue] $\alpha_{2,1,3}$ (F);
[blue,ultra thick] (F) edge node [right,blue] $id_2\otimes \sigma_{1,3}$ (D);

[Triangle, pentagon and hexagon identities](a) Triangle Identity, (b) pentagon identity, (c) hexagon identity

§.§ Ribbon Category

The first Reidemeister move is to straighten out the twist in a strand, as shown in Fig. <ref> (a). The Reidemeister moves 1, 2, and 3 are for unframed tangles.
To show the relation between duality and braiding, we need two more arrows, as given by
e^'_V: V⊗^*V →1,     i^'_V : 1 →^*V ⊗V.
These are called the left duals and are shown graphically in Fig. <ref> (b). These identities are used to swap the inputs of $i_V$ and $e_V$.
To get rid of the twist, a natural family of isomorphisms is introduced $\theta_V: V\xrightarrow{}V$, such that
θ_V⊗W = σ_W,Vσ_V,W(θ_V θ_W),

(θ_V⊗id_V^*)i_V = (id_V⊗θ_V^*)i_V,

θ_1 = id,

θ_V^* = (θ_V)^*.
The first of these relations is shown in Fig. <ref> (c). The $\theta$ is the actual $2\pi$ twist in a ribbon. For a balanced category, we can write $^*V = V^*$, which means the left, and the right duals are equal.

Now we will introduce the ribbon category $(\mathcal{C},\otimes,1,\sigma,\theta)$ which is a braided monoidal category with a compatible twist.
A ribbon category is a rigid, balanced, and braided tensor category. By compensating for the twist, the ribbon diagrams remain invariant under 3d isotopy. Two ribbon diagrams are isotopic when they are topologically deformable into each other.

[use Hobby shortcut,scale=1.6]
consider self intersections=true,

ignore endpoint intersections=false,
flip crossing=2,
only when rendering/.style=

[ultra thick,blue] (0.2,1.2) .. (0.25,1) .. (0.7,0.9) .. (0.7,1.05) .. (0.33,1);
[ultra thick,blue] (0,1.1) ..(0.55,0.57).. (0.95,0.96) .. (0.5,1.37) .. (0.25,1.23);
[ultra thick,blue] (0,1) .. (0,1.8);
[ultra thick,blue] (0.2,1.2) .. (0.2,1.8);
[ultra thick,blue] (0,0) .. (0,0.8);
[ultra thick,blue] (0.2,0) .. (0.2,0.6);
[ultra thick,blue] (0,1.8) – (0.2,1.8);
[ultra thick,blue] (0,0) – (0.2,0);
[below] at (0.1,0) $V$;
[above] at (0.1,1.8) $V$;

[] at (-0.2,1) $ = $;
[ultra thick,blue] (0,0) – (0,0.2);
[ultra thick,blue] (0.2,0) – (0.2,0.2);
[ultra thick,knot=blue] (0,0.2) – (0.2,0.6);
[ultra thick,knot=blue] (0.2,0.2) – (0,0.6);
[ultra thick,blue] (0,0.6) – (0,1.2);
[ultra thick,blue] (0.2,0.6) – (0.2,1.2);
[ultra thick,knot=blue] (0,1.2) – (0.2,1.6);
[ultra thick,knot=blue] (0.2,1.2) – (0,1.6);
[ultra thick,blue] (0,1.6) – (0,1.8);
[ultra thick,blue] (0.2,1.6) – (0.2,1.8);
[ultra thick,blue] (0,0) – (0.2,0);
[ultra thick,blue] (0,1.8) – (0.2,1.8);
[below] at (0.1,0) $V$;
[above] at (0.1,1.8) $V$;

[] at (-0.2,1) $ \ne $;
[ultra thick,blue] (0,0) – (0,1.8)–(0.2,1.8)–(0.2,0)–cycle;
[below] at (0.1,0) $V$;
[above] at (0.1,1.8) $V$;

[ultra thick,->,blue] (5.5,4) – (5.5,4.2);
[ultra thick,knot=blue] (5.5,3) – (5.5,4);
[ultra thick,knot=blue] (5.5,1) – (5.5,2);
[] at (5.5,2.5) $ \theta $;
[above] at (5.5,4.2) $ V $;
[below] at (5.5,1) $ V $;
[ultra thick,red] (5,2) rectangle (6,3);
at (6.5,2.5) $= \theta$;
[ultra thick,knot=blue] (7,1) – (7,4);
[ultra thick,->,blue] (7,4) – (7,4.2);
[] at (5.5,2.5) $ \theta $;
[above] at (7,4.2) $ V $;
[below] at (7,1) $ V $;

[->,ultra thick,blue] (2.5,3)–(2.5,4);
[ultra thick,blue] (2.5,1) – (2.5,2);
[->,ultra thick,blue] (3.5,3)–(3.5,4);
[ultra thick,blue] (3.5,1) – (3.5,2);
[ultra thick,red] (2,2) rectangle (4,3);
[] at (3,2.5) $ \theta $;
[above] at (2.5,4) $ V $;
[below] at (2.5,1) $ V $;
[above] at (3.5,4) $ W $;
[below] at (3.5,1) $ W $;
[] at (4.5,2.5) $ = $;
[ultra thick,->,blue] (7,4) – (7,4.2);
[ultra thick,->,blue] (5.5,4) – (5.5,4.2);
[ultra thick,knot=blue] (5.5,3) – (7,4);
[ultra thick,knot=blue] (7,3) – (5.5,4);
[ultra thick,knot=blue] (5.5,1) – (7,2);
[ultra thick,knot=blue] (7,1) – (5.5,2);
[] at (7,2.5) $ \theta $;
[above] at (7,4.2) $ W $;
[below] at (7,1) $ W $;
[] at (5.5,2.5) $ \theta $;
[above] at (5.5,4.2) $ V $;
[below] at (5.5,1) $ V $;
[ultra thick,red] (5,2) rectangle (6,3);
[ultra thick,red] (6.5,2) rectangle (7.5,3);

Representation of a twist in a ribbon category.

§ SEMISIMPLE RIBBON CATEGORIES

Suppose an object $U$ of a category does not contain nontrivial subobjects, then the object is called simple. An Abelian category is semisimple if any object in ${\cal C}$ is isomorphic to the direct sum of simple objects $V_i$ [194].
V = ⊕_i∈I N_iV_i,     V_i ⊗V_j ≃⊕_k ∈Π_C N_i,j^k V_k.
These formulas are called the fusion rules, and $N_{ij}^k$ is called the fusion coefficient. These are positive integers and are given by
N_i,j^k = dim(Hom_C(V_k,V_i⊗V_j))= dim(Hom(1,V_i⊗V_j⊗V_k)),
N_ij^k = N_ji^k=N_ik^*^j^*=N_i^*j^*^k^*,  N_ij^0 = δ_ij^8.
We also have
θ_V = θ_i id_V_i,  dimV_i = d_i,  θ_0 = 1,  θ_i = θ_i^*,

d_0 = 1, d_i = d_i^*,  d_id_j = ∑_k N_ij^kd_k.
The fusion matrix $N_i$ of $V$ is defined as $(N_i)_{k,j} = N_{i,j}^k$ with the non-negative entries. The largest eigenvalue is called Frobenus-Perron dimension of ${\cal C}$.

A fusion category is a semisimple rigid category with a unit object [197], a finite-dimensional morphism, and a finite number of simple objects. The braided fusion category with a ribbon structure is called a ribbon fusion, or a premodular category. A premodular category is a modular category if the S-matrix of ${\cal C}$ is defined as follows
S_ij = (σ_V_j,V_i∘σ_V_i,V_j)     for  i,j ∈Π_C,
S_ij:V_j⊗V_j →V_i ⊗V_i^*.
The $S$-matrix is the basis change between the components of the links. This $S$-matrix is a topological analogue of the $S$-matrix in the scattering theory we will discuss in the Appendix <ref>. A modular category is a semisimple ribbon category if the matrix $S$ is invertible and $\mathcal{C}$ has only a finite number of isomorphism classes of objects.
The $S$-matrix is an invariant of a Hopf link colored by $a$ and $b$, which are the entries of the $S$-matrix.
A single loop can be thought of as a loop linking vacuum, which give $S_{a0} = S_{0a}$.
For the normalization, we take the loop of vacuum linking vacuum $S_{00}$, so
d_a = S_a0/S_00.
The total quantum dimension is a number $D = \sqrt{\sum_{a \in \Pi_{\cal C}}d_a^2}$. It is an invariant of the category and is called the quantum order.
The columns of the $S$-matrix can be seen to be simultaneous eigenvectors for the fusion matrix so that S diagonalizes all $N_a$ simultaneously. This leads to Verlinde formula [198].
N_ab^c = ∑_x S_ax S_bxS_cx/S_1x.
See [79, 199] for the proof of the Verlinde formula. By the Verlinde formula, $S$ matrix contains the information about the braiding as well as the fusion.

Let, $V_{ab}^c$ be $Hom(a\otimes b,c)$, the Hilbert space that is assigned to a surface when there are three objects. When there are four simple objects in a semisimple category, the F-matrix can be written as

§.§ Modular Transformation

The S-matrix is the change of basis between the two links shown in Fig. <ref>.
This transformation is called the modular transformation. The topological twist $\theta_a$ is encoded in a diagonal matrix $T= (\delta_{ab} \theta_a)$. $S_{ab}$ is an invariant of the Hopf link of two charges colored by $a$ and $b$. The $S$ and $T$ are called the modular data of the category.

The $S$ and $T$ matrices define a projective representation of a modular group or mapping class group (MCG).
The mapping class group on a torus is $SL(2,\mathbb{Z})$, the special linear group on $\mathbb{Z}$. The Lie algebra and special linear groups are discussed in the Appendix <ref>. The mapping class group is obtained from the deformation of spaces. Taking an example of a torus, which is the identification of a sheet at two opposite edges. The modular transformation of the torus is cutting the torus, putting one twist, called the Dehn twist, and then joining again. A torus remains invariant under non-trivial transformation of the modular group.

A torus is made up of a lattice in the form of a parallelogram in complex plane with two opposite sides identified. The parallelogram is generated by two complex numbers $w_1$ and $w_2$. Modular parameter $\tau=w_1/w_2 \in \mathbb{H}$, where $\mathbb{H}=\{z\in \mathbb{C}|z>0\}$ is the complex upper half plane. We can denote the lattice as $L(w_1,w_2)$. Let another torus denoted by a lattice $L(u_1,u_2)$ with the modular parameter $\tau =u_1/u_2$. The two tori are equal if their modular parameters coincide, that is when they are related by the modular transformation as

a   b

c   d

,     where
a   b

c   d
See [200] for detailed discussion. The most general modular transformation has the form
τ→aτ+ b/cτ+d,     a,b,c,d ∈ℤ; ad-bc =1.
This is also known as global conformal transformation or Möbious transformation [127]. The modular transformation has generators $S: \tau \to -\frac{1}{\tau}$ and $T:\tau \to \tau+1$. The matrices $S$ and $T$ obey $S^2=1,(ST)^3=1$. These matrices represent the rescaling of coordinates, where $S$ represents the inversion in the unit circle followed by reflection around the imaginary axis, and $T$ represents a unit translation. The projective special linear group $\text{PSL}(2,\mathbb{Z})$ is a quotient group $SL(2,\mathbb{Z})/\mathbb{Z}_2$, see Appendix <ref> for the definition of the quotient group. $\text{PSL}(2,\mathbb{Z})$ is a group of $2\cross 2$ matrices with integer coefficients and unit determinant. Some authors write the modular matrix as a bigger group $SL(2,\mathbb{Z})$ instead of $\text{PSL}(2,\mathbb{Z})$. The quotient $\mathbb{Z}_2$ is due to the fact that the above transformation is unaffected by taking $a,b,c,d$ negative.

0   -1

1   0
,     t=
1   1

0   1

The modular data satisfy the conditions:
(ST)^3 = p^+ S^2, S^2 = p^+p^- C, CT=TC, C^2 = id
$p^\pm = \sum_a d_a^2 \theta^\pm_a$ are called Gauss sums. We also have $p^+p^- = D^2 = \sum d_a^2$.

The $T$ transformation is twisting the fixed point graph by $360^\circ$ whereas an $S$ transformation is as rotating the fixed point graph by $90^\circ$. The modular transformation would become $S: \tau \to -\frac{1}{\tau}$ and $T:(\alpha,\beta) \to (\alpha,\beta-\alpha)$.

[thick,fill= teal!40] (0,0) rectangle (1.5,2);
[ultra thick,dashed,fill= cyan!40,opacity=0.6] (0,0) rectangle (2,1.5);
[ultra thick,fill= orange!40,opacity=0.7] (0,0) rectangle (1.5,1);

Modular transformation of a torus.

The modular transformation is an alternative to the fusion coefficients $N_{ab}^c$. This transformation encodes the braiding statistics and fusion rule of anyons [11].
A braid is a special case of MCG when there are $n$ punctures. The non-Abelian Berry phase is obtained from the transformations $T$ and $S$ on a torus. A ground state subspace form a unitary representation of MCG [201].
For example, let there be charges in an Ising anyon model given as $\ket{1}_m,\ket{\sigma}_m,\ket{\psi}_m$. In this case, the $S$ matrix would be 3 by 3, and the effect of its application would be to change the basis to a new basis $\ket{1}_l,\ket{\sigma}_l,\ket{\psi}_l$. The $S$ matrix is a change of basis, so this matrix should be unitary.

The topological spin or a twist factor $\theta_a = e^{2 \pi i h_a}$ is a twist when an anyon is rotated by $2\pi$, where $h_a$ is a scaling dimension or conformal dimension. See Fig. <ref>. It is an integer for bosons and gives a spin factor identity. It is a half-integer for fermions that would give the spin factor as $-1$. For vacuum, $h_0 = 0$. The exchange of two particles without twisting is governed by the braid group for anyons. In general, the left over right exchange is not equal to the right over the left exchange. The product of two left over right exchanges is called monodromy. It returns the excitations to their original position but results in a nontrivial effect on the state of the system. This effect is given by the ribbon equation pictorially shown in Fig. <ref>,
e^i θ_ab^c = e^2 πi h_c/e^2 πi h_a e^2 πi h_b.
This is the same as the full twist of fusion products combined with the full twist of charges in the opposite direction. The effect of monodromy introduces the phase factor
θ_c/θ_a θ_b = e^2 πi(h_c - h_a -h_b)= [R^ab_c]^2.
This is shown in Fig. <ref>. This relation is used to derive the entries of the $R$-matrix for Fibonacci and Ising anyons. The trace of monodromy on $a$ and $b$ gives $S_{ab}$.

The classification of topological phases of matter is actually the classification of modular categories [202, 203]. For example, the Fibonacci and Ising categories given below are the phases of matter in which Fibonacci and Ising anyons are present [204]. These two types of anyonic models are the most widely used in topological quantum computing. We will discuss these types of anyons in Chapter <ref>.

[] at (-2.3,-1) $ s_{ij} = $;
[ultra thick, ->,red] (0.15,-0.1) arc (50:-285:1);
[ultra thick, ->,blue] (-0.1,-1.6) arc (230:-110:1);
[below] at (-1,-2) $ i $;
[below] at (1,-1.9) $ j $;

Modular transformation between two colored links

[use Hobby shortcut,scale=1]
[consider self intersections=true,
ignore endpoint intersections=false,
flip crossing=2,
only when rendering/.style=]
[ultra thick,blue] (0,1) ..(0.5,0.5).. (1,1) .. (0.5,1.5) .. (0,1);
[ultra thick,blue] (0,-0.5) .. (0,1);
[ultra thick,blue,->] (0,1) .. (0,2.5);
[below] at (0,-0.5) $a$;

at (0,1) $=\theta_a$;
at (1,1) $=$;
[ultra thick,blue,->] (0.5,-0.5) – (0.5,2.5);
[below] at (0.5,-0.5) $a$;

[use Hobby shortcut,scale=1]
[consider self intersections=true,
ignore endpoint intersections=false,
flip crossing=2,
only when rendering/.style=]
[ultra thick,blue,->] (0,1) .. (0,2.5);
[ultra thick,blue] (0,-0.5) .. (0,1);
[ultra thick,blue] (0,1) ..(-0.5,0.5).. (-1,1) .. (-0.5,1.5) .. (0,1);
[below] at (0,-0.5) $a$;

[use Hobby shortcut,scale=1]
[consider self intersections=true,
ignore endpoint intersections=false,
flip crossing=2,
only when rendering/.style=]
[ultra thick,blue] (0,-0.5) .. (0,1);
[ultra thick,blue,->] (0,1) .. (0,2.5);

[ultra thick,blue] (0,1) ..(0.5,0.5).. (1,1) .. (0.5,1.5) .. (0,1);
[below] at (0,-0.5) $a$;

at (0,1) $=\theta^*_a$;
at (1,1) $=$;
[ultra thick,blue,->] (0.5,-0.5) – (0.5,2.5);
[below] at (0.5,-0.5) $a$;

[use Hobby shortcut,scale=1]
[consider self intersections=true,
ignore endpoint intersections=false,
flip crossing=2,
only when rendering/.style=]

[ultra thick,blue] (0,1) ..(-0.5,0.5).. (-1,1) .. (-0.5,1.5) .. (0,1);
[ultra thick,blue] (0,-0.5) .. (0,1);
[ultra thick,blue,->] (0,1) .. (0,2.5);
[below] at (0,-0.5) $a$;

[Removing a twist is equivalent to adding a phase.]Removing a twist is equivalent to adding a phase (a) $\theta_a = e^{2\pi i h_a}$ and (b) $\theta^*_a = e^{-2\pi ih_a}$.

[ultra thick,knot=blue] (0.2,0)–(0.2,0.5);
[ultra thick,knot=blue](-0.2,0)–(-0.2,0.5);
[ultra thick,knot=blue](0.2,0.5)–(-0.2,1);
[ultra thick,knot=blue](-0.2,0.5)–(0.2,1);
[ultra thick,knot=blue](0.2,1)–(-0.2,1.5);
[ultra thick,knot=blue](-0.2,1)–(0.2,1.5);
[ultra thick,knot=blue](-0.2,1.5)–(-0.2,2);
[ultra thick,knot=blue](0.2,1.5)–(0.2,2);
[ultra thick,knot=blue](-0.2,2)–(-1,3);
[ultra thick,knot=blue](0.2,2)–(1,3);

[ultra thick,knot=blue](-1,3)–(-1,3.5);
[ultra thick,knot=blue](1,3)–(1,3.5);

[ultra thick,blue](0,2.3)–(0.6,3)–(0.6,3.5);
[ultra thick,blue](0,2.3)–(-0.6,3)–(-0.6,3.5);
[ultra thick,knot=blue](0.6,3.5)–(1,4);
[ultra thick,knot=blue](1,3.5)–(0.6,4);
[ultra thick,knot=blue](-1,3.5)–(-0.6,4);
[ultra thick,knot=blue](-0.6,3.5)–(-1,4);
[ultra thick,knot=blue](0.6,4)–(1,4.5);
[ultra thick,knot=blue](1,4)–(0.6,4.5);
[ultra thick,knot=blue](-1,4)–(-0.6,4.5);
[ultra thick,knot=blue](-0.6,4)–(-1,4.5);
[ultra thick,blue](-1,4.5)–(-1,5);
[ultra thick,blue](-0.6,4.5)–(-0.6,5);
[ultra thick,blue](1,4.5)–(1,5);
[ultra thick,blue](0.6,4.5)–(0.6,5);
[ultra thick,blue](-0.2,0)–(0.2,0);
[ultra thick,blue](-1,5)–(-0.6,5);
[ultra thick,blue](0.6,5)–(1,5);
at (1.5,2.5) $=$;

[ultra thick,knot=blue](-0.2,0)–(-0.2,0.5);
[ultra thick,knot=blue](0.2,0)–(0.2,0.5);
[ultra thick,knot=blue](-0.2,0.5)–(-1,1.5);
[ultra thick,knot=blue](0.2,0.5)–(1,1.5);
[ultra thick,blue](0,0.8)–(0.6,1.5);
[ultra thick,blue](0,0.8)–(-0.6,1.5);

[ultra thick,knot=blue](0.6,1.5)–(-1,3.5);
[ultra thick,knot=blue](1,1.5)–(-0.6,3.5);
[ultra thick,knot=blue](-0.6,1.5)–(1,3.5);
[ultra thick,knot=blue](-1,1.5)–(0.6,3.5);

[ultra thick,knot=blue](0.6,3.5)–(-1,5.5);
[ultra thick,knot=blue](1,3.5)–(-0.6,5.5);

[ultra thick,knot=blue](-1,3.5)–(0.6,5.5);
[ultra thick,knot=blue](-0.6,3.5)–(1,5.5);

[ultra thick,knot=blue](-0.6,5.5)–(-0.6,6);
[ultra thick,knot=blue](-1,5.5)–(-1,6);
[ultra thick,knot=blue](0.6,5.5)–(0.6,6);
[ultra thick,knot=blue](1,5.5)–(1,6);
[ultra thick,blue](-0.2,0)–(0.2,0);
[ultra thick,blue](-1,6)–(-0.6,6);
[ultra thick,blue](0.6,6)–(1,6);

Braid-twist correspondence.

§.§.§ Fibonacci Modular Tensor Category

Anyon types: $1,\tau$

Fusion rules: $\tau^2 = 1+ \tau$

Quantum dimensions: $d_1 = 1$, $d_\tau = \phi$

Twists: $\theta_1=1$, $\theta_\tau = e^{4i\pi/5}$

Braiding: $R_1^{\tau \tau}= e^{-4i\pi/5}$, $R^{\tau\tau}_\tau = e^{3i\pi/5}$

S-matrix: $S= \frac{1}{\sqrt{2+\phi}} \begin{pmatrix}
1 & \phi\\
\phi & 1
\end{pmatrix}$

F-matrices: $F^{\tau\tau\tau}_\tau = \begin{pmatrix}
\phi^{-1} & \phi^{-1/2}\\
\phi^{-1/2} & -\phi^{-1}
\end{pmatrix}$

§.§.§ Ising Modular Tensor Category

Anyon types: $1,\sigma,\psi$

Fusion rules: $\sigma^2 = 1+ \psi, \sigma \psi = \psi \sigma = \sigma, \psi^2 = 1$

Quantum dimensions: $d_1 = 1$, $d_\sigma = \sqrt{2}, d_\psi = 1$

Twists: $\theta_1=1$, $\theta_\sigma = e^{i\pi/8}, \theta_\psi = -1$

Braiding: $R_1^{\sigma\sigma}= e^{-i\pi/8}, R^{\psi \psi}_1=-1, R^{\psi\sigma}_\sigma = -i, R^{\sigma\sigma}_\psi = e^{3i\pi/8}$

S-matrix: $S= \frac{1}{2} \begin{pmatrix}
1 & \sqrt{2} & 1\\
\sqrt{2}&0 & -\sqrt{2}\\
1 & -\sqrt{2} & 1
\end{pmatrix}$

F-matrices: $F^{\sigma\sigma\sigma}_\sigma = \frac{1}{2} \begin{pmatrix}
1 & 1\\
1 & -1
\end{pmatrix}, F^{\psi\sigma\psi}_\sigma = (-1), F^{\sigma\psi\sigma}_\psi =(-1)$

§ CATEGORY THEORY AND TOPOLOGICAL QUANTUM

There are three steps to implement quantum computation: the creation of anyons from the vacuum, braiding these anyons by moving them around, and measuring the anyon type of the pair of neighboring particles.
In category theory, the first step is the implementation of $Hom(\textbf{1}, X^{\otimes n})$ and the third step is $Hom(X^{\otimes n}, \textbf{1})$. After the braiding, neighboring anyons may have a different total charge besides vacuum. The computing result is a probability distribution on anyon types obtained by repeating the same process polynomially many times. Usually, the probability of neighboring pairs of anyons returning to vacuum is taken as the computing answer, which approximates some topological invariant of links obtained from the braiding process [193].

A group of particles in the configuration space makes a quantum state in the Hilbert space. The objects in a category correspond to particles, and dual objects correspond to antiparticles. The twist is a topological spin of an anyon. A loop is a trace of a worldline and gives the dimension of a particle. If an object $X$ has $d_X=1$, then it is an Abelian anyon, whereas the object is a non-Abelian anyon when the dimension $d_X>1$.
The decompositions are called the fusion rules.
The comparison of the anyon model to the category theory is given in [205, 193].

§ CATEGORICAL ASPECT OF TQFT

The functorial view of the quantum field theory is a mapping from geometry to algebra. More specifically, from a geometric and dynamical structure of spacetime to the algebraic structure of observables and states.
The TQFT is a symmetric monoidal functor, can be written as [188]
Z = nCob →Vect ,
where $\textbf{nCob}$ is a category whose objects are closed $n-1$-dimensional manifold $\Sigma$ and whose morphisms $M:\Sigma_1 \to \Sigma_2$ are cobordism, that is, $n$-dimensional manifolds having an input boundary $\Sigma_1$ and an output boundary $\Sigma_2$. The concept of cobordism is explained in Chapter <ref>.
Physically, that $Z$ is a functor means $Z(MM^{'}) = Z(M)Z(M^{'})$, and when $Z$ is monoidal means that the two non-interacting systems have a state-space which is the tensor product of the states of individual systems. $Z$ is a symmetric functor when we are dealing with a bosonic statistic.

[commutative diagrams/every diagram]
(P0) at (90:2.3cm) $((V_1\otimes V_2)\otimes V_3)\otimes V_4$;
(P1) at (90+72:2cm) $(V_1\otimes (V_2\otimes V_3))\otimes V_4$ ;
(P2) at (90+2*72:2cm) [5ex][r]$V_1\otimes ((V_2\otimes V_3)\otimes V_4)$;
(P3) at (90+3*72:2cm) [5ex][l]$V_1\otimes (V_2\otimes (V_3\otimes V_4))$;
(P4) at (90+4*72:2cm) $(V_1\otimes V_2)\otimes (V_3\otimes V_4)$;
[commutative diagrams/.cd, every arrow, every label]
(P0) edge node[swap] $\alpha_{1,2,3}\otimes id_4$ (P1)
(P1) edge node[swap] $\alpha_{1,23,4}$ (P2)
(P2) edge node $id_1\otimes \alpha_{2,3,4}$ (P3)
(P4) edge node $\alpha_{1,2,34}$ (P3)
(P0) edge node $\alpha_{12,3,4}$ (P4);

[column sep=2cm,between origins, row sep=1.5cm,between origins]
(V_1⊗1 ⊗V_2) [rr, "α"]rr [dr, "id ⊗λ"']                  V_1 ⊗(1 ⊗V_2) [dl, "ρ⊗id"]

V_1 ⊗V_2

[column sep=2cm,between origins, row sep=1.5cm,between origins]
V_1⊗(V_2 ⊗V_3) [rr, "σ_1,23"]        (V_2⊗V_3) ⊗V_1 [rd, "α_2,3,1"]

(V_1⊗V_2) ⊗V_3 [rd, "σ_1,2⊗id_3"'][ru, "α_1,2,3"']                 V_2⊗(V_3 ⊗V_1)

(V_2⊗V_1) ⊗V_3 [rr, "α_2,1,3"']        V_2⊗(V_1 ⊗V_3) [ru, "id_2 ⊗σ_1,3"']

CHAPTER: TOPOLOGICAL QUANTUM COMPUTATION

Topological quantum computing is a fault-tolerant quantum computing, proposed by Alexei Kitaev [9], manifested by manipulating quantum information using anyons. A quantum computation model involves three main steps; initialization, unitary evolution, and measurement [31].
In quantum theory, the time evolution of a state is represented by the unitary time evolution operator $U(t)$. When the initial state $\ket{\psi_i}$ evolves unitarily to the final state, it is written as $\ket{\psi_f}=U(t)\ket{\psi_i}$.
The initial state is an input and the final state is an output of a quantum gate, and the readout is a measurement in a certain basis to get a classical result [32]. See Chapter <ref> for further details.
The gate operation $U(t)$ is equivalent to the rotation of states in the Hilbert space. Analogous to conventional quantum computing, topological quantum computing has three steps; the creation of pairs of anyons from the vacuum, their braiding, and their fusion. The result of fusion corresponds to the measurement, and the braiding corresponds to the unitary transformation $\Psi_f = (Braid) \Psi_i$. The braids cause rotation within degenerate N-particles space. The change of state by braiding can be explained by geometric phases. The braid group and the geometric phase are discussed in Chapters <ref> and <ref>.
To see how topological quantum computing is a fault-tolerant quantum computing, let two spacetime histories $\ket{1}$ and $\ket{2}$ in Fig. <ref> have time reversed states $\bra{1}$ and $\bra{2}$. By the Kauffman bracket we have $\bra{1}\ket{1} = \bra{2}\ket{2} = d^2$, $\bra{2}\ket{1} = d$. The number $d$ is assigned to a loop as we discussed in context of the Kauffman bracket in Chapter <ref>. So $\ket{1}$ and $\ket{2}$ are distinct states as long as $\abs{d} \ne 1$ [206]. The states $\ket{1}$ and $\ket{2}$ locally look the same, but the outcomes of their fusion with $\ket{1}$ or $\ket{2}$ are different. Therefore, disturbing one of the particles would not affect the outcome if the topology of a spacetime trajectory is not changed. The state $\ket{1}$ can also be interpreted as the creation of two pairs of anyons and the state $\bra{1}$ as a fusion of the two pairs of anyons.

For a basic introduction on topological quantum computation, see [207, 28]. The topological gates with Ising anyons, using the Kauffman version of the recoupling theory [208], are proposed by [209]. For the implementation of gates with Ising anyons in the quantum Hall phase, see [210, 211] and for gates with Fibonacci anyons, see [212].

[ultra thick,fill=cyan!20] (-4.5,0) ellipse (4 and 0.7);
[ultra thick,rotate=180, style strands=1red,style strands=2blue,style strands=3red,style strands=4blue,style strands=5red,style strands=6blue,style strands=7red,style strands=8blue]s_5s_7S_4^-1s_1s_3S_6^-1s_2;
[ultra thick,fill=cyan!20] (-4.5,7.9) ellipse (4 and 0.7);
[blue] (-8,0) circle(6pt);
[red] (-7,0) circle(6pt);
[blue] (-6,0) circle(6pt);
[red] (-5,0) circle(6pt);
[blue] (-4,0) circle(6pt);
[red] (-3,0) circle(6pt);
[blue] (-2,0) circle(6pt);
[red] (-1,0) circle(6pt);
[] at (-8.5,4) 90Time;
[] at (1.2,8) Measure;
[left] at (1.5,4) Braid;
[] at (1.2,0) Initialize;

[Creation, fusion and braiding of anyons.]Creation of anyon pairs, braiding them by dragging around each other, then fusing them and getting the result of fusion [206]

[ultra thick,fill=cyan!30,opacity=0.5] (0,0) ellipse (1.7 and 0.7);
[draw,ultra thick,red,postaction=decorate,decoration=markings,mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,0) .. controls(0,3) ..(1,0);
[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];](-0.2,-0.2) .. controls(0,2.2) ..(0.5,0.3);
[blue] (-1,0) circle(5pt);
[blue] (1,0) circle(5pt);
[red] (-0.2,-0.2) circle(5pt);
[red] (0.5,0.3) circle(5pt);
[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,-2.5) .. controls(0,-5) ..(1,-2.5);
[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];] (-0.2,-2.8) .. controls(0,-4.5) ..(0.5,-2.2);
[ultra thick,fill=cyan!30,opacity=0.5] (0,-2.5) ellipse (1.7 and 0.7);
[blue] (-1,-2.5) circle(5pt);
[blue] (1,-2.5) circle(5pt);
[red] (-0.2,-2.8) circle(5pt);
[red] (0.5,-2.2) circle(5pt);
[left] at (-2,0) $\bra{1}$;
[left] at (-2,-2.5) $\ket{1}$;
[ultra thick,fill=cyan!30,opacity=0.5] (0,0) ellipse (1.7 and 0.7);
[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,0) .. controls(-0.5,3) ..(-0.2,-0.3);

[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];] (0.2,0.2) .. controls(0.5,3) ..(1,0);
[blue] (-1,0) circle(5pt);
[red] (-0.2,-0.3) circle(5pt);
[red] (0.2,0.2) circle(5pt);
[blue] (1,0) circle(5pt);
[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,-2.3) .. controls(-1,-5) ..(-0.3,-2.8);
[draw,ultra thick,red,postaction=decorate,decoration=markings, mark=between positions 0.15 and 1 step 0.25 with [blue];] (0.2,-2.1) .. controls(0,-5) ..(0.9,-2.6);
[ultra thick,fill=cyan!30,opacity=0.5] (0,-2.5) ellipse (1.7 and 0.7);
[blue] (-1,-2.3) circle(5pt);
[red] (-0.3,-2.8) circle(5pt);
[red] (0.2,-2.2) circle(5pt);
[blue] (0.9,-2.6) circle(5pt);
[left] at (-2,0) $\bra{2}$;
[left] at (-2,-2.5) $\ket{2}$;
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,0) .. controls(0,-3) ..(1,0);
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-0.2,-0.2) .. controls(0,-2.2) ..(0.5,0.3);
[ultra thick,fill=cyan!30,opacity=0.5] (0,0) ellipse (2 and 0.8);
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,0) .. controls(0,3) ..(1,0);
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-0.2,-0.2) .. controls(0,2.2) ..(0.5,0.3);
[teal] (-1,0) circle(5pt);
[teal] (1,0) circle(5pt);
[teal] (0.5,0.3) circle(5pt);
[teal] (-0.2,-0.2) circle(5pt);
[left] at (-2,0) $\bra{1}\ket{1}=$;
[right] at (2.5,0) $=d^2$;
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,-5) .. controls(-1,-8) ..(-0.2,-5.5);
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (0.5,-5) .. controls(0,-8) ..(1,-5);
[ultra thick,fill=cyan!30,opacity = 0.5] (0,-5.1) ellipse (2 and 0.8);
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-1,-5) .. controls(0,-2) ..(1,-5);
[draw,ultra thick,red,postaction=decorate,decoration=markings,
mark=between positions 0.15 and 1 step 0.25 with [blue];] (-0.2,-5.5) .. controls(0,-2.5) ..(0.5,-5);
[teal] (-1,-5) circle(5pt);
[teal] (-0.2,-5.5) circle(5pt);
[teal] (0.5,-5) circle(5pt);
[teal] (1,-5) circle(5pt);
[left] at (-2,-5) $\bra{1}\ket{2}=$;
[right] at (2.5,-5) $=d$;

[States of spacetime histories.]States of spacetime histories [206].

§ HILBERT SPACE

The types of anyons are categorized by quantum numbers attached to them called the topological charges. When the charges are created from the vacuum, their total charge must be zero. Therefore, the value of the topological charge is assigned with respect to its fusion with other anyons. The topological charge zero is assigned to the vacuum. The vacuum is also called a trivial charge.
For example, in Ising anyon model, there are three charges $\left\{\textbf{1},\sigma,\psi \right\}$. In some anyonic models, the vacuum is represented by $0$ or $I$. The $\textbf{1}$ represents a vacuum, or a trivial particle, whereas the $\sigma$ and $\psi$ are nontrivial particles. Since the anyons are created from the vacuum, to conserve the total charge, they must be fused to vacuum. The number of ways these anyons are fused to vacuum is called the fusion channels or fusion trees. The number of the fusion channels is equal to the ground state degeneracy of the system. The fusion space is a shared property of a collection of non-Abelian anyons regardless of where they are located. Therefore, local perturbations do not affect the degeneracy of the system. In Chapter <ref>, we identified the anyons on a two-dimensional manifold as the punctures or holes on a sphere.

Let several anyons be created from the vacuum, and let us consider a subtree consisting of two anyons $a$ and $b$. A fusion tree diagram is equivalent to the anyons' creation tree if the time direction is reversed.
When these anyons are Abelian, they fused to only one outcome and the dimension of Hilbert space for two anyons is one. But when the particles $a$ and $b$ are non-Abelian, there is more than one fusion outcome, that is $a\cross b = \sum_c N_{ab}^c$.
Their Hilbert space is denoted as $V_{ab}^{c}$. The dimension of Hilbert space is given as $N_{ab}^c = dim(V_{ab}^c)$. The numbers $N_{ab}^c$ are also called the fusion rules. These fusion rules appear in conformal field theory and category theory in the form as $\phi_a\cross \phi_b = \sum_c N_{ab}^c\phi_c$. The fusion rules put some restrictions on what types of anyons a particular anyonic model can have. As the fusion of two non-Abelian anyons can result in several anyons, in general, $N_{ab}^c$ could have more values than one. Most of the anyonic models are built by considering only the two fusion outcomes; vacuum or another anyon, that is $N_{ab}^c=0,1$. When the fusion of $a$ and $b$ gives the topological charge $c$ then $N_{ab}^c=1$, but when $a$ and $b$ cannot be fused to $c$ then $N_{ab}^c=0$.

The dimension of Hilbert space increases with the number of anyons, analogous to the addition of spins $1/2 \otimes 1/2 = 0 \oplus 1 $. This analogy is not exact, because the anyons are not elementary particles, but they have internal degrees of freedom. The dimension of Hilbert space of $N$ particles of type $a$ is roughly $\sim d_a^N$, where $d$ is the dimension of Hilbert space of one anyon and is called the quantum dimension. It is a measure of how much the Hilbert space is increased by adding this anyon. Therefore, it is an asymptotic degeneracy per particle. It needs not to be an integer. The quantum dimension of vacuum is one. In terms of knot theory, the $d$ is a number assigned to a loop. For the process $a\cross b = \sum_c N_{ab}^c c$, the quantum dimension can be defined to satisfy $d_ad_b = \sum_c N_{ab}^c d_c$.

[blue, ultra thick] (2,0) – (2,1);
[blue, ultra thick] (2,1) – (4,3);
[blue, ultra thick] (2,1) – (0,3);
[blue, ultra thick] (1,2) –(2,3);
[above] at (0,3) $a$;
[above] at (2,3) $b$;
[above] at (4,3) $c$;
[below] at (2,0) $d$;
[below left] at (1.5,1.5) $i$;
[] at (6,1) $= \sum_{j}(F_{acb}^d)^i_j $;
[blue, ultra thick] (10,0) – (10,1);
[blue, ultra thick] (10,1) – (12,3);
[blue, ultra thick] (10,1) – (8,3);
[blue, ultra thick] (11,2) –(10,3);
[above] at (8,3) $a$;
[above] at (10,3) $b$;
[above] at (12,3) $c$;
[below] at (10,0) $d$;
[below right] at (10.5,1.5) $j$;
[blue, ultra thick] (1,0) – (1,1);
[blue, ultra thick] (1,1) – (0,2);
[blue, ultra thick] (1,1) – (2,2);
[blue, ultra thick] (0,2) – (2,4);
[blue, ultra thick] (2,2) – (1.1,2.9);
[blue, ultra thick] (0.9,3.1) – (0,4);
[below] at (1,0) $i$;
[above] at (0,4) $a$;
[above] at (2,4) $b$;
[] at (4,1) $= R_{ab}^{i}$;
[blue, ultra thick] (6,0) – (6,1);
[blue, ultra thick] (6,1) – (7,4);
[blue, ultra thick] (6,1) – (5,4);
[below] at (6,0) $i$;
[above] at (5,4) $a$;
[above] at (7,4) $b$;

F and R moves.

Let there be a situation when the fusion outcome of three non-Abelian anyons $a,b,c$ is $d$. (It can be a subtree of another fusion tree in which $d$ can be fused with another anyon and the outcome is vacuum or some other anyons).
There are more different fusion channels than one. For example, $a$ can be fused to $b$ first, then their outcome $i$ is fused with $c$ to get $d$. Or $b$ can be fused to $c$ first, then their fusion outcome $j$ can be fused with $a$ to get $d$. The fusion channels $i$ and $j$ make two sets of basis. The transformation between these bases $i$ and $j$ is given by $F$-symbols or $F$-moves. As for the non-Abelian anyons, $i$ and $j$ occur in more ways than one, the $F$-moves between different $i$'s and $j$'s will be a matrix called the $F$-matrix shown in Fig. <ref> (a). The vector space for these anyons can be written as $V_{abc}^d=\bigoplus_i V_{ab}^i \otimes V_{ic}^d=\bigoplus_j V_{bc}^j \otimes V_{ja}^d$. The fusion diagrams which can be continuously deformed into each other are equivalent and represent the same state.
For a diagram of $n$ anyons shown in Fig. <ref>, the dimension of Hilbert space can be written in terms of the fusion rules as $N_{a_1a_2}^{e_1}N_{e_1a_3}^{e_2}...N_{e_{n-3}a_{n-1}}^{a_n}$.

[ultra thick,blue] (0,0)–(3,0);
[ultra thick,blue] (1,0)–(1,1);
[ultra thick,blue] (2,0)–(2,1);
[left] at (0,0) $ a $;
[above] at (1,1) $ b $;
[below] at (1.5,0) $ i $;
[above] at (2,1) $ c $;
[right] at (3,0) $ d $;

[ultra thick,blue] (0,0)–(6,0);
[ultra thick,blue] (1,0)–(1,1);
[ultra thick,blue] (2,0)–(2,1);
[ultra thick,blue] (4,0)–(4,1);
[ultra thick,blue] (5,0)–(5,1);
[left] at (0,0) $ a_1 $;
[above] at (1,1) $ a_2 $;
[below] at (1.5,0) $ e_1 $;
[above] at (2,1) $ a_3 $;
[below] at (2.5,0) $ e_2 $;
[above] at (4,1) $ a_{n-2} $;
[below] at (4.5,0) $ e_{n-3} $;
[above] at (5,1) $ a_{n-1} $;
[right] at (6,0) $ a_n $;

[Fusion space of multiple anyons.]Fusion space of anyons [28] for (a) four anyons, (b) $n$ anyons.

Now let us braid two anyons $a$ and $b$ by exchanging their places. Abelian anyons get a complex phase that depends on the types of anyons and on whether the exchange is clockwise or counterclockwise. But it does not depend on the order of exchange. Therefore, for a Abelian anyon we have the $R$-move given as $R_{ab} = e^{i\theta_{ab}}$. But for non-Abelian anyons, the $R$-move is a matrix $R_{ab}^{i} = e^{i \theta_{ab}^{i}}$ and it depends on the order of the exchange [19]. Topologically equivalent braids have the same outcomes. The braid matrix is shown in Fig. <ref> (b).

The fusion channel of a pair of anyons cannot be changed by $R$-move only. In other words, the system would not evolve from one ground state to the other by exchanging the two anyons of the same pair. To change their fusion channel, we need to braid $b$ and $c$. For this braiding, we have to transform $i$ bases to the $j$ bases by an $F$-move. Therefore, having only two anyons is not enough for making a topological qubit.
To see the effect of the exchange of $b$ and $c$ in the basis $i$, first an $F$-matrix is applied to change the basis from $i$ to $j$, then an $R$ matrix is applied, and then an $F^{-1}$-matrix is applied to change the basis back to $i$. This process is shown in Fig. <ref> and can be written as
B_ab = F_acb^d^-1 R_ab F_acb^d.
The matrix $B$ creates the superposition of the fusion channels, with a distinct phase factor for different fusion channels.

[ultra thick,blue] (0.5,0)–(2.5,0);
[ultra thick,blue] (1,0)–(1,1);
[ultra thick,blue] (2,0)–(2,1);
[below] at (0.5,0) $ a $;
[above] at (1,2) $ c $;
[below] at (1.5,0) $ i $;
[above] at (2,2) $ b $;
[below] at (2.5,0) $ d $;
[ultra thick,knot=blue] (2,1)–(1,2);
[ultra thick,knot=blue] (1,1)–(2,2);
[] at (4.5,0) $= \sum_{j}(F_{abc}^d)^i_j $;
[ultra thick,blue] (6.5,0)–(7.5,0);
[ultra thick,blue] (7,0)–(7,1);
[ultra thick,knot=blue] (7.5,1)–(6.5,2);
[ultra thick,knot=blue] (6.5,1)–(7.5,2);
[ultra thick,blue] (6.5,1)–(7.5,1);
[below] at (6.5,0) $ a $;
[above] at (6.5,2) $ c $;
[right] at (7,0.5) $ j $;
[above] at (7.5,2) $ b $;
[below] at (7.5,0) $ d $;
[right] at (8,0) $= \sum_{j}R_{bc}^j(F_{abc}^d)^i_j $;
[ultra thick,blue] (12,0)–(13,0);
[ultra thick,blue] (12.5,0)–(12.5,1);
[ultra thick,knot=blue] (12,1)–(12,2);
[ultra thick,knot=blue] (13,1)–(13,2);
[ultra thick,blue] (12,1)–(13,1);
[below] at (12,0) $ a $;
[above] at (12,2) $ c $;
[right] at (12.5,0.5) $ j $;
[above] at (13,2) $ b $;
[below] at (13,0) $ d $;

[right] at (13.5,0) $= \sum_{j}(F_{abc}^{d^{-1}})^i_jR_{bc}^j(F_{abc}^d)^i_j $;
[ultra thick,blue] (19,0)–(21,0);
[ultra thick,blue] (19.5,0)–(19.5,1);
[ultra thick,blue] (20.5,0)–(20.5,1);
[below] at (19,0) $ a $;
[above] at (19.5,1) $ c $;
[below] at (20,0) $ i $;
[above] at (20.5,1) $ b $;
[below] at (21,0) $ d $;

Braiding for the superposition of the two fusion channels.

$F$ and $R$ have to satisfy some consistency conditions, which restrict the multiplicity of possible models to finitely many. Such conditions are called pentagon and Hexagon equations [195] due to their geometric interpretation. These conditions are also studied algebraically in Chapter <ref> on category theory.
In the Fig. <ref>, the fusion process of four anyons $1,2,3,4$ is written as
(F_12c^5)_a^d (F_a34^5)_b^c = ∑_e (F_234^d)_e^c (F_1e4^5)_b^d (F_123^b)_a^e .
By the addition of braiding $R$-matrices, we get the relation pictorially shown in Fig. <ref>
∑_b (F_231^4)_b^c R_1b^4 (F_123^4)_a^b = R_13^c(F_213^4)_a^c R_12^a .

The twist factor or spin factor of an anyon is a phase corresponding to the rotation of a charge around its own magnetic flux. It can be thought of as a twist in a framed ribbon, as discussed in Chapter <ref>. This factor is written as $\theta_a = e^{2 \pi i h_a}$ when an anyon is rotated by $2\pi$, where $h_a$ is the topological spin of an anyon $a$. It is an integer for bosons and gives spin factor identity. It is a half-integer for fermions that would give the spin factor $-1$. Its value is between 0 and 1 for an anyon. For vacuum, $h_0 = 0$. The twist factor is also discussed in Chapter <ref> and is shown in Fig. <ref>. Through the ribbon equation, the spin factor can also be used to derive the entries of an $R$-matrix for a particular anyonic model. For example, when two anyons $a$ and $b$ are fused to $c$, the spin factor is given by the ribbon equation pictorially shown in Fig. <ref> [28, 213]
[R_ab^c]^2=θ_c/θ_a θ_b= e^2 πi h_c/e^2 πi h_a e^2 πi h_b = e^2 πi(h_c - h_a -h_b).
This is interpreted as the full twist of fusion product $c$ combined with the full twist of the charges $a$ and $b$ in the opposite direction, and is equal to the double exchange of the $a$ and $b$.

(P0) at (1.5,1) $\tik{[scale = 0.3]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (3,4);
\draw [blue, ultra thick] (0,1) -- (-3,4);
\draw [blue, ultra thick] (-2,3) --(-1,4);
\draw [blue, ultra thick] (-1,2) --(1,4);
\node [above] at (-3,4) {1};
\node [above] at (-1,4) {2};
\node [above] at (1,4) {3};
\node [above] at (3,4) {4};
\node [below] at (0,0) {5};
\node [below] at (-2,3) {a};
\node [below] at (-1,2) {b};}$;
(P1) at (4.5,3) $\tik{[scale = 0.3]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (3,4);
\draw [blue, ultra thick] (0,1) -- (-3,4);
\draw [blue, ultra thick] (-2,3) --(-1,4);
\draw [blue, ultra thick] (2,3) --(1,4);
\node [above] at (-3,4) {1};
\node [above] at (-1,4) {2};
\node [above] at (1,4) {3};
\node [above] at (3,4) {4};
\node [below] at (0,0) {5};
\node [below] at (-2,3) {a};
\node [below] at (2,3) {c};}$;
(P2) at (7.5,1) $\tik{[scale = 0.3]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (3,4);
\draw [blue, ultra thick] (0,1) -- (-3,4);
\draw [blue, ultra thick] (1,2) --(-1,4);
\draw [blue, ultra thick] (2,3) --(1,4);
\node [above] at (-3,4) {1};
\node [above] at (-1,4) {2};
\node [above] at (1,4) {3};
\node [above] at (3,4) {4};
\node [below] at (0,0) {5};
\node [below] at (1,2) {d};
\node [below] at (2,3) {c};}$;
(P3) at (3,-2) $\tik{[scale = 0.3]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (3,4);
\draw [blue, ultra thick] (0,1) -- (-3,4);
\draw [blue, ultra thick] (0,3) --(-1,4);
\draw [blue, ultra thick] (-1,2) --(1,4);
\node [above] at (-3,4) {1};
\node [above] at (-1,4) {2};
\node [above] at (1,4) {3};
\node [above] at (3,4) {4};
\node [below] at (0,0) {5};
\node [below] at (0,3) {e};
\node [below] at (-1,2) {b};}$;
(P4) at (6,-2) $\tik{[scale = 0.3]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (3,4);
\draw [blue, ultra thick] (0,1) -- (-3,4);
\draw [blue, ultra thick] (1,2) --(-1,4);
\draw [blue, ultra thick] (0,3) --(1,4);
\node [above] at (-3,4) {1};
\node [above] at (-1,4) {2};
\node [above] at (1,4) {3};
\node [above] at (3,4) {4};
\node [below] at (0,0) {5};
\node [below ] at (0,3) {e};
\node [below ] at (1,2) {d};}$;
[ultra thick,red,->] (P0) – (P1);
[ultra thick,red,->] (P1) – (P2);
[ultra thick,red,->] (P0) – (P3);
[ultra thick,red,->] (P3) – (P4);
[ultra thick,red,->] (P4) – (P2);
at (2.5,2.5) $(F^5_{a34})_b^c$;
at (6,2.5) $(F^5_{12c})_a^d$;
at (1.5,-0.6) $(F^b_{123})_a^e$;
at (4.5,-2.5) $(F^5_{1e4})_b^d$;
at (7.4,-0.7) $(F^d_{234})_e^c$;

The pentagon identity.

(P1) at (0,0) $\tik{[scale = 0.4]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (-2,3);
\draw [blue, ultra thick] (0,1) -- (2,3);
\draw [blue, ultra thick] (-1,2) --(0,3);
\node [above] at (-2,3) {$1$};
\node [above] at (0,3) {$2$};
\node [above] at (2,3) {$3$};
\node [below] at (-1,2) {$a$};
\node [below] at (0,0) {$4$};}$;
(P2) at (3,-1.5) $\tik{[scale = 0.4]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (-2,3);
\draw [blue, ultra thick] (0,1) -- (2,3);
\draw [blue, ultra thick] (-1,2) --(0,3);
\node [above] at (-2,3) {$1$};
\node [above] at (0,3) {$2$};
\node [above] at (2,3) {$3$};
\node [below] at (-1,2) {$a$};
\node [below] at (0,0) {$4$};}$;
(P3) at (6,-1.5) $\tik{[scale = 0.4]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (-2,3);
\draw [blue, ultra thick] (0,1) -- (2,3);
\draw [blue, ultra thick] (1,2) --(0,3);
\node [above] at (-2,3) {$1$};
\node [above] at (0,3) {$2$};
\node [above] at (2,3) {$3$};
\node [below] at (1,2) {$c$};
\node [below] at (0,0) {$4$};}$;
(P4) at (9,0) $\tik{[scale = 0.4]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (-2,3);
\draw [blue, ultra thick] (0,1) -- (2,3);
\draw [blue, ultra thick] (1,2) --(0,3);
\node [above] at (-2,3) {$1$};
\node [above] at (0,3) {$2$};
\node [above] at (2,3) {$3$};
\node [below] at (1,2) {$c$};
\node [below] at (0,0) {$4$};}$;
(P5) at (3,1.5) $\tik{[scale = 0.4]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (-2,3);
\draw [blue, ultra thick] (0,1) -- (2,3);
\draw [blue, ultra thick] (1,2) --(0,3);
\node [above] at (-2,3) {$1$};
\node [above] at (0,3) {$2$};
\node [above] at (2,3) {$3$};
\node [below] at (1,2) {$b$};
\node [below] at (0,0) {$4$};}$;
(P6) at (6,1.5) $\tik{[scale = 0.4]
\draw [blue, ultra thick] (0,0) -- (0,1);
\draw [blue, ultra thick] (0,1) -- (-2,3);
\draw [blue, ultra thick] (0,1) -- (2,3);
\draw [blue, ultra thick] (-1,2) --(0,3);
\node [above] at (-2,3) {$1$};
\node [above] at (0,3) {$2$};
\node [above] at (2,3) {$3$};
\node [below] at (-1,2) {$b$};
\node [below] at (0,0) {$4$};}$;
[ultra thick,red,->] (P1) – (P2);
[ultra thick,red,->] (P1) – (P5);
[ultra thick,red,->] (P2) – (P3);
[ultra thick,red,->] (P3) – (P4);
[ultra thick,red,->] (P5) – (P6);
[ultra thick,red,->] (P6) – (P4);
at (1.5,1.3) $(F^4_{123})_a^b$;
at (1.4,-1.3) $R^a_{12}$;
at (4.5,2) $R^4_{1b}$;
at (4.5,-2) $(F^4_{213})_a^c$;
at (7.5,1.3) $(F^4_{231})_b^c$;
at (7.5,-1.3) $R^c_{13}$;

The hexagon identity.

§ TOPOLOGICAL QUBIT

Now we will summarize what we discussed in the last section and build a qubit from this discussion. A pair of non-Abelian anyons cannot be used directly as a qubit, because the two states belong to different topological charge sectors $\ket{ab;i_1}$ and $\ket{ab;i_2}$, and cannot be superposed by braiding.
Let three anyons $a$, $b$ and $c$ be fused to $d$. The first two are fused to $i$, then their outcome is fused with the third gives $d$.
We can write the two different fusion channels as the two states of our qubit as
|i⟩ = |a,b →i⟩ |i,c →d⟩,
where the tensor product symbol is omitted. Alternatively, when the last two anyons $b$ and $c$ are fused to $j$, the $j$ is fused with $c$ to make $d$. $i$ and $j$ are two sets of bases. This is shown in the Fig. <ref>. A qubit can also be formed as
|j⟩ = |b,c →j⟩|j,a →d⟩.
The change of basis is performed by using the $F$-matrices as
|i⟩ = ∑_j (F_abc^d)_j^i |j⟩.
The $(F^d_{abc})_j^i$ are the matrix elements of $F_{abc}^d$ summed over $j$. $F$ and $R$ matrices are obtained for a particular anyon model from the solution of the pentagon and hexagon Eqs. (<ref>) and (<ref>) [28]. Ising and Fibonacci anyons are the most popular systems to make the topological quantum computing logic gates. These anyons are found as quasiparticles in non-Abelian fractional quantum Hall effect and topological superconductors.

A possible error in topological quantum computation is due to the braiding or fusion with some unattended anyon in the system. This error can be avoided by carefully accounting for the charges participating in the encoding. Another type of error could be due to the energy of the system, such that the gap between the ground state and the excited state gets filled. This error can be minimized by keeping the system at a very low temperature. The measurement of outcome is either interference or the projective measurement as discussed in Section <ref>, see also [11, 130].

§.§ Example 1: Fibonacci Anyon

The simplest non-Abelian anyon model consists of only two particles $\textbf{1}$ and $\tau$ [214, 205, 215]. The fusion rules for this anyon model are
ττ= 1 + τ
The basis states can be written as
|0⟩ = |τ,τ→1⟩,     |1⟩ = |τ,τ→τ⟩
The dimension is the different number of ways the fusion of all anyons can result in topological charge $\textbf{1}$ or $\tau$. In the fusion outcome of two $\tau$ we get $\textbf{1}$ with probability $p_0 = 1/\phi^2$ and $\tau$ with probability $p_1 = \phi/\phi^2 = 1/\phi$ [215].
The dimension grows as the Fibonacci series, in which the next number is a sum of the last two numbers. The quantum dimension $d_\tau = \phi = (1+ \sqrt{5})/2$ is the golden mean.
τττ   = 1 + 2τ

ττττ   = 2 ·1 + 3 ·τ

τττττ   = 3 ·1 + 5 ·τ
As we discussed above, no amount of braiding can change one qubit state to the other. Therefore, we need more than two $\tau$ particles for the qubit. Also, topological quantum computation has no tensor product structure. That means, if three $\tau$ anyons are used to make a qubit then the six anyons have only the five dimensional fusion space. Therefore, only a subspace is used to encode a qubit. Three Fibonacci anyons are required for the qubit and the fusion of four particles results in the vacuum [19]. $F$ and $R$ matrices for this model are obtained by consistency conditions, as in Ref. [28] are given as
1/ϕ     1/√(ϕ)

1/√(ϕ)     -1/ϕ

R_ττ =
R^1_ττ     0

0     R^τ_ττ
e^4πi/5     0

0     -e^3πi/5

Quantum computing with Fibonacci anyons is done as follows.
The fusion of two $\tau$ particles gives either $\bm{1}$ or $\tau$. These two orthogonal states are represented as $\ket{(\bullet, \bullet)_\mathbf{1}}$ and $\ket{(\bullet, \bullet)_\tau}$. The addition of the third $\tau$ particle to the state $\ket{(\bullet, \bullet)_\mathbf{1}}$ gives $\tau$ and is denoted as $\ket{((\bullet, \bullet)_\mathbf{1}, \bullet)_\tau} \equiv \ket{0}$. But when the third particle is added to the state $\ket{(\bullet, \bullet)_\mathbf{1}}$, we get either $\bm{1}$ or $\tau$. These states are represented as $\ket{((\bullet,\bullet)_\tau,\bullet)_\tau}\equiv\ket{1}$ and
$\ket{((\bullet,\bullet)_\tau,\bullet)_{\bm{1}}}\equiv \ket{N}$, here $\ket{N}$ stands for the non-computational state. The amplitude in this state is considered as the leakage error [11, 212]. The states $\ket{((\bullet, \bullet)_\mathbf{1}, \bullet)_\tau} \equiv \ket{0}$ and $\ket{((\bullet,\bullet)_\tau,\bullet)_\tau}\equiv\ket{1}$ are the basis states for a qubit and which are interchanged by an $F$ matrix (<ref>). The braiding of these particles is represented by an $R$ matrix (<ref>). These basis states and the fusion of Fibonacci anyons is shown in Fig.<ref>. The set of gates required to build any kind of circuit is called the universal quantum gate set. The Fibonacci anyonic model is the universal for quantum computing. These kinds of anyons are proposed to be found in the Read-Rezayi state $\nu = 12/5$ which is a very fragile state, so other anyon models are also under consideration [11].

[ultra thick,blue] (0,0)–(3,0);
[ultra thick,blue] (1,0)–(1,1);
[ultra thick,blue] (2,0)–(2,1);
[left] at (0,0) $ \tau $;
[above] at (1,1) $ \tau $;
[below] at (1.5,0) $ i $;
[above] at (2,1) $ \tau $;
[right] at (3,0) $ \tau $;

[ultra thick,blue] (0,0)–(6,0);
[ultra thick,blue] (1,0)–(1,1);
[ultra thick,blue] (2,0)–(2,1);
[ultra thick,blue] (4,0)–(4,1);
[ultra thick,blue] (5,0)–(5,1);
[left] at (0,0) $ \tau $;
[above] at (1,1) $\tau $;
[below] at (1.5,0) $ e_1 $;
[above] at (2,1) $ \tau $;
[below] at (2.5,0) $ e_2 $;
[above] at (4,1) $ \tau $;
[below] at (4.5,0) $ e_{n-3} $;
[above] at (5,1) $ \tau $;
[right] at (6,0) $ \tau $;

[Fusion space of Fibonacci anyons.](a) Two orthogonal qubit states, $i$ would be either $1$ or $\tau$. (b) fusion space for $n$ Fibonacci anyons.

at (-1.5,0) $\ket{0} = \ket{((\bullet,\bullet)_1,\bullet)_\tau} =$;
[ultra thick,blue] (1.9,0) ellipse (1.5 and 0.5);
[ultra thick,blue] (1.55,0) ellipse (1 and 0.3);
[ultra thick,blue] (1.55,0) ellipse (1 and 0.3);
[red] at (1,0) $\bullet$;
[red] at (2,0) $\bullet$;
[red] at (3,0) $\bullet$;
[below] at (2.65,0) $\bm{1}$;
[below] at (3.55,0) $\tau$;
at (4,0) $=$;
[blue, ultra thick] (5,-1) – (5,-0.5);
[blue, ultra thick] (5,-0.5) – (6,0.5);
[blue, ultra thick] (5,-0.5) – (4,0.5);
[blue, ultra thick] (4.5,0) –(5,0.5);
[above] at (6,0.5) $\tau$;
[above] at (4,0.5) $\tau$;
[above] at (5,0.5) $\tau$;
[below] at (5,-1) $\tau$;
[below] at (4.5,0) $\bm{1}$;

at (-1.5,0) $\ket{1} = \ket{((\bullet,\bullet)_\tau,\bullet)_\tau} =$;
[ultra thick,blue] (1.9,0) ellipse (1.5 and 0.5);
[ultra thick,blue] (1.55,0) ellipse (1 and 0.3);
[ultra thick,blue] (1.55,0) ellipse (1 and 0.3);
[red] at (1,0) $\bullet$;
[red] at (2,0) $\bullet$;
[red] at (3,0) $\bullet$;
[below] at (2.65,0) $\tau$;
[below] at (3.55,0) $\tau$;
at (4,0) $=$;
[blue, ultra thick] (5,-1) – (5,-0.5);
[blue, ultra thick] (5,-0.5) – (6,0.5);
[blue, ultra thick] (5,-0.5) – (4,0.5);
[blue, ultra thick] (4.5,0) –(5,0.5);
[above] at (6,0.5) $\tau$;
[above] at (4,0.5) $\tau$;
[above] at (5,0.5) $\tau$;
[below] at (5,-1) $\tau$;
[below] at (4.4,0) $\tau$;

at (-1.5,0) $\ket{N} = \ket{((\bullet,\bullet)_\tau,\bullet)_{\bm{1}}}=$;
[ultra thick,blue] (1.9,0) ellipse (1.5 and 0.5);
[ultra thick,blue] (1.55,0) ellipse (1 and 0.3);
[red] at (1,0) $\bullet$;
[red] at (2,0) $\bullet$;
[red] at (3,0) $\bullet$;
[below] at (2.65,0) $\tau$;
[below] at (3.55,0) $\bm{1}$;
at (4,0) $=$;
[blue, ultra thick] (5,-1) – (5,-0.5);
[blue, ultra thick] (5,-0.5) – (6,0.5);
[blue, ultra thick] (5,-0.5) – (4,0.5);
[blue, ultra thick] (4.5,0) –(5,0.5);
[above] at (6,0.5) $\tau$;
[above] at (4,0.5) $\tau$;
[above] at (5,0.5) $\tau$;
[below] at (5,-1) $\bm{1}$;
[below] at (4.4,0) $\tau$;

[The orthogonal states of three Fibonacci particles.]Orthogonal states of three Fibonacci particles [11]

§.§ Example 2: Ising Anyon

This model has three anyons $\textbf{1}$, $\sigma$ and $\psi$.
The fusion rules for these anyons are;
σσ= 1 + ψ,     ψψ= 1,     ψσ= σ
Two basis states can be written as
|0⟩=|σ, σ→1⟩,     |1⟩=|σ, σ→ψ⟩
Since two fusions belong to different topological charge sectors, at least three anyons are needed that can fuse to $\sigma$ in two different ways. For every added $\sigma$ the dimension of fusion space doubles, hence for $2N$ anyons the dimension is $2^{N-1}$. This model is non-universal, so non-topological schemes are also devised in addition to topological computation. The $F$ and $R$ matrices for this model [19] are,
F_σσσ^σ= 1/√(2)
1     1

1     -1
,    R_σσ = e^-i π/8
1     0

0     i

where $R_{\sigma \sigma}^1 = e^{-i\pi/8}$ and $R_{\sigma \sigma}^\psi = e^{i3\pi/8}$. The topological charges are labeled as $0,1/2, 1$ corresponds to $\textbf{1}, \sigma, \psi$. The total charge of two particles with charges $1/2$ is either $1$ or $0$. The total charge is $0$ when both particles have charges $1$, but the total charge is $1/2$ when one particle has charge $1$ and other has $1/2$. The quantum dimensions, $d_1=d_\psi =1$ and $d_\sigma=\sqrt{2}$, for these anyons are computed in Ref. [28] using the fusion rules.

Let us consider four particles of charge $1/2$ with a total charge of $0$. The first two are fused either to $0$ or $1$. In case it is zero, then the total of the third and fourth must be zero. If the total of first and second is 1 then the total of third and fourth must be equal to 1. In this way, we have two states of four $1/2$ quasiparticles. There are $2^{n-1}$ states for $2n$ particles [116, 11].
When particles of the same pair, say $i$, are braided, only the phase is changed, but when a particle of pair $i$ is braided with the particle of other pair $j$, a NOT gate is applied [11]. Taking both particles of $i$ around both particles in $j$ then the basis state is multiplied by $+1$ if $j$ has a charge $0$, but it gets multiplied by $-1$ if it has a charge $1$. Six $\sigma$ anyons are required for two-qubit encoding. See Ref. [11, 19] for the implementation of CNOT and phase gates. The Ising anyon model is implemented by using the quantum Hall state $\nu = 5/2$ and Majorana particle in topological superconductors, see Chapters <ref> and <ref>.

[scale = 0.4]
[blue, ultra thick] (0,0) – (0,1);
[blue, ultra thick] (0,1) – (3,4);
[blue, ultra thick] (0,1) – (-3,4);
[blue, ultra thick] (-2,3) –(-1,4);
[blue, ultra thick] (-1,2) –(1,4);
[above] at (-3,4) $\sigma$;
[above] at (-1,4) $\sigma$;
[above] at (1,4) $\sigma$;
[above] at (3,4) $\sigma$;
[below] at (0,0) 1;
[below] at (-2,3) $a$;
[below] at (-1,2) $b$;

[scale = 0.4]
[blue, ultra thick] (0,0) – (0,1);
[blue, ultra thick] (0,1) – (3,4);
[blue, ultra thick] (0,1) – (-3,4);
[blue, ultra thick] (-2,3) –(-1,4);
[blue, ultra thick] (-1,2) –(1,4);
[above] at (-3,4) $\psi$;
[above] at (-1,4) $\sigma$;
[above] at (1,4) $\sigma$;
[above] at (3,4) $\sigma$;
[below] at (0,0) $\sigma$;
[below] at (-2,3) $a$;
[below] at (-1,2) $b$;

Two channels for Ising anyons

[blue, ultra thick] (2,0) – (2,1);
[blue, ultra thick] (2,1) – (4,3);
[blue, ultra thick] (2,1) – (0,3);
[blue, ultra thick] (1,2) –(2,3);
[above] at (0,3) $\sigma$;
[above] at (2,3) $\sigma$;
[above] at (4,3) $1$;
[below] at (2,0) $1$;
[below left] at (1.5,1.5) $1$;

$=[F^1_{\sigma \sigma 1}]^\sigma_1$

[blue, ultra thick] (10,0) – (10,1);
[blue, ultra thick] (10,1) – (12,3);
[blue, ultra thick] (10,1) – (8,3);
[blue, ultra thick] (11,2) –(10,3);
[above] at (8,3) $\sigma$;
[above] at (10,3) $\sigma$;
[above] at (12,3) $1$;
[below] at (10,0) $1$;
[below right] at (10.5,1.5) $\sigma$;

[blue, ultra thick] (2,0) – (2,1);
[blue, ultra thick] (2,1) – (4,3);
[blue, ultra thick] (2,1) – (0,3);
[blue, ultra thick] (1,2) –(2,3);
[above] at (0,3) $\sigma$;
[above] at (2,3) $\sigma$;
[above] at (4,3) $1$;
[below] at (2,0) $1$;
[below left] at (1.5,1.5) $1$;

$=[F^1_{\sigma \sigma \psi}]^\sigma_1$

[blue, ultra thick] (10,0) – (10,1);
[blue, ultra thick] (10,1) – (12,3);
[blue, ultra thick] (10,1) – (8,3);
[blue, ultra thick] (11,2) –(10,3);
[above] at (8,3) $\sigma$;
[above] at (10,3) $\sigma$;
[above] at (12,3) $1$;
[below] at (10,0) $1$;
[below right] at (10.5,1.5) $\sigma$;

Two trivial F-moves (a) $[F^1_{\sigma \sigma 1}]^\sigma_1=1$ (b) $[F^1_{\sigma \sigma \psi}]^\sigma_1=0$

CHAPTER: RECOUPLING THEORY

The fusion of quasiparticles is similar to the recoupling theory of addition of angular momentum in quantum mechanics [216, 217, 218]. Ternary logic gates are designed using the q-deformed version of the recoupling theory. Therefore, for the intuition, we will discuss the quantum deformation and the recoupling of angular momenta and then discuss the quantum deformation of recoupling theory [219, 220, 221]. The quantum deformed quantities are also called q-analogs.

§ QUANTUM DEFORMATION

In classical mechanics, states on phase space make a manifold represented by say $(q,p)$. Physical quantities are observables that are the functions of $(q,p)$. The Abelian algebra formed for these observables and associated geometry is commutative.
In quantum mechanics, due to the Heisenberg uncertainty principle, there is no arbitrary precision of the quantities. Algebra is non-commutative and classical mechanics is the limiting case when Planck constant $h \rightarrow 0$. Therefore, quantum mechanics is a kind of deformation of classical mechanics.
In quantum mechanics, the commuting classical observables are replaced with the noncommuting Hermitian operators. Hence we can say that we deform classical algebra and the deformation parameter is $h$.
The noncommutativity of the variables $X$ and $Y$ in the deformed space is written as
XY = qYX,
where $q$ is a complex number in general. It is called the deformation parameter. Let $q$ be a number different from $1$, and $h$ be a number different from $0$. If we take $x= qx_0$ or $x=x_0+h$ we can have the classical values when $h\to 0$ or $q\to 1$. These two are related as $q= e^h$ [222]. For $q \to 1$, we would get back the classical commuting variables.
Let us choose $q = e^{i\theta}$. Consider an example when the operators $T_\alpha$ and $G_{\theta/\alpha}$ are acting on a function $\psi(x)$ of real variable $x$, such that
T_αψ(x) = ψ(x+α),     G_θ/α ψ(x) = e^i θx / α ψ(x).
When we apply both $T_\alpha$ and $G_{\theta/\alpha}$ operators, we get
T_αG_θ/α ψ(x) = e^iθ(x+α) ψ(x+α) = e^iθ G_θ/α T_αψ(x).
With the fixed value of variables $\theta$ and $\alpha$, $T_\alpha$ and $G_{\theta/\alpha}$ become noncommuting variables that can be written as
T_αG_θ/α = e^iθG_θ/α T_α.
The q-analogs of an algebra is discussed in Appendix <ref>. See also  [222].

§.§ q-Analogs

An anyon or a pair of anyons interacts through braiding in the plane deformed by the existence of the fields of the other anyons in an abstract way. This braiding interaction may cause the twist factor that is related to the topological spin of an anyon and involves the parameter $q$. Algebraically, we can think of $q$ as a small perturbation of the usual mathematical objects. The $q$ can be generic or a root of unity. The root of unity is defined as when a complex number $q$ is raised to some power $n$ so that it equals 1 for that power, then we say that this complex number is $n$th root of unity.
Now the $q$-analog quantities give the corresponding classical quantities for the limiting case when $q \rightarrow 1$.
For $n\in \mathbb{Z}$ we define what is called a $q$-integer
[n]_q= q^n - q^-n/q-q^-1,
which is the Laurent polynomial equal to $n \in \mathbb{Z}$ for $q \rightarrow 1$.
It is done by taking the limit $q \to 1$ and applying L'Hospital's rule.
The $q$-analog of natural numbers is as follows, $[0]_q =0,\ [1]_q = 1,\ [2]_q = \frac{q^2-q^{-2}}{q-q^{-1}}$, and so on.
We can get $q$-factorial $[n]!$ that can be written as
[n]!    = [n][n-1]...[1]

=q^n - q^-n/q-q^-1·q^n-1 - q^-(n-1)/q-q^-1...q^2 - q^-2/q-q^-1·1
Sometimes, we take $[n]_q = \frac{1-q^n}{1-q}$, as in [222, 223].
In that case the $q$-analog of natural numbers and $q$-factorials are written as $[0]_q =0,\ [1]_q = 1,\ [2]_q = 1+q,\ [3]_q = 1+q+q^2$,

§ RECOUPLING OF ANGULAR MOMENTA

We will briefly present the quantum theory of angular momentum to get the idea of the $SU(2)_k$ anyonic model will be discussed in the next section. This model is the quantum deformation of recoupling theory. For more detailed study, see [218, 217, 216].
Let us consider two systems with angular momentum operators $J_1$ and $J_2$ with eigenvalues $j_1$ and $j_2$. These two systems may be the orbital angular momentum of two different particles or they may be the spin and orbital angular momentum of a single particle. The z-components of the angular momenta $J_z$ have allowed eigenvalues $-j_1 \le m_1 \le +j_1$ with $2j_1 + 1$ values, and $-j_2 \le m_2 \le +j_2$ with $2j_2 + 1$ states. The combined system is written as $j_1 \otimes j_2$.

It is like the vector spaces $V_1$ and $V_2$, with dimensions $2j_2 +1$ each, are combined as $V_1 \otimes V_2$ with dimensions $(2j_1 +1)(2j_2 +1)$. The total angular momentum operator is acting on $V_1 \otimes V_2$. This operator constitutes $SU(2)$ Lie algebra.
Two quantum numbers are needed to specify an individual system and four quantum numbers to specify the combined system.
For the whole system
J^2 = (J_1 +J_2)^2,     J_z = J_1z + J_2z.
These operators are applied to states as
J^2|j,m⟩    = j(j+1)|j,m⟩

J_z|j,m⟩    =m|j,m⟩

J_±|j,m⟩    = √((j∓m)(j±m+1))|j,m±1⟩
where $J_+ = J_x+iJ_y$ and $J_- = J_x -iJ_y$.
The uncoupled states $\ket{j_1j_2m_1m_2}$ are the eigenstates of operators $\left\{J_1^2,J_{1z},J^2_2, J_{2z}\right\}$ and the coupled state $\ket{j_1j_2jm}$ are the eigenstate of operators $\left\{J_1^2,J^2_2,J^2, J^2_z\right\}$.
The coupling of $j_1$ and $j_2$ is the construction of eigenfunctions of $J^2$ and $J_z$
We will write the total angular momentum basis $\ket{j_1j_2;jm}$ in terms of tensor product basis $\ket{j_1m_1}\ket{j_2m_2}$. That is done by expressing the coupled state in terms of the uncoupled state.
|(j_1 j_2)jm⟩= ∑_m_1 = -j_1^j_1∑_m_2 = -j_2^j_2 |j_1m_1⟩|j_2m_2⟩⟨j_1m_1j_2m_2||jm⟩

where $j=\abs{j_1-j_2},...,\abs{j_1+j_2}, \ m = -j,...,j$.
The coefficients $C_{mm_1m_2}^{jj_1j_2}$
∑_j_1j_2m_1m_2C_mm_1m_2^jj_1j_2 = ⟨j_1m_1;j_2m_2||j_1j_2;jm⟩ = ⟨j_1m_1;j_2m_2||jm⟩
are called Clebsch-Gordon coefficients (CGC). These are non-zero only when $m = m_1 + m_2$ and $\abs{j_1-j_2} \le j \le j_1 +j_2$.

The total angular momentum can have value $j = j_1 + j_2, j_1+j_2 -1,...,\abs{j_1 -j_2}$.
Any of the numbers $j_1,j_2,j$ can have values that are greater than or equal to the difference of the other two and less than or equal to the sum of the other two. This condition is called triangle condition and is represented by $\Delta(j_1j_2j)$.
We can also write the total angular momentum basis in terms of the product basis by using the Wigner's 3j-symbols. In that case, the coefficients are called Wigner coefficients.
The Wigner 3j-symbol is zero unless the triangle condition is satisfied.
The CGC are related to the 3j-symbols as
⟨j_1m_1;j_2m_2||j_1j_2;jm⟩ = (-1)^-j_1+j_2 -m√(2j+1)

j_1   j_2   j

m_1   m_2   -m

On similar lines, we can couple three angular momenta
$J_1,J_2,J_3$ whose total angular momentum $J= J_1+J_2+J_3$. There are two coupling schemes as
(J_1+J_2)+J_3 = J_12+J_3 = J     J_1+(J_2 +J_3) = J_1+J_23 = J.
The total coupling may be done by
first coupling $j_1$ and $j_2$ to $j_{12}$ and then $j_{12}$ and $j_3$ to $J$
= ∑_m_12 = -j_12^j_12∑_m_3 = -j_3^j_3 |(j_1j_2);j_12m_12⟩|j_3m_3⟩

Alternatively, we can first combine $j_2$ and $j_3$ to get $j_{23}$ and next $j_{23}$ can be combined with $j_1$ to make $J$
|j_1((j_2 j_3)j_23);jm⟩
= ∑_m_1 = -j_1^j_1∑_m_23 = -j_23^j_23
|j_2j_3;j_23m_23⟩ ⟨j_1 j_23;m_1m_23||j_1j_23;jm⟩.
These two coupling schemes are shown in Fig. <ref> (a). The coupling scheme results in a complete orthonormal bases for the $(2j_1 + 1)(2j_2 + 1)(2j_3+1)$-dimensional space spanned by $\ket{j_1,m_1}\ket{j_2,m_2}\ket{j_3,m_3}$, $m_1 = -j_1,...,j_1, m_2 = -j_2,...,j_2$; $m_3 = -j_3,...,j_3$.

The angular momenta in two coupling schemes are related by a unitary transformation. The matrix elements of this unitary transformation are known as recoupling coefficients. These coefficients are independent of $m$ and so we have

|((j_1j_2)j_12j_3)jm⟩    = ∑_j_23 |(j_1(j_2j_3)j_23)jm⟩⟨(j_1(j_2j_3)j_23)j||((j_1j_2)j_12j_3)j⟩.
These coefficients can be written in terms of Wigner 6j-symbols,
⟨(j_1(j_2j_3)j_23)j||((j_1j_2)j_12j_3)j⟩ = (-1)^j_1+j_2+j_3+j √((2j_12 +1)(2j_23 +1))
j_1     j_2     j_12

j_3     j     j_23
The 6j-symbols have a symmetry that permutation of columns or rows leaves it invariant. Similar to the 3j-symbols, 6j-symbols are not matrices. The Racah coefficients [224] are related to the recoupling coefficients as
W(j_1 j_2 j_3J; j_12j_23) = ⟨(j_1(j_2j_3)j_23)j||((j_1j_2)j_12j_3)j⟩/√((2J_12+1)(2J_23+1)).
Therefore, the Racah coefficients are related to the Wigner 6j-symbols by

j_1     j_2     j_12

j_3     j     j_23
= (-1)^j_1+j_2+j_3+j W(j_1j_2j_3j;j_12j_23)
If $a \equiv j_1, \ b \equiv j_2, \ c \equiv j_3, \ d \equiv j, \ e \equiv j_{12}, \ f \equiv j_{23}$, we have the triangle condition as
Δ(abc) = √((a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!).
The right hand side is zero unless the triangle condition is satisfied. This condition is satisfied by each side of the quadrilateral in Fig. <ref> (b). The Racah coefficient is a product of four of these factors
W(abcd;ef) = Δ(abe)Δ(cde)Δ(acf)Δ(bdf) ω(abcd;ef),
$$\omega(abcd;ef) = \sum_z \frac{(-1)^{z+\beta_1}(z+1)!}{(z-\alpha_1)!(z-\alpha_2)!(z-\alpha_3)!(z-\alpha_4)!(\beta_1 - z)!(\beta_2 - z)!(\beta_3 - z)!},$$

$\alpha_1 = a+b+e, \ \alpha_2 = c+d+e, \ \alpha_3 = a+c+f \ \alpha_4 = b+d+f$
$\beta_1 = a+b+c+d, \ \beta_2 = a+d+e+f, \ \beta_3 = b+c+e+f$.
The sum over $z$ is finite over the range $max(\alpha_1,\alpha_2,\alpha_3,\alpha_4) \le z \le min(\beta_1,\beta_2,\beta_3)$
See [216, 225, 226] for detailed derivation of the above equation and coupling and recoupling of angular momenta.

[blue, ultra thick] (2,0) – (2,1);
[blue, ultra thick] (2,1) – (4,3);
[blue, ultra thick] (2,1) – (0,3);
[blue, ultra thick] (1,2) –(2,3);
[above] at (0,3) $j_1$;
[above] at (2,3) $j_2$;
[above] at (4,3) $j_3$;
[below] at (2,0) $j$;
[below left] at (1.5,1.5) $j_{12}$;

[blue, ultra thick] (10,0) – (10,1);
[blue, ultra thick] (10,1) – (12,3);
[blue, ultra thick] (10,1) – (8,3);
[blue, ultra thick] (11,2) –(10,3);
[above] at (8,3) $j_1$;
[above] at (10,3) $j_2$;
[above] at (12,3) $j_3$;
[below] at (10,0) $j$;
[below right] at (10.5,1.5) $j_{23}$;

[ultra thick, blue] (0,-1) – (0,0.8);
[ultra thick, blue] (0,-1) – (1,-0.6);
[ultra thick, blue] (0,-1) – (-1,-0.6);
[ultra thick,dashed, blue] (-1,-0.6) – (1,-0.6);
[ultra thick, blue] (-1,-0.6) – (0,0.8);
[ultra thick, blue] (1,-0.6) – (0,0.8);
[left] at (0.05,0) $j_3$;
[] at (0.5,-0.45) $j_1$;
[] at (0.5,-1) $j_{12}$;
[] at (-0.5,-1) $j_2$;
[] at (0.55,0.3) $j$;
[] at (-0.55,0.4) $j_{23}$;

[Recoupling of three angular momenta and triangle condition.](a) Recoupling of three angular momenta (b) the triangle condition.

§ $SU(2)_K$ ANYON MODEL

The F-symbols in topological quantum computation can be computed using the $SU(2)_k$ model [130] and Temperley-Lieb recoupling theory [208] also called the $JK_k$ anyon model [227]. Where the $k$ is called the level of the theory. It is the coupling constant of Chern-Simons theory and is related to the number of particles present as we discussed in Chapter <ref>. These theories are the quantum analog of theory of addition of angular momentum. The $SU(2)_k$ is the q-deformed version of $SU(2)$ with $q=\exp(i2\pi/(k+2))$, for $q$ at the root of unity. The detailed derivation of the parameter $q$ in this form is given in [79]. The anyon's fusion amplitudes would be written as the recoupling coefficients. The $F$-symbols and $R$-symbols are obtained by using these two models at level $k=4$ gave identical values. The charges are half integral in $SU(2)_k$ model, but are integral in $JK_k$ model. For $F$ and $R$ symbols in the $JK_4$ model, see [227]. In our work, we will use the $SU(2)_4$ model. The topological data for this model are given as

[n]_q = q^n/2-q^-n/2/q^1/2 - q^-1/2,     𝒞 = {0, 1/2,..., k/2 },     j_1 ×j_2 = ∑_j = |j_1-j_2|^minj_1+j_2, k-j_1-j_2,

[F_j^j_1,j_2,j_3]_j_12,j_23 = (-1)^j_1 + j_2 + j_3 +j √([2j_12+1]_q [2j_23+1]_q)

j_1     j_2     j_12

j_3     j     j_23

j_1     j_2     j_12

j_3     j     j_23

×∑_z (-1)^z[z+1]_q !/[z-j_1-j_2 -j_12]_q![z-j_12-j_3 -j]_q![z-j_2-j_3 -j_23]_q![z-j_1-j_23 -j]_q!
×1/[j_1 +j_2 +j_3 +j -z]_q![j_1 +j_12 +j_3 +j_23 -z]_q![j_2 +j_12 +j +j_23 -z]_q!,

$$ \Delta(j_1,j_2,j_3) = \sqrt{\frac{[-j_1 +j_2 +j_3]_q![j_1-j_2+j_3]_q! [j_1 +j_2 -j_3]_q!}{[j_1 + j_2 +j_3 +1]_q!}}, \qquad [n]_q! \equiv \prod_{m=1}^n [m]_q,$$

$$ R_j^{j_1,j_2} = (-1)^{j-j_1-j_2} q^{\frac{1}{2}[j(j+1) - j_1(j_1+1) -j_2(j_2+1)]},$$

$$ d_j = [2j+1]_q = \frac{\sin \big[\frac{(2j+1)\pi}{k+2}\big]}{\sin \big(\frac{\pi}{k+2}\big)}, \quad {\mathcal{D}} = \frac{\sqrt{\frac{k+2}{2}}}{\sin \big(\frac{\pi}{k+2}\big)},$$

$$ \theta_j = q^{j(j+1)} = e^{i 2\pi \frac{j(j+1)}{k+2}}, \qquad S_{j_1j_2} = \sqrt{\frac{2}{k+2}} \sin \big[ \frac{(2j_1 +1)(2j_2 +1) \pi}{k+2} \big].$$

Where $d$, $\mathcal{D}$, and $S_{j_1,j_2}$ are quantum dimension, total quantum dimension and topological $S$-matrix respectively. The data in $SU(2)_k$ theory are used to compute the $[F_j^{j_1,j_2,j_3}]_{j_{12},j_{23}}$ and $R^{j_1j_2}_j$ matrices [228].

For $SU(2)_4$, the charges are $0,1,2,3,4$, and the fusion rules are

$$0\cross 0 = 0, \ 0\cross 1 = 1, \ 0\cross 2 = 2, \ 0\cross 3 = 3, \ 0\cross 4 = 4,$$

$$1\cross 1 = 0+2, \ 1\cross 2 = 1+3, \ 1\cross 3 = 2+4, \ 1\cross 4 = 3,$$

$$2\cross 2 = 0+2+4, \ 2\cross 3 = 1+3, \ 2\cross 4 = 2,$$

$$3\cross 3 = 0+2, \ 3\cross 4 = 1, \ 4\cross 4 = 0$$

The $F$ and $R$ symbols are calculated using these fusion rules and from the $F$ and $R$ matrices, the $\sigma$ matrices are obtained in the next Chapter <ref> [228, 227]. The $F$ and $R$ symbols for $SU(2)_k$ anyonic model are given as

$[F^{abc}_d]_{ef}=1$ when any of the $a,b,c,d$ is zero,

$F_4^{114} = F_4^{123} = F_3^{124} = F_4^{132} = F_3^{133} = F_2^{134} = F_4^{141} = F_3^{142} = F_2^{143} = F_1^{144} = F_4^{213} = F_3^{214} = F_4^{222} = F_2^{224} = F_4^{231} = F_1^{234} = F_3^{241} = F_2^{242} = F_1^{243} = F_4^{312} = F_3^{313} = F_2^{314} = F_4^{321} = F_1^{324} = F_3^{331} = F_1^{331} = F_4^{334} = F_2^{341} = F_1^{342} = F_4^{343} = F_3^{344} = F_4^{411} = F_3^{412} = F_2^{413} = F_1^{414} = F_3^{421} = F_2^{422} = F_1^{423} = F_2^{431} = F_1^{432} = F_4^{433} = F_3^{434} = F_1^{441} = F_3^{443} = -1$,

$F_1^{111} = F_3^{131} = F_1^{313} = F_3^{333} = \begin{pmatrix}
-\frac{1}{\sqrt{3}} & \sqrt{\frac{2}{3}} \\
\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}}
\end{pmatrix},$

$F_2^{112} = F_1^{122} = F_3^{122} = F_2^{132} = F_2^{211} = F_2^{213} = F_1^{221} = F_3^{221} = F_1^{223} = F_2^{231} = F_2^{312} = F_1^{322} = \begin{pmatrix}
-\frac{1}{\sqrt{2}} & \sqrt{\frac{1}{2}} \\
\sqrt{\frac{1}{2}} & \frac{1}{\sqrt{2}}
\end{pmatrix},$

$F_3^{113} = F_1^{133} = F_3^{311} = F_1^{331} = \begin{pmatrix}
-\frac{2}{\sqrt{3}} & \sqrt{\frac{1}{3}} \\
\sqrt{\frac{1}{3}} & \frac{2}{\sqrt{3}}
\end{pmatrix},$

$F_2^{121} = F_1^{212} = \begin{pmatrix}
-\frac{1}{2} & \frac{\sqrt{3}}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{pmatrix},$

$F_2^{123} = F_3^{212} = F_1^{232} = F_2^{321} = \begin{pmatrix}
-\frac{\sqrt{2}}{3} & \frac{1}{2} \\
\frac{1}{2} & \frac{\sqrt{2}}{3}
\end{pmatrix},$

$F_3^{223} = F_2^{233} = F_3^{322} = F_2^{332} = \begin{pmatrix}
\frac{1}{\sqrt{2}} & -\sqrt{\frac{1}{2}} \\
-\sqrt{\frac{1}{2}} & -\sqrt{\frac{1}{2}}
\end{pmatrix},$

$F_3^{232} = F_2^{323} = \begin{pmatrix}
\frac{1}{2} & -\frac{\sqrt{3}}{2} \\
-\frac{\sqrt{3}}{2} & -\frac{1}{2}
\end{pmatrix},$

$F_2^{222} = \begin{pmatrix}
\frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2} \\
-\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2}
\end{pmatrix},$

$R_0^{00} = R_1^{01} = R_2^{02} = R_3^{03} = R_4^{04} = R_1^{10} = R_2^{20} = R_3^{30} = R_4^{40} = R_0^{44} = 1,$
$R_0^{11} = e^{3i\pi/4},$
$R_2^{11} = e^{i \pi/12},$
$R_1^{12} = R_1^{21} = R_2^{22} = R_3^{23} = R_3^{32} = e^{2i\pi/3},$

$R_3^{12} = R_3^{21} = e^{i\pi/6},$
$R_2^{13} = R_2^{31} = e^{7i\pi/12},$
$R_4^{13} = R_4^{31} = e^{i\pi/4},$
$R_3^{14} = R_3^{41} = i,$
$R_0^{22} = e^{-2i\pi/3},$

$R_4^{22} = e^{i\pi/3},$
$R_1^{23} = R_1^{32} = e^{-5i\pi/6},$
$R_2^{24} = R_2^{42} = -1,$
$R_0^{33} = e^{-i\pi/4},$
$R_2^{33} = e^{-11i\pi/12},$
$R_1^{34} = R_1^{43} = -i$.

[scale = 0.7]
[blue, ultra thick] (0,0) – (0,1);
[blue, ultra thick] (0,1) – (3,4);
[blue, ultra thick] (0,1) – (-3,4);
[blue, ultra thick] (-2,3) –(-1,4);
[blue, ultra thick] (2,3) –(1,4);
[above] at (-3,4) a;
[above] at (-1,4) b;
[above] at (1,4) c;
[above] at (3,4) d;
[below] at (0,0) (ab)(cd);
[below] at (-2,2.5) (ab);
[below] at (2,2.5) (cd);
[scale = 0.7]
[blue, ultra thick] (0,0) – (0,1);
[blue, ultra thick] (0,1) – (3,4);
[blue, ultra thick] (0,1) – (-3,4);
[blue, ultra thick] (0,3) –(-1,4);
[blue, ultra thick] (-1,2) –(1,4);
[above] at (-3,4) a;
[above] at (-1,4) b;
[above] at (1,4) c;
[above] at (3,4) d;
[below] at (0,0) (a(bc)d);
[below] at (0.2,2.8) (bc);
[below] at (-1.3,1.7) (a(bc));

[ultra thick, blue] (0,0) – (0,1);
[ultra thick, blue] (0,0) – (1,-0.6);
[ultra thick, blue] (0,0) – (-1,-0.6);
[ultra thick, blue] (-1,-0.6) – (1,-0.6);
[ultra thick, blue] (-1,-0.6) – (0,1);
[ultra thick, blue] (1,-0.6) – (0,1);
[left] at (0,0.4) $a$;
[] at (-0.4,-0.4) $b$;
[] at (0.4,-0.4) $e$;
[below] at (0,-0.6) $c$;
[right] at (0.4,0.4) $d$;
[left] at (-0.4,0.4) $f$;

CHAPTER: TERNARY LOGIC DESIGN WITH METAPLECTIC ANYONS

Quantum computation is performed with metaplectic anyons which are simple objects in weakly integral categories. The term metaplectic is for a braid group that is in the metaplectic representation. These representations are the symplectic analog of spinor representation. [228, 229].

The metaplectic anyons can be studied from the category theory. Anyons are simple objects in a unitary modular category. See Chapter <ref> for the introduction to category theory and the use of category theory in topological quantum computation. A category is integral when the quantum dimension or Frobenius-Perron dimension of a simple object is an integer, whereas a category is called weakly integral if the squares of the quantum dimensions of all the simple objects are integer. Weakly integral categories are a class of metaplectic categories [203, 230].
There are five anyons $\left\{1,Z,X,X^{'},Y\right\}$ in the theory of metaplectic anyons with fusion rules, quantum dimensions, and topological twists given as
X ⊗X    = 1 +Y,

Y ⊗Y    = 1 + Z + Y,

X ⊗Z    = X',

X ⊗X'    = Z+Y,
d_1 = d_Z = 1,  d_X = d_X'= √(3),  d_Y = 2,
θ_0 = θ_4 = 1,  θ_1 = θ_3 = e^iπ/4,  θ_2 = e^i2π/3.

There is a non-Abelian boson quasiparticle $Z$. These theories also have a fundamental particle $X$. This particle is also a non-Abelian. It is a vortex for the $Z$ boson. This $X$ particle is fused with another $X$ particle to give $Y_i$ or vacuum, where $i = 1,2,...,r$ and $r= (m-1)/2$. These non-Abelian particles $Y_i$ have the quantum dimension $2$. When $X$ and $Z$ are fused, the result is the particle $X'$ [231, 230].
A collection of $N$ quasiparticles $X$ at a fixed position has an $n_N$-dimensional degenerate subspace with $n_N \sim m^{N/2}$.
The proposed metaplectic anyon systems are the quantum Hall effect and Majorana zero modes [172, 173, 174, 175, 176].

§ ONE-QUTRIT BRAIDING GATES

Let us consider four $X$ anyons. The first two of the four are fused to $c_{12}$ and the last two are fused to $c_{34}$ as shown in Fig. <ref> (a).
With the constraint $c_{14} = Y$, we get three fusion trees [228, 229]
(c_12,c_34) ∈{-(YY),(1Y),(Y1)}.
These are corresponding to the three states of qutrit $\ket{0},\ket{1},\ket{2}$. The minus sign is just to make the algebra nicer later.
Let $\sigma_1$ be a braid matrix for the first two particles, and $\sigma_2$ corresponds to a braid of the second with the third, and $\sigma_3$ is a braid matrix for the third and fourth as shown in Fig. <ref> and <ref>.
The associated Hilbert space is represented by $V_y^{\epsilon \epsilon \epsilon \epsilon}$, for $\epsilon=X$. Under the basis $\left\{-\ket{YY},\ket{\bm{1}Y}, \ket{Y\bm{1}}\right\}$, the generators of the braid group ${\mathcal{B}}_4$ for the representation $V_y^{\epsilon \epsilon \epsilon \epsilon}$ are
σ_1 = γ
1     0     0

0     ω    0

0     0     1
,     σ_3 = γ
1     0     0

0     1     0

0     0     ω,
σ_2    = γ^3/√(3)
1     ω    ω

ω    1     ω

ω    ω    1
= γ
1/2 + √(3)i/6     -1/2 + √(3)i/6     -1/2 + √(3)i/6

-1/2 + √(3)i/6     1/2 + √(3)i/6     -1/2 + √(3)i/6

-1/2 + √(3)i/6     -1/2 + √(3)i/6     1/2 + √(3)i/6
where $\omega = e^{2 \pi i/3}$ and $\gamma = e^{\pi i/12}$.
Ignoring the $\gamma$ in front, let us define [228]
$p = \sigma_1 \sigma_2 \sigma_1$, $q = \sigma_2 \sigma_3 \sigma_2$,
p^2 = -
0     1     0

1     0     0

0     0     1
,     q^2 = -
0     0     1

0     1     0

1     0     0
,     -(q^2pq^2)^2 =
1     0     0

0     0     1

0     1     0

(q^2pq^2)^2Z^*((q^2pq^2)^2)^* =
0     0     1

1     0     0

0     1     0
,     (q^2pq^2)^2Z((q^2pq^2)^2)^* =
0     1     0

0     0     1

1     0     0
These gates correspond to one-qutrit gates $Z_3(+1)$, $Z_3(+2)$, $Z_3(01)$, $Z_3(12)$, $Z_3(02)$ in conventional quantum computing discussed in Chapter <ref>. The phase gate $Z=\sigma_1\sigma_3^{-1}=\sigma_1\sigma_3^2$ and the ternary Hadamard gate $H=q^2pq^2$ can be written in matrix form as
Z =
1     0     0

0     ω    0

0     0     ω^2
, 	H = 1/√(3)i
1     1     1

1     ω    ω^2

1     ω^2     ω.

§ TWO-QUTRIT BRAIDING GATES

The two-qutrit model would consist of eight $X$ anyons with the final fusion outcome $Y$ as shown in Fig. <ref> (b).
The braid matrices for two qutrits are written as $\sigma_1,\sigma_2,\sigma_3,\sigma_4,\sigma_5,\sigma_6,\sigma_7$.
Let us define
s_1 = σ_2σ_1σ_3σ_2,  s_2 = σ_4σ_3σ_5σ_4,  s_3 = σ_6σ_5σ_7σ_6.
From these matrices, we can calculate a matrix
Λ(Z) = s_1^-1s_2^2 s_1 s_3^-1s_2^2s_3.
The two-qutrit encoding is obtained when restricting the vector space $V_y^{\epsilon \epsilon \epsilon \epsilon \epsilon \epsilon \epsilon \epsilon}$ to nine dimensional subspace $V_y^{\epsilon \epsilon \epsilon \epsilon}\otimes V_y^{\epsilon \epsilon \epsilon \epsilon} \subset V_y^{\epsilon \epsilon \epsilon \epsilon \epsilon \epsilon \epsilon \epsilon}$
with $c_{14}=c_{58}= Y$. This nine-dimensional restriction of the $\Lambda (Z)$ is the Controlled-Z gate (CZ).
The SUM gate is a generalization of the CNOT gate [228, 229]. It is related to CZ as
SUM = (I⊗H)CZ(I ⊗H^-1).
This $SUM$ gate will be combined with the topological charge measurement to build arithmetic circuits in the next section.

[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (1,3) – (2,4);
[blue, ultra thick] (5,3) –(4,4);
[below left] at (1.5,2.5) $c_{12}$;
[below right] at (4.5,2.5) $c_{34}$;
[below] at (3,0) $c_{14}$;
[above] at (0,4) X;
[above] at (2,4) X;
[above] at (4,4) X;
[above] at (6,4) X;

[scale = 0.3]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[above] at (0,8) X;
[above] at (2,8) X;
[above] at (4,8) X;
[above] at (6,8) X;
[above] at (8,8) X;
[above] at (10,8) X;
[above] at (12,8) X;
[above] at (14,8) X;
[below left] at (2,6) $c_{12}$;
[below right] at (4,6) $c_{34}$;
[below left] at (10,6) $c_{56}$;
[below right] at (12,6) $c_{78}$;
[below left] at (5,3) $c_{14}$;
[below right] at (9,3) $c_{58}$;
[below] at (7,0) Y;

[One-qutrit and two-qutrit gates.](a) One-qutrit gate (b) Two-qutrit gate [229].

[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (1,3) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,4) $a$;
[above] at (4,4) $a$;
[above] at (6,4) $a$;
[below left] at (2,2.5) $x_i$;
[below right] at (4,2.5) $y_i$;

[scale = 0.3]
[knot=blue, ultra thick] (2,4) –(0,6);
[knot=blue, ultra thick] (0,4) – (2,6);
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (1,3) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) $b$;
[above] at (2,6) $a$;
[above] at (0,6) $a$;
[above] at (4,4) $a$;
[above] at (6,4) $a$;
[below left] at (2,2.5) $x_i$;
[below right] at (4,2.5) $y_i$;

[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (1,3) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,4) $a$;
[above] at (4,4) $a$;
[above] at (6,4) $a$;
[below left] at (2,2.5) $x_i$;
[below right] at (4,2.5) $y_i$;

[scale = 0.3]
[knot=blue, ultra thick] (6,4) –(4,6);
[knot=blue, ultra thick] (4,4) – (6,6);
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (1,3) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) $b$;
[above] at (2,4) $a$;
[above] at (0,4) $a$;
[above] at (4,6) $a$;
[above] at (6,6) $a$;
[below left] at (2,2.5) $x_i$;
[below right] at (4,2.5) $y_i$;

[One-qutrit braid matrices $\sigma_1$ and $\sigma_3$.]One-qutrit braid matrices (a) $\sigma_1$ and (b) $\sigma_3$.

[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (1,3) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,4) $a$;
[above] at (4,4) $a$;
[above] at (6,4) $a$;
[below left] at (2,2.5) $x_i$;
[below right] at (4,2.5) $y_i$;

[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (4,2) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,4) $a$;
[above] at (4,4) $a$;
[above] at (6,4) $a$;
[below right] at (3,1.5) $c$;
[below right] at (4,2.5) $y_i$;

[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (4,2) –(2,4);
[blue, ultra thick] (3,3) –(4,4);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,4) $a$;
[above] at (4,4) $a$;
[above] at (6,4) $a$;
[below right] at (3,1.5) $c$;
[below left] at (3.45,3.3) $d$;

[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (4,2) –(2,4);
[blue, ultra thick] (3,3) –(4,4);
[blue, ultra thick] (2,4) –(4,6);
[blue, ultra thick] (4,4) –(3.2,4.8);
[blue, ultra thick] (2.8,5.2) –(2,6);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,6) $a$;
[above] at (4,6) $a$;
[above] at (6,4) $a$;
[below right] at (3,1.5) $c$;
[below left] at (3.45,3.3) $d$;

$=\sum_{cde}[F^{aay_i}_b]_{x_ic}[F^{aaa}_c]^{-1}_{dy_i}R^{aa}_d [F^{aaa}_c]_{de}$
[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (4,2) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[blue, ultra thick] (2,4) –(4,6);
[blue, ultra thick] (4,4) –(3.2,4.8);
[blue, ultra thick] (2.8,5.2) –(2,6);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,6) $a$;
[above] at (4,6) $a$;
[above] at (6,4) $a$;
[below right] at (4.5,3) $e$;
[below right] at (3,1.5) $c$;

$=\sum_{cdef}[F^{aay_i}_b]_{x_ic}[F^{aaa}_c]^{-1}_{dy_i}R^{aa}_d [F^{aaa}_c]_{de}
[scale = 0.3]
[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (1,3) –(2,4);
[blue, ultra thick] (5,3) –(4,4);
[blue, ultra thick] (2,4) –(4,6);
[blue, ultra thick] (4,4) –(3.2,4.8);
[blue, ultra thick] (2.8,5.2) –(2,6);
[below] at (3,0) $b$;
[above] at (0,4) $a$;
[above] at (2,6) $a$;
[above] at (4,6) $a$;
[above] at (6,4) $a$;
[below right] at (4,2.5) $e$;
[below left] at (2,2.5) $f$;

[One-qutrit braid matrix $\sigma_2$.]One-qutrit braid matrix $\sigma_2$ [228].

[scale = 0.2]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (2,8) –(1.2,8.8);
[blue, ultra thick] (0.8,9.2) –(0,10);
[blue, ultra thick] (0,8) –(2,10);
[above] at (0,10) $a$;
[above] at (2,10) $a$;
[above] at (4,8) $a$;
[above] at (6,8) $a$;
[above] at (8,8) $a$;
[above] at (10,8) $a$;
[above] at (12,8) $a$;
[above] at (14,8) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

[scale = 0.2]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (4,8) –(3.2,8.8);
[blue, ultra thick] (2.8,9.2) –(2,10);
[blue, ultra thick] (2,8) –(4,10);
[above] at (0,8) $a$;
[above] at (2,10) $a$;
[above] at (4,10) $a$;
[above] at (6,8) $a$;
[above] at (8,8) $a$;
[above] at (10,8) $a$;
[above] at (12,8) $a$;
[above] at (14,8) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

[scale = 0.2]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (6,8) –(5.2,8.8);
[blue, ultra thick] (4.8,9.2) –(4,10);
[blue, ultra thick] (4,8) –(6,10);
[above] at (0,8) $a$;
[above] at (2,8) $a$;
[above] at (4,10) $a$;
[above] at (6,10) $a$;
[above] at (8,8) $a$;
[above] at (10,8) $a$;
[above] at (12,8) $a$;
[above] at (14,8) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

[scale = 0.2]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (8,8) –(7.2,8.8);
[blue, ultra thick] (6.8,9.2) –(6,10);
[blue, ultra thick] (6,8) –(8,10);
[above] at (0,8) $a$;
[above] at (2,8) $a$;
[above] at (4,8) $a$;
[above] at (6,10) $a$;
[above] at (8,10) $a$;
[above] at (10,8) $a$;
[above] at (12,8) $a$;
[above] at (14,8) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

[scale = 0.2]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (10,8) –(9.2,8.8);
[blue, ultra thick] (8.8,9.2) –(8,10);
[blue, ultra thick] (8,8) –(10,10);
[above] at (0,8) $a$;
[above] at (2,8) $a$;
[above] at (4,8) $a$;
[above] at (6,8) $a$;
[above] at (8,10) $a$;
[above] at (10,10) $a$;
[above] at (12,8) $a$;
[above] at (14,8) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

[scale = 0.2]

[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (12,8) –(11.2,8.8);
[blue, ultra thick] (10.8,9.2) –(10,10);
[blue, ultra thick] (10,8) –(12,10);
[above] at (0,8) $a$;
[above] at (2,8) $a$;
[above] at (4,8) $a$;
[above] at (6,8) $a$;
[above] at (8,8) $a$;
[above] at (10,10) $a$;
[above] at (12,10) $a$;
[above] at (14,8) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

[scale = 0.2]
[blue, ultra thick] (7,0) – (7,1);
[blue, ultra thick] (7,1) – (0,8);
[blue, ultra thick] (7,1) – (14,8);
[blue, ultra thick] (3,5) –(6,8);
[blue, ultra thick] (11,5) –(8,8);
[blue, ultra thick] (1,7) –(2,8);
[blue, ultra thick] (5,7) –(4,8);
[blue, ultra thick] (9,7) –(10,8);
[blue, ultra thick] (13,7) –(12,8);
[blue, ultra thick] (14,8) –(13.2,8.8);
[blue, ultra thick] (12.8,9.2) –(12,10);
[blue, ultra thick] (12,8) –(14,10);
[above] at (0,8) $a$;
[above] at (2,8) $a$;
[above] at (4,8) $a$;
[above] at (6,8) $a$;
[above] at (8,8) $a$;
[above] at (10,8) $a$;
[above] at (12,10) $a$;
[above] at (14,10) $a$;
[below left] at (2,7) $a_1$;
[below right] at (4.5,7) $b_1$;
[below left] at (10,7) $a_2$;
[below right] at (12.5,7) $b_2$;
[below left] at (5,4) $c_1$;
[below right] at (9.5,4) $c_2$;
[below] at (7,0) 2;

Two-qutrit braid matrices.

[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (1,3) – (2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) 0/2/4;
[above] at (0,4) 2;
[above] at (2,4) 2;
[above] at (4,4) 2;
[above] at (6,4) 2;

$= (-1)$

[blue, ultra thick] (3,0) – (3,1);
[blue, ultra thick] (3,1) – (0,4);
[blue, ultra thick] (3,1) – (6,4);
[blue, ultra thick] (1,3) – (2,4);
[blue, ultra thick] (5,3) –(4,4);
[below] at (3,0) 0/2/4;
[above] at (0,4) 2;
[above] at (2,4) 2;
[above] at (4,4) 2;
[above] at (6,4) 2;

§ TERNARY ARITHMETIC CIRCUITS

The most important challenge in circuit design is reducing the number of gates. The more the gates, the harder it is to implement the circuit.
The MS gates in conventional quantum computing are designed by keeping the controlling value 2. Since, in topological quantum computation, any anyon can be braided to another anyon at any stage of the implementation, we can have a controlling value of $0,1,2$. Therefore, we can create a more general methodology of designing topological circuits presented here. We redesigned the qutrit arithmetic circuits that can be implemented with one-qutrit and two-qutrit gates made by metaplectic anyons [232] described in the last section. The universal set of gates cannot be made by braiding alone, it is to be combined with the topological charge measurement [228].

The gates that can be obtained by braiding alone are the Clifford gates, whereas the non-Clifford gates cannot be implemented by braiding alone. In this work, we have the Clifford gate $SUM$ that can be implemented by braiding alone, whereas $C_c(X)$ is a non-Clifford gate that is implemented by the measurement of the topological charge.
As in <ref>, one-qutrit ternary gates are represented as $Z_3(+1)$, $Z_3(+2)$, $Z_3(01)$, $Z_3(12)$, and $Z_3(02)$, where the first two are increment gates and the last three are permutation gates.
The non-Clifford gate, that is $C_c(X)$, applies $X$ when the controlling value is $c=0,1,2$, where $X$ can be a permutation or increment gate. The gates $C_c(X)$ are $9\cross 9$ matrices can be written as $diag(I_3,I_3,X)$ for control $\ket{2}$, $diag(I_3,X,I_3)$ for control $\ket{1}$, and $diag(X,I_3,I_3)$ for control $\ket{0}$, where $I_3$ is $3\cross 3$ identity matrix. In Ref. [232], the $SUM$ gates are called soft-controlled whereas the $C_c(U)$ are called hard-controlled gates.

The measurement can be projective or based on interference [228, 11, 26].
In case of interferometric measurement, a probe charge is sent through the paths around some region. The interference between different paths is related to the total topological charge in that region. The charge of the region can be found by measuring the charge of the probe. This kind of measurement can distinguish charge of fusion channel from an overall collection. This is non-demolition measurement but the fusion channels evolve in non-universal manner.
Local measurement for topological charge of a single quasiparticle is performed by bringing two charges close to each other and find their charge from their fusion outcome.
Let the measurement $\mathcal{M}_1 = \left\{\Pi_1,\Pi'_1\right\}$ correspond to the topological charge measurement of the first pair of anyons spanned by $\ket{\textbf{1}Y}$ and its orthogonal complements $\ket{-YY},\ket{Y\textbf{1}}$. The topological charge of the first pair of anyons is found by this measurement. If it is $\textbf{1}$ or $Y$ then the second pair is still in a coherent superposition of $\textbf{1}$ and $Y$. This measurement allows us to find whether an anyon is trivial or not.
For the hard-controlled gates $C_c(U)$, braiding gate is applied only when the controlling value of topological charge is 'c', the one mentioned on the gate, otherwise go to previous step or start over. The process is repeated several times until we get the required result.
The braiding supplemented with the projective measurement provides the universal set of gates for anyonic quantum computation [228].

As discussed in previous section, the Clifford gate $SUM = (I\otimes H)\Lambda (Z)(I \otimes H^{-1})$ in Eq. <ref> is a generalization of the CNOT gate. It can also be written as $SUM= \ket{0}\bra{0} \otimes I + \ket{1}\bra{1} \otimes X + \ket{2}\bra{2} \otimes X^2$. Here, $X$ is an increment gate $Z_3(+1)$ or $Z_3(+2)$. The two-qutrit $9\cross 9$ matrix for the $SUM$ is written as $diag(I_3, X, X^2)$. Let us call this gate $SUM_1$. But if we use $\Lambda (Z^{-1})$ in Eq. <ref>, we get the matrix form as $diag(I_3, X^2, X)$. Let us represent this form as $SUM_2$. The braiding implementation of $SUM_1$ and $SUM_2$ is equivalent. The $SUM_1$ and $SUM_2$ are used for designing the two-qutrit braiding gates $Z_3(+1)$ and $Z_3(+2)$. These gates are shown in Fig. <ref> and their matrices are shown in Eq. <ref>. We can also note that the matrix $diag(I_3, X^2, X)$ is the square of the matrix $diag(I_3, X, X^2)$. When control is the second qutrit and target is the first qutrit, then the $SUM$ gates are written in the form $SUM=I \otimes \ket{0}\bra{0} + X \otimes \ket{1}\bra{1} + X^2 \otimes \ket{2}\bra{2}= (H\otimes I)\Lambda (Z)(H^{-1} \otimes I)$.

Two qutrits can be swapped by the $SWAP$ gate, as discussed in <ref>. This gate can be formed by braiding alone [229] with the use of the permutation gates. One of the realizations of the $SWAP$ gate $SWAP: \ket{i,j} \to \ket{j,i}$ is obtained by using the gate $Z_3(12)$ [229] and can be written as
SWAP = (Z_3(12) ⊗I)SUM_1,2SUM_2,1SUM_2,1SUM_1,2,
where $SUM_{j,k}$ is a two-qutrit $SUM$ gate applied to $k$th qutrit when $j$th qutrit is the control qutrit. Two other non-Clifford gates used in this paper are Honer gate and controlled-SUM gate given as
Horner = Λ(Λ(X)): |i,j,k⟩ →|i,j,ij+k⟩

C(SUM) = C_c(SUM): |i,j,k⟩ →|i,j,jδ_i,c+k⟩
These gates are generalization of Toffoli gate [232, 233].

In Fig. <ref>, the non-Clifford gates are represented by the filled circles at the controlling values with $c=0,1,2$, whereas for the $SUM$ gates, the hollow circles are drawn at the controlling values. To avoid cluttering, labels of the control is omitted when its value is 2.
The increment gates will be represented by $+1$ and $+2$ and the permutation gates will be represented by $01$, $12$, and $02$. The gates in the boxes are related to the sum or product column of truth tables. Blue and orange colors of gates correspond to Clifford gates and non-Clifford gates respectively. A circuit is read from left to right but when it is written as matrices it is read from right to left, the same as the matrix multiplication.

[ultra thick,cyan] (0,0)–(0,1);

[ultra thick,cyan] (0,0)–(0,1);

[ultra thick,cyan] (0,0)–(0,1);

[ultra thick,cyan] (0,0)–(0,2);

[ultra thick,cyan] (0,0)–(0,1);
[ultra thick,cyan] (1.3,0)–(1.3,1);
[ultra thick,cyan] (2.6,0)–(2.6,1);
[ultra thick,cyan] (3.9,0)–(3.9,1);
[ultra thick,cyan] (5.2,0)–(5.2,1);

[Graphical representation of two-qutrit ternary gates.]The graphical representation of two-qutrit ternary gates [232].

1     0     0

0     1     0

0     0     1

0     0     1

1     0     0

0     1     0

0     1     0

0     0     1

1     0     0,  SUM_2=
1     0     0

0     1     0

0     0     1

0     1     0

0     0     1

1     0     0

0     0     1

1     0     0

0     1     0

Λ(Z) = 1     0     0     0     0     0     0     0     0
0     1     0     0     0     0     0     0     0
0     0     1     0     0     0     0     0     0
0     0     0     1     0     0     0     0     0
0     0     0     0     1     0     0     0     0
0     0     0     0     0     1     0     0     0
0     0     0     0     0     0     1     0     0
0     0     0     0     0     0     0     e^- 2 i π/3     0
0     0     0     0     0     0     0     0     e^2 i π/3,     SUM=
1     0     0     0     0     0     0     0     0

0     1     0     0     0     0     0     0     0

0     0     1     0     0     0     0     0     0

0     0     0     0     0     1     0     0     0

0     0     0     1     0     0     0     0     0

0     0     0     0     1     0     0     0     0

0     0     0     0     0     0     0     0     1

0     0     0     0     0     0     1     0     0

0     0     0     0     0     0     0     1     0

§.§ Ternary Adder

The adder circuit is the most important arithmetic circuit used in almost all circuits, especially in algorithms such as Grover, Shor, and HHL algorithms.
Binary adder circuits are proposed by  Ref. [234, 235, 236] and their ternary counterparts are given in Ref. [47, 237, 56].
The adder circuit of Ref. [29] consists of 14 MS and shift gates and the circuit from Ref. [238] obtained using the genetic algorithm, has 13 MS and shift gates. In Ref. [55], half adder is designed by 5 MS gates and one shift gate. An output that remains unused and thrown out is called the garbage output. Most of these designs used the Toffoli gate for their implementation, but the Toffoli gate cannot be built by braiding alone [228]. Our circuit design for half adder consist of four gates. Only one braiding gate is used to implement the sum and three non-Clifford gates are used for the implementation of carry. The constant inputs and the garbage outputs are the same for our designs as in the existing designs.

When we add two one-digit numbers, then we get the half adder, whereas the full adder circuit adds three one-digit numbers. The third digit can be a carry from the previous half adder.
The truth table for a half adder is shown in Table <ref> and the circuit realization is shown in Fig. <ref>. Let us discuss the cases when there is a nonzero carry. For example, the case when the input $A$ has value 1 and input $B$ has value 2. For the first gate, the control value is not 2 so it would not be applied. For the second gate, control value is 2 but third input is zero. The second gate will also remain ineffective. Third gate will be applied and it will give the carry 1. At the fourth gate, withing the box, $S$ will be zero as $B=2$ will add 2 to $A=1$.

When $A=2$ and $B=2$, the first gate will change the third qutrit from 0 to 2. At the second gate, third qutrit will be changed to 1. The third gate will not be applied. The fourth gate will add 2 to $A=2$ and give the sum $S=1$. The garbage bits at the end of the computation will be ignored.

[fill=red!20,opacity=0.5] (4.6,0.6) rectangle (5.8,2.6);
[ultra thick,cyan] (1.3,0)–(1.3,2);
[ultra thick,cyan] (2.6,0)–(2.6,1);
[ultra thick,cyan] (3.9,0)–(3.9,2);
[ultra thick,cyan] (5.2,1)–(5.2,2);

figureTernary half adder circuit realization.

$A$     $B$     $S$     $c_{out}$

0     0     0     0

0     1     1     0

0     2     2     0

1     0     1     0

1     1     2     0

1     2     0     1

2     0     2     0

2     1     0     1

2     2     1     1

tableTruth table of the ternary half adder.

The full adder adds three qutrits $A$, $B$, $C$ as shown in Fig. <ref>. The truth table for the ternary full adder is given in Table <ref>. The sum of $A$ and $B$ is obtained that is added to the third input qutrit $C$ to get the output $\bm{S}$. The input $C$ can be a carry from the previous sum of two qutrits. The garbage outputs $g_1$ and $g_2$ are ignored.

The addition of two-qutrit numbers and its circuit realization are shown in Fig. <ref> (a), (b). A half adder and a full adder can be used.
The first qutrit $A_0$ of $A_0B_0$ is added by the first half adder and the first digit of the sum $\bm{S}_0$ and $g_1$ are obtained. Their carry $c_0$ is to be added with the sum of the second qutrits $A_1$ and $B_1$. This $c_0$ corresponds to the input $C$ of the full adder. The addition of $A_1+B_1 +c_0$ gives the second digits of the output as $\bm{S}_1$ and a carry $\bm{c}_{out}$. The $SWAP$ gate in Eq. <ref> is used to exchange the qutrits $S$ and $c_0$, and the garbage qutrits are thrown out.

[fill=red!20,opacity=0.5] (3.3,1.7) rectangle (4.5,3.6);
[fill=red!20,opacity=0.5] (8.5,0.7) rectangle (9.7,3.6);
[ultra thick,cyan] (0,0)–(0,3);
[ultra thick,cyan] (1.3,0)–(1.3,2);
[ultra thick,cyan] (2.6,0)–(2.6,3);
[ultra thick,cyan] (3.9,2)–(3.9,3);
[ultra thick,cyan] (5.2,0)–(5.2,3);
[ultra thick,cyan] (6.5,0)–(6.5,3);
[ultra thick,cyan] (7.8,0)–(7.8,3);
[ultra thick,cyan] (9.1,1)–(9.1,3);

figure[Ternary full adder circuit realization.]Ternary full adder circuit realization.

$A$     $B$     $C$     $S$     $c_{out}$

0     0     0     0     0

0     0     1     1     0

0     0     2     2     0

0     1     0     1     0

0     1     1     2     0

0     1     2     0     1

0     2     0     2     0

0     2     1     0     1

0     2     2     1     1

1     0     0     1     0

1     0     1     2     0

1     0     2     0     1

1     1     0     2     0

1     1     1     0     1

1     1     2     1     1

1     2     0     0     1

1     2     1     1     1

1     2     2     2     1

2     0     0     2     0

2     0     1     0     1

2     0     2     1     1

2     1     0     0     1

2     1     1     1     1

2     1     2     2     1

2     2     0     1     1

2     2     1     2     1

2     2     2     0     2

tableTruth table for ternary full adder.

$A_1$   $A_0$

$+$   $B_1$   $B_0$

$c_{out}$   $S_1$   $S_0$

[fill=red!20,opacity=0.5] (3.3,4.7) rectangle (4.5,6.6);
[ultra thick,cyan] (0,4)–(0,6);
[ultra thick,cyan] (1.3,4)–(1.3,5);
[ultra thick,cyan] (2.6,4)–(2.6,6);

[ultra thick,cyan] (3.9,5)–(3.9,6);

[fill=red!20,opacity=0.5] (3.3,1.7) rectangle (4.5,3.6);
[fill=red!20,opacity=0.5] (9.8,0.7) rectangle (11,3.6);

[ultra thick,cyan] (0,0)–(0,3);
[ultra thick,cyan] (1.3,0)–(1.3,2);
[ultra thick,cyan] (2.6,0)–(2.6,3);

[ultra thick,cyan] (3.9,2)–(3.9,3);
[ultra thick,cyan] (6.5,0)–(6.5,3);
[ultra thick,cyan] (7.8,0)–(7.8,3);
[ultra thick,cyan] (9.1,0)–(9.1,3);
[ultra thick,cyan] (10.4,1)–(10.4,3);
[ultra thick, red] (4.5,4)–(5.2,4)–(5.2,1)–(5.9,1);

[Ternary two-qutrit addition and its circuit realization.]Ternary two-qutrit (a) addition and (b) circuit realization by using one half adder and one full adder.

§.§ Bocharov's Adder for Multiqutrit

[ibmqxD]11-90  $AdjC_0$
[ibmqxD]1-3-90  $AdjC$
[ibmqxC]3-3-90    Merger
[ibmqxD]41-90  $AdjC_0^{-1}$
[ibmqxD]4-3-90  $AdjC^{-1}$
[ultra thick,cyan] (6.5,0) – (6.5,2);
[ultra thick,cyan] (6.5,-3) – (6.5,-1);

Ternary multiqutrit adder

§.§ Ternary Subtractor

A ternary subtractor gives an output as a difference between two inputs. The subtractor circuit takes two inputs $A$ and $B$ and one ancilla. The difference between two inputs and the borrow is obtained at the output. For ternary subtractor in conventional quantum computing, see Ref. [56]. The half subtractor truth table is shown in Table <ref> and the circuit realization is shown in Fig. <ref>. Let us discuss the case when $A=1,B=2$. Since the controlling, value needs to be at 0, but we have $B=2$, the first $C(SUM)$ gate would not be applied. For the second gate, the controlling value is 2, therefore the MS permutation gate would be applied, but the value at the third qutrit is 0 so this gate will remain ineffective. The gate for the carry will give 1, because the controlling values are 2. The fourth gate changes $A$ from 1 to 2. Therefore, at the output, we get a difference $D=2$ and the borrow $b_{out} = 1$. This design of half subtractor circuit consists of only one braiding gate and three non-Clifford gates.

[fill=red!20,opacity=0.6] (3.3,0.5) rectangle (4.5,2.6);
[ultra thick,cyan] (0,0)–(0,2);
[ultra thick,cyan] (1.3,0)–(1.3,1);

[ultra thick,cyan] (2.6,0)–(2.6,2);
[ultra thick,cyan] (3.9,1)–(3.9,2);

figureTernary half subtractor circuit realization.

A     B     D     $b_{out}$

0     0     0     0

0     1     2     1

0     2     1     1

1     0     1     0

1     1     0     0

1     2     2     1

2     0     2     0

2     1     1     0

2     2     0     0

tableTruth table for ternary half subtractor.

When taking the difference of two numbers and a borrow is needed, then have a full subtractor. It has three inputs $A$, $B$, and $C$, where $C$ is the borrow-in. The truth table for full subtractor is shown in Table <ref> and the circuit realization is shown in Fig. <ref>.

[fill=red!20,opacity=0.5] (3.3,0.4) rectangle (5.8,3.6);
[ultra thick,cyan] (0,0)–(0,2);
[ultra thick,cyan] (1.3,0)–(1.3,3);
[ultra thick,cyan] (2.6,0)–(2.6,3);

[ultra thick,cyan] (3.9,2)–(3.9,3);

[ultra thick,cyan] (5.2,1)–(5.2,3);

[ultra thick,cyan] (6.5,0)–(6.5,3);
[ultra thick,cyan] (7.8,0)–(7.8,3);
[ultra thick,cyan] (9.1,0)–(9.1,3);

figureTernary full subtractor circuit realization.

A     B     C     D     $b_{out}$

0     0     0     0     0

0     0     1     2     1

0     0     2     1     1

0     1     0     2     1

0     1     1     1     1

0     1     2     0     1

0     2     0     1     1

0     2     1     0     1

0     2     2     2     2

1     0     0     1     0

1     0     1     0     0

1     0     2     2     1

1     1     0     0     0

1     1     1     2     1

1     1     2     1     1

1     2     0     2     1

1     2     1     1     1

1     2     2     0     1

2     0     0     2     0

2     0     1     1     0

2     0     2     0     0

2     1     0     1     0

2     1     1     0     0

2     1     2     2     1

2     2     0     0     0

2     2     1     2     1

2     2     2     1     1

tableTruth table for ternary full subtractor.

[ibmqxD]14-90  HALF ADDER
[ibmqxD]11-90  HALF ADDER
[ibmqxC]32-90  HALF ADDER

§.§ Ternary Multiplier

In a two-qutrit multiplier $A_0B_0 \cross A_1B_1$, each digit of the first number is multiplied by each digit of the second number. Then all the partial products are added in the way shown in Fig. <ref>. Therefore, we need ternary partial product generation (TPPG) circuits and adder circuits for the two-qutrit multiplier. This kind of multiplier is discussed in Ref. [239]. The outputs of the two numbers are $P_0$, $P_1$, $P_2$, and $P_3$ and the carry is $c_{out}$.

$A_1$   $A_0$

$\cross$   $B_1$   $B_0$

$c_2$   $cp_1$   $cp_0$

$c_1$   $cp_2$   $A_1B_0$   $A_0B_0$

+   $cp_3$   $A_1B_1$   $A_0B_1$   0

$c_{out}$   $P_3$   $P_2$   $P_1$   $P_0$

Two-qutrit multiplication.

The carries produced by addition are represented as $c_i$, whereas the $cp_i$ are the carries we get as a result of one-digit multiplication. The first digit of multiplication is $P_0=A_0B_0$ and its carry is represented by $cp_0$. To compute $P_1,P_2,P_3$ and $c_{out}$, adder circuits are needed. As each of the digits $A_0,B_0,A_1,B_1$ can have values $0,1,2$, not all the partial products produce carries in multiplication. We do not need extra input lines and gates corresponding to the carries, and therefore, it is cheaper to implement the circuit without full adders. Instead, we are using the Adder Blocks as in Panahi [239]. The one-digit TPPG circuit is shown in Fig. <ref> and the truth table is shown in Fig. <ref>. The carry-out appears only when both the input values are at value $2$ in the one-digit multiplication. The existing realization in Ref. [239] has 13 MS and shift gates.
P_0   =A_0B_0

P_1   =cp_0+A_1B_0+A_0B_1

P_2   =c_0+cp_1+cp_2+A_1B_1

P_3   =c_2+c_1+cp_3

[ultra thick,cyan] (1.3,0)–(1.3,3);
[ultra thick,cyan] (2.6,0)–(2.6,2);
[ultra thick,cyan] (3.9,1)–(3.9,2);
[ultra thick,cyan] (6.5,2)–(6.5,3);
[ultra thick,cyan] (7.8,1)–(7.8,2);
[ultra thick,cyan] (9.1,2)–(9.1,3);
[ultra thick,cyan] (13,1)–(13,3);
[ultra thick,cyan] (14.3,0)–(14.3,2);
[ultra thick,cyan] (15.6,0)–(15.6,3);

TPPG circuit realization in ref. [239].

[ultra thick] (3.2,0)–(13.6,0);

[ultra thick,cyan] (3.9,1)–(3.9,2);
[ultra thick,cyan] (5.2,2)–(5.2,3);

[ultra thick,cyan] (6.5,1)–(6.5,2);
[ultra thick,cyan] (7.8,2)–(7.8,3);

[ultra thick,cyan] (9.1,1)–(9.1,2);
[ultra thick,cyan] (10.4,0)–(10.4,1);
[ultra thick,cyan] (11.7,1)–(11.7,2);

[ultra thick,cyan] (13,1)–(13,3);

[fill=red!20,opacity=0.5] (2,0.4) rectangle (5.8,3.6);
[ultra thick,cyan] (2.6,1)–(2.6,2);

[ultra thick,cyan] (3.9,1)–(3.9,3);

[ultra thick,cyan] (5.2,1)–(5.2,3);

[ultra thick,cyan] (6.5,0)–(6.5,3);
[ultra thick] (1.95,0)–(6,0);

figureCircuit design of the TPPG component.

A     B     $P$     $cp$

0     0     0     0

1     0     0     0

2     0     0     0

0     1     0     0

1     1     1     0

2     1     2     0

0     2     0     0

1     2     2     0

2     2     1     1

tableTruth table for a two-qutrit partial product.

From the Fig. <ref>, we have $P_1=cp_0+A_1B_0+A_0B_1$. Adder Block 1 adds the partial products $A_1B_0,A_0B_1$ and the carry $cp_0$ of the partial products. As we can see from the truth table <ref> of the partial product, the carry in the first partial product is never 2. The input values can be $0$, $1$ and $2$ but carry is only $0$ or $1$. That is, the $cp_0$, $cp_1$, $cp_2$, and $cp_3$ are 1 and the carry is only for the case when inputs are at values 2. The truth table for the Adder Block 1 is shown in Table <ref>, and the implementation is shown in Fig. <ref>. The input lines with label $0$ are ancilla lines, and $g$ are the garbage outputs.
Adder Block 2 adds three qutrits $A_1B_1$, $cp_1$ and $cp_2$ to get $P_2$. The sum of these three qutrits will be added to $c_0$ using Block 4. The first input has values $0,1,2$, but the second and third inputs have values $0,1$. The partial product $A_1B_1$ is added to the two input carry numbers, which have values $0,1$. The truth table and implementation of Block 2 is shown in Table <ref> and Fig. <ref>.
Adder Block 3 adds qutrits $c_1$, $c_2$ and $cp_3$ to get $P_3$. The truth table is shown in Table <ref> and the implementation is shown in Fig. <ref>. All the inputs have values $0$ or $1$. The output carry appears only when all the inputs have values $1$.
Adder Block 4 is used to add the sum of Adder Block 2 and $c_0$. The half adder is used for its implementation. The output carry is only for one of the cases. The input carry $c_{in}$ is either $0$ or $1$ and input $A=0,1,2$. The truth table of Block 4 is shown in Table <ref> and its implementation is shown in Fig. <ref>. The Block 1 is designed with two Clifford and 4 non-Clifford gates, Block 2 has 2 Clifford and 2 non-Clifford gates, Block 3 consists of 2 Clifford and one non-Clifford gates, whereas Block 4 is implemented with only one Clifford gate and 1 non Clifford gate.

[fill=red!20,opacity=0.5] (4.6,0.5) rectangle (7.1,3.6);
[ultra thick,cyan] (1.3,0)–(1.3,2);
[ultra thick,cyan] (2.6,0)–(2.6,3);
[ultra thick,cyan] (3.9,0)–(3.9,3);
[ultra thick,cyan] (5.2,2)–(5.2,3);
[ultra thick,cyan] (6.5,1)–(6.5,3);

[ultra thick,cyan] (7.8,0)–(7.8,3);

figureImplementation for the ternary Block 1.

A     B     $c_{in}$     $Sum$     $c_{out}$

0     0     0     0     0

1     0     0     1     0

2     0     0     2     0

0     1     0     1     0

1     1     0     2     0

2     1     0     0     1

0     2     0     2     0

1     2     0     0     1

2     2     0     1     1

0     0     1     1     0

1     0     1     2     0

2     0     1     0     1

0     1     1     2     0

1     1     1     0     1

2     1     1     1     1

0     2     1     0     1

1     2     1     1     1

2     2     1     2     1

tableTruth table for ternary Adder Block 1.

[fill=red!20,opacity=0.5] (4.6,0.5) rectangle (7,3.7);
[ultra thick,cyan] (3.9,0)–(3.9,3);
[ultra thick,cyan] (5.2,2)–(5.2,3);

[ultra thick,cyan] (6.5,1)–(6.5,3);

[ultra thick,cyan] (7.8,0)–(7.8,3);

figureImplementation of Block 2.

A     B     $c_{in}$     $Sum$     $c_{out}$

0     0     0     0     0

1     0     0     1     0

2     0     0     2     0

0     1     0     1     0

1     1     0     2     0

2     1     0     0     1

0     0     1     1     0

1     0     1     2     0

2     0     1     0     1

0     1     1     2     0

1     1     1     0     1

2     1     1     1     1

tableTruth table for ternary Adder Block 2.

[fill=red!20,opacity=0.5] (0.6,0.5) rectangle (3.2,3.6);
[ultra thick,cyan] (1.3,2)–(1.3,3);

[ultra thick,cyan] (2.6,1)–(2.6,3);

[ultra thick,cyan] (3.9,0)–(3.9,3);

figureImplementation of ternary Adder Block 3.

A     B     $c_{in}$     $Sum$     $c_{out}$

0     0     0     0     0

1     0     0     1     0

0     1     0     1     0

1     1     0     2     0

0     0     1     1     0

1     0     1     2     0

0     1     1     2     0

1     1     1     0     1

tableTruth table for ternary Adder Block 3.

[fill=red!20,opacity=0.6] (2,0.5) rectangle (3.2,2.6);
[ultra thick,cyan] (1.3,0)–(1.3,2);
[ultra thick,cyan] (2.6,1)–(2.6,2);

figureImplementation of Block 4.

A     $c_{in}$     $Sum$     $c_{out}$

0     0     0     0

1     0     1     0

2     0     2     0

0     1     1     0

1     1     2     0

2     1     0     1

tableTruth table for ternary Adder Block 4.

Now we will combine the TPPG circuits and the Adder Blocks to get a full two-digit multiplier circuit. From the two-digit qutrit multiplication in Fig. <ref>, the first digits of two numbers give the first partial product and first multiplication digit $P_0$. The second multiplication digit is the addition of partial products $A_0B_1$ and $A_1B_0$ and added to the carry from the first partial product. These partial products also create the carries. The carries from these partial products and the carries of additions are added to the partial product $A_1B_1$. These give the third digit of multiplication. The additions generate two carry digits.

The full two-digit qutrit multiplication circuit is shown in Fig. <ref>. When the line goes on the top of a Block, then it is non-interacting, but when a line goes below the Block, then it is given to that Block as the input line.
In the Fig., the upper TPPG at stage 1 gives a partial product $P_0=A_0B_0$ and the carry $cp_0$.
The lower TPPG at the stage 1 computes $A_1B_1$, and we have the carry $cp_3$.
At stage 2, the first TPPG computes partial product $A_1B_0$ and produces the carry $cp_1$, whereas the second TPPG computes $A_0B_1$ and produces the carry $cp_2$. Inputs of the first TPPG are $B_0$ and $A_1$ and two ancillae, whereas the inputs of the second TPPG are $A_0$ and $B_1$.
The stages 3, 4 and 5 consist of Adder Blocks. As we can see in Fig. <ref>, we need to add $cp_0$, $A_0B_1$ and $A_1B_0$ to get $P_1$ and the carry $c_0$. This is done by Block 1 at stage 3. At this stage, the Adder Block 2 adds $cp_1,A_1B_1$ and $cp_2$. The output of this block is $s_1$ and the carry is $c_1$. At stage 4, the sum $s_1$ is added to $c_0$ by using the Block 4 and the output $P_2$ is obtained. The Block 4 has a carry $c_2$. At stage 5, $cp_3$, $c_1$ and $c_2$ are added using the Block 3 to get an output qutrit $P_3$ and the carry $c_{out}$.

[dashed,fill=cyan!10] (1.8,9.7) rectangle (3.35,10.5);
[dashed,fill=cyan!10] (4.45,9.7) rectangle (7.2,10.5);
[dashed,fill=cyan!10] (8.35,9.7) rectangle (11.1,10.5);
[dashed,fill=cyan!10] (12.25,9.7) rectangle (13.7,10.5);
[dashed,fill=cyan!10] (14.85,9.7) rectangle (16.3,10.5);
at (2.5,10.1) $1$;
at (5.8,10.1) $2$;
at (9.7,10.1) $3$;
at (12.95,10.1) $4$;
at (15.6,10.1) $5$;
[dashed,fill=cyan!10] (1.8,-0.5) rectangle (3.35,9.5);
[dashed,fill=cyan!10] (4.45,-2.5) rectangle (7.2,9.5);
[dashed,fill=cyan!10] (8.35,-4.5) rectangle (11.1,9.5);
[dashed,fill=cyan!10] (12.25,-5.5) rectangle (13.7,9.5);
[dashed,fill=cyan!10] (14.85,-6.5) rectangle (16.3,9.5);
[ibmqxC]7-1-90$\ \ Block \ 1$

[ibmqxC]8-2-90$\ \ Block \ 2$
[ibmqxF]10-3-90$\ Block \ 4$
[ibmqxB]12-5-90 $ Block \ 3$

[Ternary two-qutrit multiplier is implemented using the Adder Blocks and TPPG circuits.]Ternary two-qutrit multiplier is implemented using the Adder Blocks and TPPG circuits [239].

CHAPTER: CONCLUSION

Topological quantum computing is a promising candidate for fault-tolerant quantum computing. The main focus of this dissertation was to find the ternary arithmetic circuit implementation in topological quantum computation, and the fundamentals on which topological quantum computation is based. It is also explained how to build fault-tolerant gates by using the ideas based on quantum topology and braiding of anyons.

Topological quantum computation is a wide field of research. The areas of topology and knot theory, topological phases of matter, quantum field theory, and quantum computing are merged in this field. The modular categories and the quantum group are the algebraic mathematical models for the topological phases of matter. To make this dissertation self-contained, we discussed all of these areas and made their connections with quantum computing.

First, we discussed the quantum binary and ternary logic design in conventional quantum computing and how the quantum computer can work better in terms of complexity.
Topology and knot theory as a mathematical background were presented. The physical insight, behind the knots and braids, is explained through the geometric phases in quantum physics. These phases also explain the existence of the degenerate state in the topological phases of matter and the evolution of the system in these states.
There are several proposals for the physical systems for topological quantum computation. Among these, the non-Abelian anyons in the quantum Hall effect and Majorana fermions in topological superconductors are the most popular. We discussed the quantum Hall effect in detail and its similarities and differences to the other topological materials like the topological insulator. Superconductivity and the appearance of Majorana fermions in topological superconductors were explained. The Chern-Simons theory is used as an effective field theory to explain the charge-flux composites as the non-Abelian anyons in such systems and their connection to knots and braids.
The fusion and braiding matrices, topological spin, and Hilbert space are discussed in terms of topological quantum field theory and category theory.

Both binary and ternary logic gates design are presented. Theoretical models for a binary logic design are based on the Fibonacci and Ising anyons. For ternary logic design, metaplectic anyons are used. The basic ternary logic gates are reproduced. The topological data for metaplectic anyons are obtained by the quantum deformation of recoupling theory. The metaplectic anyons are defined in terms of the category theory that also provides their topological data. The fusion matrices $F$ and braiding matrices $R$ are discussed in category theory and quantum deformation of the recoupling theory.

Our arithmetic circuits are based on a combination of Clifford gates and non-Clifford gates. The quantum cost of the implementations of the ternary adder, ternary subtractor, and ternary multiplier circuits is smaller than the existing realization. We proposed the implementation of the arithmetic circuits by using the topological charge measurement gates $C_c(X)$ and the braiding gates $SUM$ and $SWAP$. We modified the existing realizations to have more controlled increment gates than the controlled permutation gates so that these circuits can be realized by braiding.

As a future plan, we will use some optimized schemes to minimize the number of gates, input lines, garbage outputs, and compute the topological complexity of the circuits.

§ ACHIEVEMENTS

* The comprehensive background of topological quantum computation is presented and the ternary logic design by braiding of the metaplectic anyons is explained.

* The quantum cost of the ternary adder, ternary subtractor, and ternary multiplier circuits is reduced.

* Since the ternary two-qutrit increment gates can be implemented by braiding, the equivalent circuits for the ternary arithmetic circuits are proposed in which there are more controlled increment gates than the controlled permutation gates.

* The ternary arithmetic circuits are designed with braiding of the metaplectic anyons combined with the topological charge measurement gates.

CHAPTER: ABSTRACT ALGEBRA

§ GROUP

The group theory is indispensable in physics for the study of symmetry. But group representation theory is of direct use in quantum physics and other branches of physics.

A set $G$ is a group when it fulfills the following axioms:

* Closure: For any two elements $A$ and $B$ of $G$, their composition $A\circ B$ is also in $G$. Here, $\circ$ is addition or multiplication.

* Associativity: The composition of any three elements $A,B,C$ $\in$ $G$, is associative. That is, $A\circ(B\circ C) = (A\circ B)\circ C$

* Identity: There exist an identity element $I$ $\in$ $G$, such that, for any element $A$ $\in$ $G$, $I\circ A = A \circ I = A$. The identity element is $0$ for a group under addition, and $1$ for a group under multiplication.

* Inverse: For any element $A$ $\in$ $G$, there exist a unique element $A^{-1}$ $\in$ $G$ such that $A\circ A^{-1} = A^{-1}\circ A = I$. The additive inverse is the negative of an element, whereas the multiplicative inverse is the reciprocal of an element of the group.

The matrix groups are of particular interest in physics. The set of all matrices form a group under addition. The null matrix is the additive identity. The additive inverse of a matrix $A$ is $-A$. The group is non-commutative when the order of the composition matters, that is, when $A\circ B \neq B \circ A$. The commutative group is called an Abelian group, whereas the non-commutative group is called the non-Abelian group. Matrices form a non-Abelian group under multiplication, as the matrix multiplication is not commutative in general.

An ordered pair is a pair of objects in which one element of the pair is designated as the first entry and the other element is designated as the second entry. It is denoted as $(a,b)$. The order of entries matters in the ordered pair, that is $(a,b)\ne (b, a)$. The Cartesian product of two sets is denoted as $A\cross B=\left\{(a,b)| a\in A \ \text{and} \ b\in B\right\}$. It is the set of all possible ordered pairs such that the first entry is from the set $A$ and the second entry is from the set $B$.