# Universal topological quantum computing via double-braiding in SU(2) Witten-
Chern-Simons theory

Adrian L. Kaufmann  West Lafayette Junior/Senior High School, West Lafayette,
IN 47906<EMAIL_ADDRESS>and  Shawn X. Cui  Department of
Mathematics, Department of Physics and Astronomy
Purdue University
West Lafayette, IN
47907
USA<EMAIL_ADDRESS>
(Date: December 2023)

###### Abstract.

We study the problem of universality in the anyon model described by the
$SU(2)$ Witten-Chern-Simons theory at level $k$. A classic theorem of
Freedman-Larsen-Wang states that for $k\geq 3,\ k\neq 4$, braiding of the
anyons of topological charge $1/2$ is universal for topological quantum
computing. For the case of one qubit, we prove a stronger result that double-
braiding of such anyons alone is already universal.

##  1\. Introduction

Topological quantum computing is an approach to building a fault-tolerant
quantum computer using some quasi-particles in two dimensions, called anyons.
The physical systems that host anyons are 2-dimensional topological phases of
matter. Topological phases are gapped quantum phases that go beyond Landau’s
theory of symmetry breaking and local order parameters; instead, they obey a
new order called topological quantum order [16]. Such phases exhibit several
remarkable properties including robust ground state degeneracy depending on
the topology of the underlying system, long-range entanglement, protected
gapless edge modes, fractionalized quasi-particle excitations, and exotic
exchange statistics. The robustness of the ground/excited state space provides
an ideal place to store quantum information as logical qubits. Braiding of
anyons, a process of adiabatically interchanging anyons, induces a unitary
transformation on the state space. These unitary transformations remain
unchanged under local deformations of the braiding world-lines, and hence
serve as logical quantum gates. This method of encoding and manipulating
information in global degrees of anyons is called topological quantum
computing [7, 10], and it has the advantage of achieving fault-tolerence at
the ‘hardware’ level.

An important family of topological phases are described by the Witten-Chern-
Simons theory associated with the Lie group $SU(2)$ and a level
$k\in\mathbb{Z}_{\geq 0}$ [19]. Denote the theory by $\textbf{SU}(2)_{k}$. It
is among the earliest-studied anyon models. By a classic result of [8], except
for a few values of $k$, the model is universal for topological quantum
computing, i.e., it is equivalent to the standard circuit model. Furthermore,
the theory has various potential realizations in fractional quantum Hall (FQH)
systems. For example, $\textbf{SU}(2)_{2}$ contains the Ising anyon expected
to exist in FQH with filling factor $\nu=5/2$, and $\textbf{SU}(2)_{3}$
contains the Fibonacci anyon expected to exist in $\nu=12/5$ FQH.

We elaborate a bit more on the universality of $\textbf{SU}(2)_{k}$. It
contains an anyon type, which we denote by $\tau$, with topological charge
$1/2$. By iteratively fusing $\tau$ with itself, any other anyon types in this
model can be produced. In this sense, $\tau$ is the most critical anyon type
in $\textbf{SU}(2)_{k}$. Denote by $V^{\tau^{\otimes 3}}_{\tau}$ the space of
three $\tau$ anyons with total type equal to $\tau$, or equivalently, the
space of four type-$\tau$ anyons (with total type trivial). This space has
dimension two, and hence is a qubit. Braiding of the $\tau$ anyons induces a
unitary representation of the braid group $B_{3}$ on the qubit. The theorem of
[8] states that, for all $k\geq 3$, $k\neq 4,\ 8$, this representation has a
dense image in $U(V^{\tau^{\otimes 3}}_{\tau})$, implying universal quantum
computing on one qubit by braiding. In fact, [8] shows that for any $n\geq 3$,
the image of the braid group representation is dense in $V^{\tau^{\otimes
n}}_{\tau}$ 111This result also holds for $k=8$, but $n$ needs to be at least
$5$.. These results set the theoretical foundation for universal topological
quantum computing.

In this paper, we focus on the qubit $V^{\tau\tau\tau}_{\tau}$ and study the
representation from braiding,

$\rho_{k}\colon B_{3}\to U(V^{\tau\tau\tau}_{\tau}).$

Recall that the braid group on $n$ strands $B_{n}$ has the standard generators
$\sigma_{1},\cdots,\sigma_{n-1}$ (see Sec. 2). $B_{n}$ acts on the space of
$n$ identical anyons of certain type where $\sigma_{i}$ corresponds to a
counterclockwise braiding the $i$-th and $(i+1)$-th anyon. We call
$\sigma_{i}^{2}$ a _double-braiding_ , and it corresponds to moving the $i$-th
anyon counterclockwise around the $(i+1)$-th anyon and returning to its
original position in the end. See Figure 1. More generally, any braid in the
group generated by the $\sigma_{i}^{2}\ {}^{\prime}$s is also called a double-
braiding. Specializing to the case of one qubit $V^{\tau\tau\tau}_{\tau}$ in
$\textbf{SU}(2)_{k}$ with $k\geq 3$, $k\neq 4,\ 8$, while [8] states that the
image of $B_{3}$ under $\rho_{k}$ is dense in $U(V^{\tau\tau\tau}_{\tau})$, we
prove that the image of the subgroup of double-braidings alone is dense in
$U(V^{\tau\tau\tau}_{\tau})$.

Figure 1. (Left) A counterclockwise braiding of two anyons; (Right) A
counterclockwise double-braiding of two anyons.

Our result is of significance in several aspects. Firstly, it is
mathematically stronger than the theorem of [8] adapted to one qubit, and may
review some hidden structure in the representations of the braid group.
Secondly, it generalizes the work of [5] where the Fibonacci anyon was shown
to be universal by double-braidings, which corresponds to the case $k=3$ here.
Moreover, using the double-braiding-universality of the Fibonacci anyon, the
authors in [5] provided an elegant, exponentially fast algorithm to produce
entangling leakage-free 2-qubit gates which are necessary for universal
quantum computing on multi-qubits. With the result in the current paper, the
algorithm of [5] can be straightforwardly adapted to other
$\textbf{SU}(2)_{k}$. Thirdly, from a physical point of view, double-braidings
have the advantage that at each step only one anyon needs to move, it needs to
move at most to the vicinity of its nearest neighbour, and it returns to its
own position immediately after the move. Hence, there is no need to track the
positions of the involved anyons. It should be noted the group of double-
braidings is strictly smaller than the pure braid group which consists of
braids where the anyons return to their own positions _eventually_.

It is now a well-established dictionary that 2-dimensional topological phases
of matter are characterized by the structure of a unitary modular tensor
category which is a braided category satisfying additional conditions. A
double-braiding is also called a _twine_ structure in the braided category
introduced by [3]. The twine structure can be formalized and defined on non-
braided categories. There exist monoidal categories without a braiding
structure, but with a twine structure. An example of this is the fermionic
Moore-Read fusion category [11]. While most FQH states are expected to fit in
the framework of modular tensor categories, some do not seem so. Instead, they
might be described by twine fusion categories [11, 2]. Our work on double-
braidings could provide insight on exploring the power of quantum computing in
those systems.

We conjecture that our result on the universality of double-braidings also
hold for the case of of multi-qubits, i.e., the space of more than three
anyons. For that generalization, the techniques utilized in this paper may not
apply. Instead, it is possible to make use of the Lie-theoretical tools on the
two-eigenvalue problem in [8]. We leave this as a future direction.

The rest of the paper is organized as follows. Section 2 gives a brief
overview on the algebraic theory of anyons and the setup in topological
quantum computing, with a more detailed explanation in Appendix A. In Section
3.1 \- 3.2, we provide explicit data for $\textbf{SU}(2)_{k}$ and explicit
calculations of the braid group representations. Section 3.3 contains the main
result.

##  2\. Topological quantum computing with anyons

In this section, we provide a very brief introduction to topological quantum
computing (TQC). There are many references with more comprehensive discussions
on this subject, such as [15, 12, 6, 13].

Mathematically, an anyon model is characterized by the structure of a unitary
modular tensor category (MTC). An MTC can be described either in terms of
abstract categorical language or by a set of concrete data. We take the second
approach and provide a partial set of data for the purpose of fixing the
convention. See Appendix A for a more detailed discussion of MTCs.

An anyon model has a finite set

$L=\\{a,b,c,\cdots\\}$

consisting of all the possible anyon types in a topological phase. Each anyon
type $a$ has a dual anyon type $\bar{a}$. The ground state is considered as a
special trivial anyon type, usually denoted by $\mathbf{1}$. The fusion rule
describes the possible outcomes when fusing two anyons. Given $a,b\in L$, we
formally write,

$a\otimes b=\bigoplus_{c\in L}\ N_{ab}^{c}\ c,$

where $N_{ab}^{c}$ denotes the number of different channels of fusing $a$ and
$b$ to result in the output $c$. If there is no way to obtain $c$ from the
fusion, then $N_{ab}^{c}=0$. If $N_{ab}^{c}>0$, we say $c$ is a _total type_
or _total charge_ of $a$ and $b$, and call the triple $(a,b;c)$ _admissible_.
For simplicity, in the following discussions we will assume $N_{ab}^{c}$ is
either $0$ or $1$, i.e., the anyon model is multiplicity-free.

For anyon types $c,a_{1},\cdots,a_{n}$, denote by $V_{c}^{a_{1}a_{2}\cdots
a_{n}}$ the space of states representing $n$ anyons $a_{1},\cdots,a_{n}$ with
total charge $c$. A basis for such a space can be described as follows. Choose
an upward-growing binary tree with one root at the bottom and $n$ leaves at
the top. See Figure 7 for an illustration. Label the root by $c$ and the
leaves, from left to right, by $a_{1},a_{2},\cdots,a_{n}$. Now label each
internal edge $e$ by an anyon type $b_{e}$ such that at each fork, the
relevant triple of labels are admissible. Then the binary tree with all
possible labels $\\{b_{e}\\}$ of internal edges form a basis of
$V_{c}^{a_{1}a_{2}\cdots a_{n}}$, called a _splitting-tree basis_. For each
labeled binary tree, one can interpret the state it represents as a splitting
process. For example, the state represented by the tree in Figure 7 is
obtained by splitting $c$ into $b_{n-2}$ and $a_{n}$, followed by splitting
$b_{n-2}$ into $b_{n-3}$ and $a_{n-1}$, $\cdots\cdots$, followed by splitting
$b_{1}$ into $b_{0}=a_{1}$ and $a_{2}$.

For $n=2$, there is one splitting tree as shown in Figure 6 (Right), which has
no internal edges. Hence $\dim(V_{c}^{ab})=N_{ab}^{c}$. For $n=3$, there are
two splitting trees, with one internal edge, as shown on both sides of the
equation below. The basis corresponding to the tree on the left side of the
equation consists of all possible labelings $m$ of the internal edge so that
$(a,b;m)$ and $(m,c;d)$ are both admissible. Similarly, the basis for the tree
on the right side consists of labelings $n$ of the internal edge so that
$(b,c;n)$ and $(a,n;d)$ are both admissible. Denote the matrix change between
the two bases by $F_{d}^{abc}$. More explicitly,

$d$$a$$c$$b$$m$$=\,\sum\limits_{n}\,F^{abc}_{d;nm}$$a$$d$$c$$b$$n$

where $F^{abc}_{d;nm}$ is the $(n,m)$-entry of $F_{d}^{abc}$, and the sum is
over all labelings $n$ as described above. Note that, here the anyon types $n$
and $m$ are used as the indices of the entries of $F_{d}^{abc}$. We call
$F_{d}^{abc}$ an $F$-matrix, its entries $F$-symbols or $6j$-symbols, and the
identity in the above equation an $F$-move. The $F$-symbols need to satisfy
the Pentagon Equations 6. Using $F$-moves one can relate arbitrary two
splitting-tree bases of the state spaces.

The process of swapping positions of anyons is called a braiding. A braiding
induces a unitary transformation on the state space. Consider two anyons $a$
and $b$ with total type $c$. A counterclockwise braiding of $a$ and $b$ maps a
state in $V_{c}^{ab}$ to one in $V_{c}^{ba}$. Since both spaces have dimension
one, there exists a phase $R^{ba}_{c}$ such that the following equality holds,

$a$$b$$c$$b$$a$$=$$R^{ba}_{c}$$b$$a$$c$

The above equality is called an $R$-move, and $R^{ba}_{c}$ is called an
$R$-symbol. The $R$-symbols need to satisfy the Hexagon Equations 7 8. The set
of $F$\- and $R$-symbols are crucial for calculations in the anyon model.

Now, we discuss how to perform quantum computing with anyons. The state space
of multi-anyons is the logical space to store information. Typically, one
chooses multiple identical anyons, say $n$ type-$a$ anyons with total type $c$
for some $n$. Denote this space by $V_{c}^{a^{\otimes n}}:=V_{c}^{aa\cdots
a}$. The type $a$ needs to be non-Abelian so that the dimension of
$V_{c}^{a^{\otimes n}}$ approaches to infinity as $n\to\infty$. In general,
$V_{c}^{a^{\otimes n}}$ does not have a natural tensor product structure, and
we need to choose a subspace which does have a tensor product structure as
multi-qubits or multi-qudits. The computational basis for the qudits can be
chosen to be any splitting-tree basis.

Braiding of anyons induces a representation of the braid group and acts as
quantum gates on the logical space $V_{c}^{a^{\otimes n}}$. Recall that the
$n$-strand braid group $B_{n}$ has a presentation as

$\displaystyle B_{n}=\langle\sigma_{1},\cdots,\sigma_{n-1}\,|\,$
$\displaystyle\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},$
$\displaystyle\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|>1\,\rangle,$

where $\sigma_{i}$ (resp. $\sigma_{i}^{-1}$) corresponds to the braid diagram
in Figure 2(Left) (resp. (Right) ).

$\cdots$$\cdots$$1$$i+1$$i$$n$$\cdots$$\cdots$$1$$i+1$$i$$n$ Figure 2. (Left)
the braid diagram $\sigma_{i}$; (Right) the braid diagram $\sigma_{i}^{-1}$.

We have a representation,

$\rho\colon B_{n}\to U(V_{c}^{a^{\otimes n}}),$

where $\sigma_{i}$ (resp. $\sigma_{i}^{-1}$) acts on $V_{c}^{a^{\otimes n}}$
by counterclockwise (resp. clockwise) braiding of the $i$-th with the
$(i+1)$-th anyon. With a chosen splitting-tree basis, a matrix for each braid
generator can be computed using $F$\- and $R$-symbols. The set of quantum
gates obtained from braiding is the image of this braid group representation.

##  3\. Universality in $\textbf{SU}(2)_{k}$ anyons

###  3.1. $\textbf{SU}(2)_{k}$ anyons

Recall that the $SU(2)$ Witten-Chern-Simons theory at level
$k\in\mathbb{Z}_{\geq 0}$ is denoted by $\textbf{SU}(2)_{k}$. The MTC
corresponding to $\textbf{SU}(2)_{k}$ is constructed from finite-dimensional
representations of the quantum group $U_{q}(\text{sl}_{2})$ for
$q=e^{\frac{2\pi i}{k+2}}$. It also describes the Wess–Zumino–Witten conformal
field theory [17, 18].

Below we describe the data of the $\textbf{SU}(2)_{k}$ MTC obtained from Sec.
5.4 of [2], though the data also appeared in earlier literature in different
forms. The MTC has $k+1$ anyon types (i.e. simple objects) labeled by half-
integers $0,\frac{1}{2},\frac{2}{2},\dots,\frac{k}{2}$, where $0$ denotes the
trivial anyon type. The fusion rule is given by,

$j_{1}\otimes
j_{2}=\bigoplus_{j=|j_{1}-j_{2}|}^{\min\\{j_{1}+j_{2},k-j_{1}-j_{2}\\}}j,$

where $j$ has an increment of 1 in the above sum, implying that for admissible
$(j_{1},j_{2};j)$, $j_{1}+j_{2}+j$ is always an integer. Throughout the
context, fix $q=e^{\frac{2\pi i}{k+2}}$, and denote by $q^{x}:=e^{\frac{2\pi
ix}{k+2}}$. For $n\in\mathbb{Z}_{\geq 0}$, the quantum integer $[n]$ and the
quantum factorial are defined by,

$\displaystyle[n]_{q}:=\frac{q^{\frac{n}{2}}-q^{-\frac{n}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}},\quad\quad[n]_{q}!=([n-1]_{q}!)\
[n],\quad\quad[0]_{q}!=1.$

The $R$-symbols are given by,

$R_{j}^{j_{1},j_{2}}=(-1)^{j-j_{1}-j_{2}}q^{\frac{1}{2}(j(j+1)-j_{1}(j_{1}+1)-j_{2}(j_{2}+1))}.$

The $F$-symbols are given by,

$F_{j;j_{12},j_{23}}^{j_{1},j_{2},j_{3}}=[F_{j}^{j_{1},j_{2},j_{3}}]_{j_{12},j_{23}}=(-1)^{j_{1}+j_{2}+j_{3}+j}\sqrt{[2j_{12}+1]_{q}[2j_{23}+1]_{q}}\left\\{\begin{matrix}j_{1}&j_{2}&j_{12}\\\
j_{3}&j&j_{23}\end{matrix}\right\\}_{q},$

where

$\displaystyle\left\\{\begin{matrix}j_{1}&j_{2}&j_{12}\\\
j_{3}&j&j_{23}\end{matrix}\right\\}_{q}=\Delta(j_{1},j_{2},j_{12})\Delta(j_{12},j_{3},j)\Delta(j_{2},j_{3},j_{23})\Delta(j_{1},j_{23},j)\times$
$\displaystyle\sum_{z}\Bigg{\\{}\frac{(-1)^{z}[z+1]_{q}!}{[z-j_{1}-j_{2}-j_{12}]_{q}![z-j_{12}-j_{3}-j]_{q}![z-j_{2}-j_{3}-j_{23}]_{q}![z-j_{1}-j_{23}-j]_{q}!}$
$\displaystyle\times\frac{1}{[j_{1}+j_{2}+j_{3}+j-z]_{q}![j_{1}+j_{12}+j_{3}+j_{23}-z]_{q}![j_{2}+j_{12}+j+j_{23}-z]}\Bigg{\\}},$

where the sum is over $z$ with an increment of $1$ for which all the
$[\cdot]_{q}!$ in the sum are defined, and

$\Delta(j_{1},j_{2},j_{3})=\sqrt{\frac{[-j_{1}+j_{2}+j_{3}]_{q}![j_{1}-j_{2}+j_{3}]_{q}![j_{1}+j_{2}-j_{3}]}{[j_{1}+j_{2}+j_{3}+1]_{q}!}}.$

A fact that will not be used in this paper is that a close cousin of the
$\textbf{SU}(2)_{k}$ MTC is the Temperley-Lieb-Jones MTC obtained from skein
theory [14]. Under a proper translation between the level $k$ and the Kauffman
variable $A$ in the skein theory, the $\textbf{SU}(2)_{k}$ MTC and the
Temperley-Lieb-Jones MTC are equivalent as braided fusion categories, but
differ by a ribbon twist.

###  3.2. The 1-qubit model

For $k\geq 2$, we consider the anyon type labeled by $\tau:=\frac{1}{2}$ in
the $\textbf{SU}(2)_{k}$ model. For $k=2$, $\tau$ is the Ising anyon, while
for $k=3$, $\tau$ is closely related to the Fibonacci anyon 222It is the
composite of the Fibonacci anyon with a semion..

There are two standard ways to obtain a qubit using $\tau$ anyons. They are
the dense encoding and the sparse encoding corresponding to the spaces
$V^{\tau\tau\tau}_{\tau}$ and $V^{\tau\tau\tau\tau}_{0}$, respectively. That
is, the dense encoding takes, as a qubit, the space of three $\tau$ anyons
with total type $\tau$, while the sparse encoding takes the space of four
$\tau$ anyons with total type $0$. From the fusion rule,

$\displaystyle\frac{1}{2}\otimes\frac{1}{2}=0\oplus 1,\quad\frac{1}{2}\otimes
1=\frac{1}{2}\oplus\frac{3}{2},\quad 0\otimes j=j,$

both $V^{\tau\tau\tau}_{\tau}$ and $V^{\tau\tau\tau\tau}_{0}$ are of two
dimensions with a splitting-tree basis given in Figure 3. Denote the
splitting-tree basis in Figure 3 (Left) by $\\{|x\rangle:\ x=0,1\\}$ and the
one in Figure 3 (Right) by $\\{|x\rangle^{\prime}:\ x=0,1\\}$.

$\tau$$\tau$$\tau$$\tau$$a$$\tau$$b$$\tau$$b$$\tau$$\tau$$0$ Figure 3. (Left)
A splitting-tree basis for $V_{\tau}^{\tau\tau\tau}$; (Right) A splitting-tree
basis for $V_{0}^{\tau\tau\tau\tau}$, where $\tau$ is the anyon of type
$\frac{1}{2}$. $a,b=0,1$.

The braiding of the $\tau$ anyons induces representations of the braid groups
with the action of the generator $\sigma_{i}$ given by the counter-clockwise
swap of the $i$-th and the $(i+1)$-th anyon,

$\displaystyle\rho_{k}\colon$ $\displaystyle B_{3}\to
U(V^{\tau\tau\tau}_{\tau}),$ $\displaystyle\rho_{k}^{\prime}\colon$
$\displaystyle B_{4}\to U(V^{\tau\tau\tau\tau}_{0}).$

For the dense encoding, under the basis $\\{|x\rangle:\ x=0,1\\}$ of
$V^{\tau\tau\tau}_{\tau}$, the action of the generators $\sigma_{1}$ and
$\sigma_{2}$ are computed in Figure 4 and Figure 5, respectively.

$\tau$$x$$\tau$$\tau$$\tau$$=\ R^{\tau\tau}_{x}$$\tau$$x$$\tau$$\tau$$\tau$
Figure 4. The action of $\sigma_{1}$ on $V_{\tau}^{\tau\tau\tau}$.
$\tau$$x$$\tau$$\tau$$\tau$$=$$\sum\limits_{y\in\\{0,1\\}}F^{\tau\tau\tau}_{\tau;yx}$$\tau$$y$$\tau$$\tau$$\tau$$=$$\sum\limits_{y\in\\{0,1\\}}F^{\tau\tau\tau}_{\tau;yx}R^{\tau\tau}_{y}$$\tau$$y$$\tau$$\tau$$\tau$$=$$\sum\limits_{y,z\in\\{0,1\\}}F^{\tau\tau\tau}_{\tau;yx}R^{\tau\tau}_{y}(F^{\tau\tau\tau}_{\tau})^{-1}_{zy}$$\tau$$z$$\tau$$\tau$$\tau$
Figure 5. The action of $\sigma_{2}$ on $V_{\tau}^{\tau\tau\tau}$. The first
and the third equality are due to an $F$-move and an inverse $F$-move,
respectively. The second equality is due to an $R$-move.

Hence,

$\displaystyle\rho_{k}(\sigma_{1})|x\rangle=R^{\tau\tau}_{x}|x\rangle$
$\displaystyle\rho_{k}(\sigma_{2})|x\rangle=\sum_{y,z\in\\{0,1\\}}\
F^{\tau\tau\tau}_{\tau;yx}\ R^{\tau\tau}_{y}\
(F^{\tau\tau\tau}_{\tau})^{-1}_{zy}\ |z\rangle.$

Denote by,

$R=\begin{pmatrix}R^{\tau\tau}_{0}&0\\\ 0&R^{\tau\tau}_{1}\\\
\end{pmatrix},\quad\quad
F=F^{\tau\tau\tau}_{\tau}=\begin{pmatrix}F^{\tau\tau\tau}_{\tau;00}&F^{\tau\tau\tau}_{\tau;01}\\\
F^{\tau\tau\tau}_{\tau;10}&F^{\tau\tau\tau}_{\tau;11}\\\ \end{pmatrix}.$

From the data in Section 3.1,

$R=\begin{pmatrix}-q^{-\frac{3}{4}}&0\\\ 0&q^{\frac{1}{4}}\\\
\end{pmatrix},\quad\quad
F=F^{\tau\tau\tau}_{\tau}=\frac{\sqrt{q}}{q+1}\left(\begin{array}[]{cc}-1&\sqrt{q+\frac{1}{q}+1}\\\
\sqrt{q+\frac{1}{q}+1}&1\\\ \end{array}\right).$ (1)

Note that $F$ is a symmetric, real, involutory matrix. Then we have

$\displaystyle\rho_{k}(\sigma_{1})=R=\begin{pmatrix}-q^{-\frac{3}{4}}&0\\\
0&q^{\frac{1}{4}}\\\ \end{pmatrix},$
$\rho_{k}(\sigma_{2})=F^{-1}RF=\frac{q^{\frac{1}{4}}}{1+q}\left(\begin{array}[]{cc}q&\sqrt{q+\frac{1}{q}+1}\\\
\sqrt{q+\frac{1}{q}+1}&-\frac{1}{q}\\\ \end{array}\right).$

For the sparse encoding, we show in fact the image $\rho_{k}^{\prime}(B_{4})$
is the same as $\rho_{k}(B_{3})$ if we identify $|x\rangle^{\prime}\in
V^{\tau\tau\tau\tau}_{0}$ with $|x\rangle\in V^{\tau\tau\tau}_{\tau}$. Under
this identification, it is clear that
$\rho_{k}^{\prime}(\sigma_{1})=\rho_{k}^{\prime}(\sigma_{3})=\rho_{k}(\sigma_{1})$,
while $\rho_{k}^{\prime}(\sigma_{2})$ can be computed, similar to
$\rho_{k}(\sigma_{2})$ in Figure 5, as,

$\displaystyle\rho_{k}^{\prime}(\sigma_{2})|x\rangle^{\prime}=(F^{x\tau\tau}_{0})^{-1}_{\tau
x}\ \left(\sum_{y,z\in\\{0,1\\}}\ F^{\tau\tau\tau}_{\tau;yx}\
R^{\tau\tau}_{y}\ (F^{\tau\tau\tau}_{\tau})^{-1}_{zy}\right)\
F^{z\tau\tau}_{0;z\tau}\ |z\rangle^{\prime}.$

From the $F$-symbols of $\textbf{SU}(2)_{k}$ in Section 3.1, it can be checked
that $F^{j_{1}j_{2}j_{3}}_{j;j_{23}j_{12}}=1$ whenever $j=0$ and the involved
labels are admissible. Hence we have
$\rho_{k}^{\prime}(\sigma_{2})=\rho_{k}(\sigma_{2})$. This shows the equality
of $\rho_{k}^{\prime}(B_{4})$ and $\rho_{k}(B_{3})$.

Therefore, from now on, we will focus on the dense encoding only. A logical
qubit is given by the space $V^{\tau\tau\tau}_{\tau}$ whose computational
basis is $\\{|0\rangle,|1\rangle\\}$ as shown in Figure 3 (Left). The set of
1-qubit logical gates obtained from anyon braidings correspond to elements in
the image $\rho_{k}(B_{3})$. Since quantum gates are well defined only up to
global $U(1)$ phases, the gates in $\rho_{k}(B_{3})$ can be multiplied by any
phase, or they should be considered as elements of the projective unitary
group $PU(V^{\tau\tau\tau}_{\tau})$.

######  Definition 3.1.

Let $V$ be a Hilbert space, and $\mathcal{G}$ be a subset of the unitary group
$U(V)$. $\mathcal{G}$ is said to be universal on $V$ if $\mathcal{G}\cup U(1)$
generate a dense subgroup of $U(V)$.

To study the universality of the braiding gates, it is convenient to normalize
the generators of $B_{3}$ so that the image of $\rho_{k}$ lies in
$SU(V^{\tau\tau\tau}_{\tau})$. Explicitly, multiplying $-iq^{1/4}$ to the
generators, we obtain the normalized representation $\tilde{\rho}_{k}:B_{3}\to
SU(V^{\tau\tau\tau}_{\tau})$,

$\tilde{\rho}_{k}(\sigma_{1})=\tilde{R}=\begin{pmatrix}i\
q^{-\frac{1}{2}}&0\\\ 0&-i\ q^{\frac{1}{2}}\\\ \end{pmatrix},$ (2)
$\tilde{\rho}_{k}(\sigma_{2})=F^{-1}\tilde{R}F=\frac{i\sqrt{q}}{q+1}\left(\begin{array}[]{cc}-q&-\sqrt{q+\frac{1}{q}+1}\\\
-\sqrt{q+\frac{1}{q}+1}&\frac{1}{q}\\\ \end{array}\right).$ (3)

Since $\rho_{k}$ equals $\tilde{\rho}_{k}$ up to a global phase, we will use
$\rho_{k}$ and $\tilde{\rho}_{k}$ interchangeably.

As an example, when $k=2$, $\tilde{\rho}_{2}(\sigma_{1})$ and
$\tilde{\rho}_{k}(\sigma_{2})$ are given, respectively, by,

$e^{\frac{\pi i}{4}}\ \left(\begin{array}[]{cc}1&0\\\ 0&-i\\\
\end{array}\right)\quad\text{and}\quad\frac{1}{\sqrt{2}}\
\left(\begin{array}[]{cc}1&-i\\\ -i&1\\\ \end{array}\right),$

which, together with $U(1)$ phases, generate the 1-qubit Clifford group.

The multi-qubit models can be obtained by increasing the number of anyons
utilized. We will not discuss that direction. The universality of braiding the
$\tau$ anyons is a classic problem settled in [8]. We rephrase Theorem 4.1 of
[8] in the setup of one qubit.

######  Theorem 3.2 ([8]).

For any integer $k\geq 3,k\neq 4,8$, let $V^{\tau\tau\tau}_{\tau}$ be the
1-qubit space in the $\textbf{SU}(2)_{k}$ model. Then set of braiding gates
corresponding to the image of the representation $\tilde{\rho}_{k}$ defined in
Equations 2 3 is universal on $V^{\tau\tau\tau}_{\tau}$.

###  3.3. Universality of double-braiding in $\textbf{SU}(2)_{k}$

In this section, we prove a stronger result than Theorem 3.2. That is, for
those values of $k$ in Theorem 3.2, we show that the set of double-braiding
gates on $V^{\tau\tau\tau}_{\tau}$ is universal in the $\textbf{SU}(2)_{k}$
model. By a double-braiding is meant a braid generated by even powers of the
standard generators $\sigma_{i}^{\prime}$s of the braid group. We will rely on
two critical results. The content of Lemma 3.3 below first appeared in the
work of Kitaev [9] where he used the lemma to prove the universality of
certain gate sets. A proof of the lemma can be found in [1].

######  Lemma 3.3 ([9]).

Let $A$ and $B$ be two non-commuting elements of $SU(2)$ both of which are of
infinite order. Then $A$ and $B$ generate a dense subgroup of $SU(2)$.

######  Theorem 3.4 (Theorem 7, [4]).

Suppose we have at most four distinct rational multiples of $\pi$ lying
strictly between $0$ and $\frac{\pi}{2}$ for which some rational linear
combination of their cosines is rational but no proper subset has this
property. Then the appropriate linear combination is proportional to one from
the following list:
$\cos{\pi/3}=\frac{1}{3}$,
$-\cos{\phi}+\cos{(\pi/3-\phi)}+\cos{(\pi/3+\phi)}=0\;(0<\phi<\pi/6)$,
$\cos{\pi/5}-\cos{2\pi/5}=\frac{1}{2}$,
$\cos{\pi/7}-\cos{2\pi/7}+\cos{3\pi/7}=\frac{1}{2}$,
$\cos{\pi/5}-\cos{\pi/15}+\cos{4\pi/15}=\frac{1}{2}$,
$-\cos{2\pi/5}+\cos{2\pi/15}-\cos{7\pi/15}=\frac{1}{2}$,
$\cos{\pi/7}+\cos{3\pi/7}-\cos{\pi/21}+\cos{8\pi/21}=\frac{1}{2}$,
$\cos{\pi/7}-\cos{2\pi/7}+\cos{2\pi/21}-\cos{5\pi/21}=\frac{1}{2}$,
$-\cos{2\pi/7}+\cos{3\pi/7}+\cos{4\pi/21}+\cos{10\pi/21}=\frac{1}{2}$,
$-\cos{\pi/15}+\cos{2\pi/15}+\cos{4\pi/15}-\cos{7\pi/15}=\frac{1}{2}$.

The following is the main result of the paper.

######  Theorem 3.5.

For any integer $k\geq 3,k\neq 4,8$, let $\tau$ be the anyon of type
$\frac{1}{2}$ in the anyon model $\textbf{SU}(2)_{k}$, and $\rho_{k}:B_{3}\to
U(V^{\tau\tau\tau}_{\tau})$ be the representation of $B_{3}$ on the dense-
encoding 1-qubit $V^{\tau\tau\tau}_{\tau}$. Then the images of
$\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ under $\rho_{k}$, together with phases,
generate a dense subgroup of $U(V^{\tau\tau\tau}_{\tau})$. That is, the
double-braiding gates alone are universal.

###### Proof.

It suffices to show, for the the normalized $\tilde{\rho}_{k}:B_{3}\to
U(V^{\tau\tau\tau}_{\tau})$ defined in Equations 2 3,
$\tilde{\rho}_{k}(\sigma_{1}^{2})$ and $\tilde{\rho}_{k}(\sigma_{2}^{2})$
generate a dense subgroup of $SU(V^{\tau\tau\tau}_{\tau})$. Let

$A:=\tilde{\rho}_{k}(\sigma_{1}^{2}\sigma_{2}^{4})=\tilde{R}^{2}F^{-1}\tilde{R}^{4}F,\quad
B:=\tilde{\rho}_{k}(\sigma_{1}^{2}\sigma_{2}^{6})=\tilde{R}^{2}F^{-1}\tilde{R}^{6}F.$

From the expressions of $\tilde{R}$ (Equation 2) and $F$ (Equation 1), the
matrices of $A$, $B$, and $W:=ABA^{-1}B^{-1}$ can be calculated as (See
Appendix B for a Mathematica implementation),

$A=\left(\begin{array}[]{cc}-\frac{q^{4}+q^{2}-q+1}{q^{3}+q^{2}}&-\frac{\sqrt{q+\frac{1}{q}+1}\left(q^{3}-q^{2}+q-1\right)}{q^{2}(q+1)}\\\
-\frac{\sqrt{q+\frac{1}{q}+1}\left(q^{3}-q^{2}+q-1\right)}{q+1}&-\frac{q^{5}+q^{2}+q+1}{q(q+1)^{2}}\\\
\end{array}\right)$
$B=\left(\begin{array}[]{cc}\frac{q^{7}+q^{6}+q^{5}+1}{q^{3}(q+1)^{2}}&\frac{\sqrt{q+\frac{1}{q}+1}\left(q^{5}-q^{4}+q^{3}-q^{2}+q-1\right)}{q^{3}(q+1)}\\\
\frac{\sqrt{q+\frac{1}{q}+1}\left(q^{5}-q^{4}+q^{3}-q^{2}+q-1\right)}{q(q+1)}&\frac{q^{6}+q+\frac{1}{q}+1}{q(q+1)^{2}}\\\
\end{array}\right)$ $\displaystyle W_{11}$
$\displaystyle=\frac{q^{13}-3q^{12}+6q^{11}-11q^{10}+16q^{9}-19q^{8}+22q^{7}-21q^{6}+19q^{5}-14q^{4}+10q^{3}-6q^{2}+3q-1}{q^{6}(q+1)}$
$\displaystyle W_{12}$
$\displaystyle=-\frac{(q-1)^{2}\sqrt{q+\frac{1}{q}+1}\left(q^{10}-q^{9}+3q^{8}-4q^{7}+5q^{6}-6q^{5}+6q^{4}-5q^{3}+4q^{2}-2q+1\right)}{q^{7}(q+1)}$
$\displaystyle W_{21}$
$\displaystyle=\frac{(q-1)^{2}\sqrt{q+\frac{1}{q}+1}\left(q^{10}-2q^{9}+4q^{8}-5q^{7}+6q^{6}-6q^{5}+5q^{4}-4q^{3}+3q^{2}-q+1\right)}{q^{4}(q+1)}$
$\displaystyle W_{22}$
$\displaystyle=\frac{-q^{13}+3q^{12}-6q^{11}+10q^{10}-14q^{9}+19q^{8}-21q^{7}+22q^{6}-19q^{5}+16q^{4}-11q^{3}+6q^{2}-3q+1}{q^{7}+q^{6}}$

We will show that, for the values of $k$ in the statement of the theorem, $A$
and $B$ are both of infinite order and they do not commute. Then by Lemma 3.3,
$A$ and $B$ generate a dense subgroup of $SU(2)$, implying the validity of the
theorem. The rest of the proof is devoted to verifying the assumptions on $A$
and $B$ mentioned above.

Proving $\mathbf{A}$ has infinite order.

Denote the eigenvalues of $A$ by $e^{\pm\theta_{k}i}$,
$0\leq\theta_{k}\leq\pi$. It suffices to show $\theta_{k}$ is not a rational
multiple of $\pi$. Recall that $q=e^{\frac{2\pi i}{k+2}}$. We have,

$\displaystyle 2\cos\theta_{k}=\operatorname{tr}(A)$
$\displaystyle=-\frac{((q-1)q+1)\left(q^{2}+1\right)}{q^{2}}$
$\displaystyle=-2-(q^{2}+\frac{1}{q^{2}})+(q+\frac{1}{q})$
$\displaystyle=-2-2\cos\frac{4\pi}{k+2}+2\cos\frac{2\pi}{k+2}.$

That is,

$\cos{\frac{4\pi}{k+2}}-\cos{\frac{2\pi}{k+2}}+\cos{\theta_{k}}=-1.$ (4)

We wish to show the above identity is not equivalent to any of the identities
in Theorem 3.4, and hence $\theta_{k}$ cannot be a rational multiple of $\pi$.
But we need to first verify the assumptions in that theorem.

We make four statements which can be verified by checking small values of $k$
and applying Theorem 3.4, noting that when $k>6$, $\frac{2\pi}{k+2}$ and
$\frac{4\pi}{k+2}$ lie strictly between 0 and $\frac{\pi}{2}$. Assume $k\geq
3$.

  1. A).

The only $k$ for which $\cos{\frac{2\pi}{k+2}}$ is rational is $k=4$,

$\displaystyle\cos{\frac{2\pi}{4+2}}=\frac{1}{2}.$

  2. B).

The only $k^{\prime}$s for which $\cos{\frac{4\pi}{k+2}}$ is rational are
$k=4,6,10$,

$\displaystyle\cos{\frac{4\pi}{4+2}}=-\frac{1}{2},\quad\cos{\frac{4\pi}{6+2}}=0,\quad\cos{\frac{4\pi}{10+2}}=\frac{1}{2}.$

  3. C).

The only $k^{\prime}$s for which neither $\cos{\frac{2\pi}{k+2}}$ nor
$\cos{\frac{4\pi}{k+2}}$ is rational but certain non-trivial rational
combinations of them is rational are $k=3,8$,

$\displaystyle-\cos{\frac{2\pi}{3+2}}-\cos{\frac{4\pi}{3+2}}=\frac{1}{2},\quad\quad\cos{\frac{2\pi}{8+2}}-\cos{\frac{4\pi}{8+2}}=\frac{1}{2}.$

  4. D).

Furthermore, the only $k^{\prime}$s for which $\cos{\theta_{k}}$ is rational
are $k=4,8$,

$\displaystyle\cos\theta_{4}=0,\quad\cos\theta_{8}=-\frac{1}{2}.$

Statement $D)$ implies that for $k=4,8$, $\theta_{k}$ is a rational multiple
of $\pi$. For all other $k^{\prime}$s, $\theta_{k}$ is not a rational multiple
of $\pi$ and in particular $\theta_{k}\neq 0,\frac{\pi}{2},\pi$.

The above statements also imply for $k\geq 7$, $k\neq 8,10$, none of
$\cos{\frac{4\pi}{k+2}}$, $\cos{\frac{2\pi}{k+2}}$, or $\cos{\theta_{k}}$ is
rational. Furthermore, for those values of $k$, for any two elements from
$\\{\cos{\frac{4\pi}{k+2}},\cos{\frac{2\pi}{k+2}},\cos{\theta_{k}}\\}$, they
cannot have a non-trivial rational combination which is rational. The above
claim for the pair $\\{\cos{\frac{4\pi}{k+2}},\cos{\frac{4\pi}{k+2}}\\}$ is
clear from Statement $C)$. For the other two pairs, say,
$\\{\cos{\frac{4\pi}{k+2}},\cos{\theta_{k}}\\}$, if they do not satisfy the
claim, then one can substitute $\cos{\theta_{k}}$ with an expression of
$\cos{\frac{4\pi}{k+2}}$ in Equation 4, yielding a rational combination of
$\cos{\frac{4\pi}{k+2}}$ and $\cos{\frac{2\pi}{k+2}}$ which is rational. That
is a contradiction.

Then for $k\geq 7$, $k\neq 8,10$, if $\theta_{k}$ is a rational multiple of
$\pi$ strictly between 0 and $\frac{\pi}{2}$, then Equation 4 is an identity
concerning a rational combination of the cosine of three distinct angles
satisfying the conditions in Theorem 3.4. However, this identity is not
equivalent to any of those in that theorem, a contradiction.

If $\theta_{k}$ is a rational multiple of $\pi$ strictly between
$\frac{\pi}{2}$ and $\pi$, then Equation 4 can be written as,

$\cos{\frac{4\pi}{k+2}}-\cos{\frac{2\pi}{k+2}}-\cos(\pi-\theta_{k})=-1.$

Applying the same argument to the new equation lead to a similar
contradiction. This shows that for $k\geq 7$, $k\neq 8,10$, $\theta_{k}$ is
not a rational multiple of $\pi$.

The remaining cases to check are $k=3,5,6,10$. The corresponding identity of
Equation 4 for these values of $k$ can be simplified below.

$\displaystyle\cos{\theta_{3}}=\frac{\sqrt{5}-2}{2},\quad\cos{\theta_{5}}=\cos{\frac{3\pi}{7}}+\cos{\frac{2\pi}{7}}-1,\quad\cos{\theta_{6}}=\frac{\sqrt{2}-2}{2},\quad\cos{\theta_{10}}=\frac{\sqrt{3}-3}{2}.$

It can be checked directly by a computer program or by applying Theorem 3.4
that the above $\theta_{k}^{\prime}$s are not rational multiple of $pi$. This
completes the proof that the eigenvalues of $A$ are of infinite order for
$k\geq 3$, $k\neq 4,8$.

Proving $\mathbf{B}$ has infinite order.

Denote the eigenvalues of $B$ by $e^{\pm\theta_{k}i}$,
$0\leq\theta_{k}\leq\pi$.

$\displaystyle 2\cos\theta_{k}=\operatorname{tr}(B)$
$\displaystyle=\frac{\left(q^{2}+1\right)\left((q-1)q\left(q^{2}+1\right)+1\right)}{q^{3}}$
$\displaystyle=-2+(q^{3}+\frac{1}{q^{3}})-(q^{2}+\frac{1}{q^{2}})+2(q+\frac{1}{q})$
$\displaystyle=-2+2\cos\frac{6\pi}{k+2}-2\cos\frac{4\pi}{k+2}+4\cos\frac{2\pi}{k+2}.$

That is,

$2\cos\frac{2\pi}{k+2}-\cos\frac{4\pi}{k+2}+\cos\frac{6\pi}{k+2}-\cos{\theta_{k}}=1.$

The rest of the proof is completely similar to the case of the matrix $A$ by
repeatedly applying Theorem 3.4. We leave the details as an exercise for
curious readers.

Proving $\mathbf{W\neq I}$.

Note that $W\neq I$ if and only if $\operatorname{Tr}(W)\neq 2$. Assume
$\operatorname{Tr}(W)=2$. Then,

$\displaystyle 2$
$\displaystyle=-\frac{((q-1)q+1)\left(q^{4}-2q^{3}-2q+1\right)}{q^{3}}$
$\displaystyle=4-(q^{3}+\frac{1}{q^{3}})+3(q^{2}+\frac{1}{q^{2}})-3(q+\frac{1}{q})$
$\displaystyle=4-2\cos\frac{6\pi}{k+2}+6\cos\frac{4\pi}{k+2}-6\cos\frac{2\pi}{k+2}.$

That is,

$\cos\frac{6\pi}{k+2}-3\cos\frac{4\pi}{k+2}+3\cos\frac{2\pi}{k+2}=1.$

That the above identity is fake can be checked directly for small values
$k\leq 10$ and by Theorem 3.4 for $k\geq 11$.

∎

##  Appendix A Anyon models

Mathematically, an anyon model is characterized by the structure of a unitary
modular tensor category (MTC). To avoid abstract categorical language, we
describe an MTC with a set of concrete data.

Label set. Associated with each anyon model is a finite set

$L=\\{a,b,c,\cdots\\}$

consisting of all the possible anyon types in a topological phase. The ground
state is considered as a special trivial anyon type, and is usually denoted by
$\mathbf{1}\in L$. For each anyon type $x\in L$, there exists $\bar{x}\in L$
corresponding to the anti-particle (i.e., the dual anyon type) of $x$. We
require that $\bar{\mathbf{1}}=\mathbf{1}$ and $\bar{\bar{x}}=x$.

Fusion rule. For $a,b\in L$, fusing $a$ and $b$ produces different possible
anyon types. Formally, it is written as,

$a\otimes b=\bigoplus_{c\in L}\ N_{ab}^{c}\ c,$

where $N_{ab}^{c}$ denotes the number of different channels of fusing $a$ and
$b$ to result in the output $c$. If there is no way to obtain $c$ from the
fusion, then $N_{ab}^{c}=0$. One can also view the equality in the above
equation from an alternative perspective. Namely, the composite type of $a$
and $b$ is a superposition of all possible anyon types with each type $c$
appearing in $N_{ab}^{c}$ copies. If $N_{ab}^{c}>0$, we say $c$ is a _total
type_ or _total charge_ of $a$ and $b$, and call the triple $(a,b;c)$
_admissible_. The collection of the integers $\\{N_{ab}^{c}\ |\ a,b,c\in L\\}$
is called the fusion rule. The fusion rule should satisfy the following
requirement.

  1. A).

The fusion rule is commutative, i.e., $a\otimes b=b\otimes a$, implying

$\displaystyle N_{ab}^{c}=N_{ba}^{c},\quad\forall a,b,c.$

  2. B).

The dual of $a\otimes b$ as a composite equals $\bar{b}\otimes\bar{a}$,
implying

$\displaystyle N_{ab}^{c}=N_{\bar{b}\bar{a}}^{\bar{c}},\quad\forall a,b,c.$

  3. C).

$\mathbf{1}\otimes a=a$, implying

$\displaystyle N_{\mathbf{1}a}^{b}=\delta_{a,b},\quad\forall a,b.$

  4. D).

$\mathbf{1}$ is a total type of $a$ and $b$ if and only if $a=\bar{b}$,
implying

$\displaystyle N_{ab}^{\mathbf{1}}=\delta_{a,\bar{b}},\quad\forall a,b.$

  5. E).

The fusion rule is associative, i.e., $(a\otimes b)\otimes c=a\otimes(b\otimes
c)$, implying

$\displaystyle\sum\limits_{p\in L}N_{ab}^{p}N_{pc}^{d}=\sum\limits_{q\in
L}N_{aq}^{d}N_{bc}^{q},\quad\forall a,b,c,d,$

For simplicity, in the following discussions we will assume $N_{ab}^{c}$ is
either $0$ or $1$, i.e., the anyon model is multiplicity-free. This already
covers a large family of anyon models including the ones considered in the
current paper.

State space. For anyon types $c,a_{1},\cdots,a_{n}$, denote by
$V_{c}^{a_{1}a_{2}\cdots a_{n}}$ the space of states representing $n$ anyons
$a_{1},\cdots,a_{n}$ with total charge $c$. The dimension and bases of these
spaces are described inductively as follows.

For $n=1$, $V_{c}^{a_{1}}=0$ if $c\neq a_{1}$, and $V_{c}^{c}$ is
$1$-dimensional with a canonical basis denoted by the left diagram of Figure
6. One can think of the diagram representing the state obtained by ‘doing
nothing’ to an existent anyon $c$.

For $n=2$, $V_{c}^{a_{1}a_{2}}=0$ if $N_{a_{1}a_{2}}^{c}=0$, and
$V_{c}^{a_{1}a_{2}}$ is $1$-dimensional otherwise, with a non-canonical basis
denoted by the right diagram in Figure 6. One can then think of the diagram
representing the state obtained from the process of splitting $c$ into the
pair $a$ and $b$. This choice of basis is not canonical as one can multiply an
arbitrary phase to it. We will use the diagrams in Figure 6 as building blocks
to describe bases of multi-anyon spaces.

$c$$c$$a_{1}$$a_{2}$ Figure 6. (Left) A canonical basis of $V_{c}^{c}$;
(Right) A (non-canonical) basis of $V_{c}^{a_{1}a_{2}}$.

For $n\geq 3$, take an upward-growing binary tree with one root at the bottom
and $n$ leaves at the top. See Figure 7 for an illustration. It is to be
understood that the tree is constructed using the two diagrams in Figure 6.
Label the root by $c$ and the leaves, from left to right, by
$a_{1},a_{2},\cdots,a_{n}$. Now label each internal edge $e$ by an anyon type
$b_{e}$ such that at each fork, the relevant labels are admissible. Then the
binary tree with all possible labels $\\{b_{e}\\}$ of internal edges form a
basis of $V_{c}^{a_{1}a_{2}\cdots a_{n}}$. For each labeled binary tree, one
can similarly interpret the state it represents as a splitting process. For
example, the state represented by the tree in Figure 7 is obtained by
splitting $c$ into $b_{n-2}$ and $a_{n}$, followed by splitting $b_{n-2}$ into
$b_{n-3}$ and $a_{n-1}$, $\cdots\cdots$, followed by splitting $b_{1}$ into
$b_{0}=a_{1}$ and $a_{2}$. Such a basis is called a _splitting-tree basis_.

$a_{1}$$a_{2}$$b_{1}$$b_{2}$$a_{3}$$a_{4}$$\cdots$$a_{n}$$c$$\cdots$ Figure 7.
A basis of $V_{c}^{a_{1}a_{2}\cdots a_{n}}$ corresponding to a binary tree

For the case of $n=3$, there are exactly two such binary trees as shown on
both sides of Equation 5, each of which provides a basis of $V_{d}^{abc}$.
Each tree has one internal edge. The basis corresponding to the tree on the
left side of the equation consists of all possible labelings $m$ of the
internal edge so that $(a,b;m)$ and $(m,c;d)$ are both admissible. Similarly,
the basis for the tree on the right side consists of labelings $n$ of the
internal edge so that $(b,c;n)$ and $(a,n;d)$ are both admissible. Denote the
matrix change between the two bases by $F_{d}^{abc}$. More explicitly,

$d$$a$$c$$b$$m$$=\,\sum\limits_{n}\,F^{abc}_{d;nm}$$a$$d$$c$$b$$n$ (5)

where $F^{abc}_{d;nm}$ is the $(n,m)$-entry of $F_{d}^{abc}$, and the sum is
over all labelings $n$ as described above. Note that, here the anyon types $n$
and $m$ are used as the indices of the entries of $F_{d}^{abc}$. We call
$F_{d}^{abc}$ an $F$-matrix, its entries $F$-symbols or $6j$-symbols, and the
identity in Equation 5 an $F$-move.

Using the $F$-move or its inverse, we can relate any two splitting-tree bases
of $V_{c}^{a_{1}a_{2}\cdots a_{n}}$. For $V_{e}^{abcd}$, there are exactly
five splitting-tree bases, and Figure 8 shows the $F$-moves connecting them.
In particular, starting from the basis labeled by 1, there are two ways of
performing $F$-moves to obtain the basis labeled by 3, namely, either via the
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\pgfsys@invoke{\lxSVG@closescope
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$\leavevmode\hbox to11.37pt{\vbox
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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induce a basis change between 1 and 3, this introduces some constraints on the
$F$-symbols, namely,

$\displaystyle F^{mcd}_{e;zn}F^{abz}_{e;ym}=\sum\limits_{x\in
L}F^{abc}_{n;xm}F^{axd}_{e;yn}F^{bcd}_{y;zx},\quad\forall a,b,c,d,e,m,n,y,z.$
(6)

Equation 6 is known as the Pentagon Equations. It is a non-trivial fact that
the Pentagon Equations guarantee that the change between splitting-tree bases
via $F$-moves for an arbitrary state space is consistent.

$a$$m$$b$$n$$c$$d$$e$1$a$$m$$b$$z$$c$$d$$e$2$a$$y$$b$$z$$c$$d$$e$3$a$$x$$b$$n$$c$$d$$e$5$a$$y$$b$$x$$c$$d$$e$4
Figure 8. The five splitting-tree bases of $V_{e}^{abcd}$ and the $F$-moves
connecting them.

Braiding. The process of swapping positions of anyons is called a braiding.
Since anyons live in 2-dimensional space, a counterclockwise braiding has a
different world-line from that of a clockwise braiding. The world-line of a
sequence of braidings of multi-anyons is a braid diagram, and hence the naming
of the process. A braiding induces a unitary transformation on the state
space. Consider two anyons $a$ and $b$ with total type $c$. A counterclockwise
braiding of $a$ and $b$ maps a state in $V_{c}^{ab}$ to one in $V_{c}^{ba}$.
Since both spaces have dimension one, there exists a phase $R^{ba}_{c}$ such
that the following equality holds,

$a$$b$$c$$b$$a$$=$$R^{ba}_{c}$$b$$a$$c$

The above equality is called an $R$-move, and $\\{R^{ab}_{c}\\}$ is called an
$R$-symbol. Since a counterclockwise braiding followed by a clockwise braiding
is equivalent to the identity process, we have,

$a$$b$$c$$b$$a$$=$$(R^{ab}_{c})^{-1}$$b$$a$$c$

Consider the space $V_{d}^{abc}$, and braid $a$ with $b$ and $c$. Each node in
Figure 9 represents a basis of $V_{d}^{bca}$, and performing an $F$-move or
$R$-move change from one basis to another. To change from basis 1 to basis 3,
one can follow either the path $\leavevmode\hbox to11.37pt{\vbox
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obtain the Hexagon Equation,

$\displaystyle R^{ba}_{m}F^{bac}_{d;nm}R^{ca}_{n}=\sum\limits_{x\in
L}F^{abc}_{d;xm}R^{xa}_{d}F^{bca}_{d;nx}.$ (7)

By replacing the counterclockwise braidings in Figure 9 with clockwise
braidings, we obtain another Hexagon Equation,

$\displaystyle(R^{ab}_{m})^{-1}F^{bac}_{d;nm}(R^{ac}_{n})^{-1}=\sum\limits_{x\in
L}F^{abc}_{d;xm}(R^{ax}_{d})^{-1}F^{bca}_{d;nx}.$ (8)

$d$$m$$b$$c$$a$1$d$$m$$b$$c$$a$6$d$$n$$b$$c$$a$5$d$$n$$b$$c$$a$4$d$$x$$b$$c$$a$3$d$$x$$b$$c$$a$$=$$d$$x$$b$$c$$a$2
Figure 9. Consistency conditions for braidings on $V_{d}^{abc}$.

Topological spin. Each anyon type $a$ has an (intrinsic) topological spin
$\theta_{a}$ which is always a root of unity. The type $a$ is said to be
bosonic if $\theta_{a}=1$, fermionic if $\theta_{a}=-1$, and semionic if
$\theta_{a}=i$. The topological spins are required to satisfy the following
conditions.

  1. A).

The trivial anyon is bosonic,

$\theta_{\mathbf{1}}=1.$

  2. B).

An anyon and its dual have equal topological spin,

$\theta_{a}=\theta_{\bar{a}},\quad\forall a\in L.$

  3. C).

Whenever $c$ is a total type of $a$ and $b$, we have

$\theta_{c}\theta_{a}^{-1}\theta_{b}^{-1}=R^{ab}_{c}R^{ba}_{c}.$

Quantum dimension. For each anyon type $a$, define an $|L|\times|L|$ matrix
$N_{a}$ whose $(b,c)$-entry is $N_{ab}^{c}=N_{ba}^{c}$. Hence the entries of
$N_{a}$ are non-negative integers. By the Perron-Frobenius theorem, $N_{a}$
has an eigenvalue $\dim(a)$, called the Frobenius-Perron dimension of $a$,
which is greater than or equal to, in absolute value, any other eigenvalues.
In the anyon model, we also call $\dim(a)$ the quantum dimension of $a$. To
get a sense of what $\dim(a)$ measures, consider the dimension of the space of
$n$ type-$a$ anyons with total type $\mathbf{1}$. We use the splitting-tree
basis in Figure 7 with $a=a_{1}=\cdots=a_{n},\ \mathbf{1}=c$ to compute the
dimension for large $n$,

$\displaystyle\sum\limits_{b_{1},\cdots,b_{n-2}}N_{aa}^{b_{1}}N_{b_{1}a}^{b_{2}}\cdots
N_{b_{n-3}a}^{b_{n-2}}N_{b_{n-2}a}^{\mathbf{1}}$
$\displaystyle=\sum\limits_{b_{1},\cdots,b_{n-3}}N_{aa}^{b_{1}}N_{b_{1}a}^{b_{2}}\cdots
N_{b_{n-3}a}^{\bar{a}}$
$\displaystyle=\left((N_{a})^{n-2}\right)_{a\bar{a}}\quad\overset{n\rightarrow\infty}{\sim}\quad\dim(a)^{n-2}.$

Thus, $\dim(a)$ measures the asymptotic size of the space of $n$ type-$a$
anyons. Apparently, $\dim(a)\geq 1$. An anyon $a$ is called _Abelian_ if
$\dim(a)=1$, and _non-Abelian_ otherwise.

$S$-matrix. Define the $|L|\times|L|$ modular $S$-matrix with entries,

$S_{ab}:=\theta_{a}^{-1}\theta_{b}^{-1}\sum_{c\in L}N_{\bar{a}b}^{c}\
\theta_{c}\dim(c).$

The $S$-matrix is required to be invertible.

To summarize, an anyon model or a unitary MTC is described by a label set,
fusion rule, $F$-symbols, $R$-symbols, and topological spins, from which one
can derive quantum dimensions and the $S$-matrix. These data should satisfy
various compatibility conditions as listed in this section.

##  Appendix B Mathematica code

Below we list the Mathematica code to compute some of the matrices used in
Section 3.2 and 3.3 including $F$, $\tilde{R}$, $A$, $B$, and $W$.

⬇

1

2Clear[q];

3QuantumInteger[n_] := (q^(n/2) - q^(-n/2))/(q^(1/2) - q^(-1/2));

4F = FullSimplify[{{-1,

5 Sqrt[QuantumInteger[3]]}, {Sqrt[QuantumInteger[3]], 1}}/

6 QuantumInteger[2]];

7Rtilde = DiagonalMatrix[{q^(-1/2), -q^(1/2)}]*I;

8A = Simplify[

9 MatrixPower[Rtilde, 2] . F . MatrixPower[Rtilde, 4] . F];

10B = Simplify[

11 MatrixPower[Rtilde, 2] . F . MatrixPower[Rtilde, 6] . F];

12W = Simplify[A . B . Inverse[A] . Inverse[B]];

Acknowledgment.  S.X.C is partly supported by NSF grant CCF-2006667, Quantum
Science Center (led by ORNL), and ARO MURI.

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