# Optimized Topological Quantum Compilation of Three-Qubit Controlled Gates in
the Fibonacci Anyon Model: A Controlled-Injection Approach

Abdellah Tounsi  Constantine Quantum Technologies,
Frères Mentouri University Constantine 1, Ain El Bey Road, Constantine, 25017,
Algeria  Laboratoire de Physique Mathématique et Subatomique, Frères Mentouri
University Constantine 1, Ain El Bey Road, Constantine, 25017, Algeria  Nacer
Eddine Belaloui  Constantine Quantum Technologies,
Frères Mentouri University Constantine 1, Ain El Bey Road, Constantine, 25017,
Algeria  Laboratoire de Physique Mathématique et Subatomique, Frères Mentouri
University Constantine 1, Ain El Bey Road, Constantine, 25017, Algeria
Mohamed Messaoud Louamri  Constantine Quantum Technologies,
Frères Mentouri University Constantine 1, Ain El Bey Road, Constantine, 25017,
Algeria  Theoretical Physics Laboratory, University of Science and Technology
Houari Boumediene, BP 32 Bab Ezzouar, Algiers, 16111, Algeria  Achour Benslama
Constantine Quantum Technologies,
Frères Mentouri University Constantine 1, Ain El Bey Road, Constantine, 25017,
Algeria  Laboratoire de Physique Mathématique et Subatomique, Frères Mentouri
University Constantine 1, Ain El Bey Road, Constantine, 25017, Algeria
Mohamed Taha Rouabah<EMAIL_ADDRESS>Constantine Quantum
Technologies,
Frères Mentouri University Constantine 1, Ain El Bey Road, Constantine, 25017,
Algeria  Laboratoire de Physique Mathématique et Subatomique, Frères Mentouri
University Constantine 1, Ain El Bey Road, Constantine, 25017, Algeria

###### Abstract

A method, termed controlled-injection, is proposed for compiling three-qubit
controlled gates within the non-abelian Fibonacci anyon model. Building on
single-qubit compilation techniques with three Fibonacci anyons, the approach
showcases enhanced accuracy and reduced braid length compared to the
conventional decomposition method for the controlled three-qubit gates. This
method necessitates only four two-qubit gates for decomposition, a notable
reduction from the conventional five. In conjunction, the study introduces a
novel class of controlled three-qubit gates and conducts a numerical
simulation of the topological $i$Toffoli gate to validate the approach. In
addition, we propose an optimization method for single-qubit gate
approximation using novel algebraic relations and numerical methods, including
distributed computing.

##  I Introduction

In the previous two decades, there has been a significant interest in
topological quantum computation (TQC) due to its potential for scalable and
fault-tolerant quantum computing. The later is crucial for fully harnessing
the capabilities of quantum systems [1, 2, 3, 4, 5, 6, 7]. The foundation of
TQC relies on the existence of anyons, which were proposed by Jon Magne
Leinaas and Jan Myrheim in 1976 [8] and quantum mechanically formulated by
Frank Wilczek in 1982 [9]. In (2+1) space-time dimensions, particles can
exhibit anyonic statistics, in contrast to the (3+1) dimensions where
particles are either bosons or fermions. Consequently, the quantum evolution
of anyons moving around each other in an effective two-dimensional space is
determined by a topological phase that is independent of the system’s dynamics
and geometry in the same spirit of Aharonov-Bohm phase of an electron moving
around a confined magnetic flux [9]. Additionally, within the framework of
topological quantum field theory (TQFT), anyon models can be described using
braided monoidal category theories [10]. In this context, there exists a
correspondence between the topological invariants of knots and the quantum
observables of the anyonic system [11]. In theory, a class of anyons can be
described within the Chern-Simons quantum field theory. This theory is
celebrated because of its successful description of the fractional quantum
Hall effect (FQHE) [12]. Specifically, the $\text{SU}(2)$ non-abelian version
of this theory produces the $\text{SU}(2)_{k}$ anyon models, which include the
most famous Ising and Fibonacci (Yang-Lee) models for $k=2$ and $k=3$,
respectively. It is proven that $\text{SU}(2)_{k}$ anyon models are Turing-
complete for $k=3$ and $k\geq 5$ [13, 14], while the $\text{SU}(2)_{2}$ model
covers only the Clifford group. On the other hand, anyons can emerge in some
types of topological lattice models. In his seminal work, Kitaev proposed the
toric code and quantum doubles as quantum error correction codes that possess
topological ground states protected by an energy gap [2]. In this framework,
abelian and non-abelian anyons emerge as excitations and serve as quantum
information processing agents. Recently, a modified version of the surface
code has been simulated in a superconducting quantum processor, providing
evidence of the non-abelian statistics of Ising anyons [15]. At the same time,
a quantum double code based on the Dihedral group $D_{4}$ implemented on a
trapped-ion quantum processor demonstrated experimental observation of non-
abelian anyon statistics [16]. Moreover, the experimental results from various
condensed matter systems provide compelling evidence in support of the
realization of topological quantum computing systems in the near future.
Specifically, it has been shown experimentally that the quasi-particles in the
fractional quantum Hall systems exhibit anyonic statistics [17, 18].
Furthermore, a recent study suggests the possibility of detecting Majorana
zero modes in semiconductor-superconductor heterostructure devices [19].

However, the compilation of quantum circuits in the TQC framework is not
generally trivial. Although the Fibonacci anyon model ensures universality and
is dense in the special unitary group, significant efforts have been devoted
to achieving an efficient and optimal compilation scheme. Numerous algorithms
are available to map $\text{SU}(2)$ quantum gates to topological braiding
operations within Fibonacci anyons, including the Solovay-Kitaev algorithm
[20], the brute-force algorithm [7, 21], the evolutionary algorithm [22], and
deep reinforcement learning [23]. Additionally, an asymptotically optimal and
systematic algorithm based on ring theory has been specifically designed for
certain $\text{SU}(2)$ quantum gates with Fibonacci anyons [24]. Moreover, a
recently developed generic Monte Carlo approach has been introduced to compile
near-optimal braid words at the single-qubit level in $\text{SU}(2)_{k}$ anyon
models [25].

Additionally, it is necessary to compile two-qubit gates, such as the CNOT
gate, to ensure universality. In this context, several procedures have been
proposed. Specifically, an injection method was introduced by Bonesteel et.
al. [26] to compose two-qubit gates from single-qubit gates of Fibonacci
anyons. Another iterative procedure has been designed to systematically
generate entangling two-qubit Fibonacci anyon braids [27]. Eventhough the
compilation of any single-qubit gate along with CNOT gate is sufficient to
demonstrate the quantum computing universality of a given anyon model, the
implementation of larger quantum circuits may require reduced compilation
schemes to decrease processing time. Specifically, the Toffoli gate, or the
controlled-controlled NOT gate, can be decomposed by concatenation of five
two-qubit controlled gates [28]. However, the pertinent query revolves around
the feasibility of reducing this requisite. In this context, a general
compilation scheme based on dense encoding was proposed to construct
controlled-controlled phase gate for a wide range of anyon models using six
successive gates acting on three qubits densely encoded on eight anyons [29].
This method does not address a wide range of controlled-controlled gates such
as the CNOT and Deutsch gates. In this study, we present a novel procedure
that we call controlled-injection to construct a class of controlled three-
qubit gates within the Fibonacci anyon model. The controlled-injection method
offers the advantage of reducing the complexity of the braids compared to the
conventional decomposition method. We demonstrate our finding by comparing the
resulting $i$Toffoli gates using both approaches. To ensure an accurate
comparison using the most optimal compilation possible, we employ the brute-
force algorithm and implement several optimizations based on algebraic and
numerical techniques, given the computational complexity of the brute-force
algorithm. Furthermore, the controlled-injection method introduces unusual
quantum logic gates that can be used in specific applications.

This paper begins by introducing the fundamentals of quantum computing with
Fibonacci anyons, along with essential notations. We then delve into the key
aspects of compiling single-qubit gates within the Fibonacci model, using the
brute-force algorithm. Thereafter, we present the controlled injection method,
supplemented by a new class of controlled three-qubit gates. We conclude our
study with a comparative analysis between the newly introduced method and the
conventional decomposition approach.

##  II Fibonacci Model

The Fibonacci anyon model is the simplest universal anyon model among
$\text{SU}(2)_{k}$ family. Fibonacci anyons can theoretically emerge as
quasiparticles in the FQHE at $12/5$ filling factor [30]. They can also appear
as defects in the so called string-net (Levin-Wen) lattice models [31, 32]. In
the Fibonacci model, we have only one non-trivial anyonic charge, referred to
as 1, along with the trivial vacuum charge, 0. The non-trivial Fibonacci
fusion rule is defined as follows [33, 21, 34]:

$\displaystyle\textbf{\scriptsize{1}}\times\textbf{\scriptsize{1}}=\textbf{\scriptsize{0}}+\textbf{\scriptsize{1}}.$
(1)

The dimension of the fusion space $\mathcal{F}$ in terms of the number of
Fibonacci anyons follows the famous Fibonacci series such that
$\dim(\mathcal{F}_{n+2})=\dim(\mathcal{F}_{n})+\dim(\mathcal{F}_{n+1})$, where
$n$ is the number of anyons, as shown in Table 1. Notice that a qubit can be
represented minimally by three or four anyons. While two anyons introduce a
trivial braid operation, three anyons form a qubit with overall charge 1 and
correspond to the minimum qubit representation with Fibonacci anyons. Four
anyons with an overall charge 0 form also a qubit. Representing a qubit with
more than four anyons is not recommended since it provokes leakage, as proven
for all anyon models [35].

To compute the matrix representation of the braid generators of a given set of
anyons, the fusion and rotation matrices must be computed. Solving the
pentagon identities reveals one non-trivial fusion matrix $F$ in Fibonacci
model given by [33, 4]:

$\displaystyle
F^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}$
$\displaystyle=\begin{pmatrix}(F^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}})^{\textbf{\scriptsize{0}}}_{\textbf{\scriptsize{0}}}&(F^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}})^{\textbf{\scriptsize{0}}}_{\textbf{\scriptsize{1}}}\\\
(F^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}})^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{0}}}&(F^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}})^{\textbf{\scriptsize{1}}}_{\textbf{\scriptsize{1}}}\end{pmatrix}=\begin{pmatrix}\frac{1}{\phi}&\frac{1}{\sqrt{\phi}}\\\
\frac{1}{\sqrt{\phi}}&-\frac{1}{\phi}\end{pmatrix},$ (2)

where $\phi$ is the golden ratio. Then, solving the hexagon identities gives
the right-handed and the left-handed solutions of the rotation matrix $R$ [33,
4]:

$\displaystyle R_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}$
$\displaystyle=\begin{pmatrix}R_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}^{\textbf{\scriptsize{0}}}&R_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}^{\textbf{\scriptsize{0}}\textbf{\scriptsize{1}}}\\\
R_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}^{\textbf{\scriptsize{1}}\textbf{\scriptsize{0}}}&R_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}^{\textbf{\scriptsize{1}}}\end{pmatrix}=\begin{pmatrix}e^{\pm
i4\pi/5}&0\\\ 0&-e^{\pm i2\pi/5}\end{pmatrix}.$ (3)

Here, the $(+)$ and $(-)$ signs in the exponents refer to right-handed and
left-handed twisting respectively. The off-diagonal components of
$R_{\textbf{\scriptsize{1}}\textbf{\scriptsize{1}}}$ are null because
exchanging two particles preserves the fusion outcome.

Braiding, a fundamental operation in anyon models, consists of exchanging two
adjacent anyons. The braiding operation between the $n$-th and $(n+1)$-th
particles is denoted by the $\sigma_{n}$ braid operator. In this work, all the
necessary braid operators matrix representations are calculated using a
systematic numerical method developed in [36]. Fortunately, the braid operator
$\sigma_{3}$ in the case of four anyons is identical to the operator
$\sigma_{1}$. That implies that all single-qubit gates acting on three anyons
yield the same evolution when applied to four anyons.

Table 1:  This table shows the size of the fusion space $\mathcal{F}_{n}$ in function of the number of anyons $n$. $\dim(\mathcal{F}_{n}^{\textbf{\scriptsize{0}}})$ is the dimension of the fusion space given that the total charge is 0, while $\dim(\mathcal{F}_{n}^{\textbf{\scriptsize{1}}})$ is the size of the fusion space given that the overall charge is 1. Finally, $\dim(\mathcal{F}_{n})$ is the total size of the fusion space. $n$ | $\dim(\mathcal{F}_{n}^{\textbf{\scriptsize{0}}})$ | $\dim(\mathcal{F}_{n}^{\textbf{\scriptsize{1}}})$ | $\dim(\mathcal{F}_{n})$
---|---|---|---
1 | 0 | 1 | 1
2 | 1 | 1 | 2
3 | 1 | 2 | 3
4 | 2 | 3 | 5
$n$ | $\sum_{i=1,2}\dim(\mathcal{F}_{n-i}^{\textbf{\scriptsize{0}}})$ | $\sum_{i=1,2}\dim(\mathcal{F}_{n-i}^{\textbf{\scriptsize{1}}})$ | $\sum_{i=1,2}\dim(\mathcal{F}_{n-i})$

The dimension of the fusion space of anyons differs from that of the Hilbert
space of qubits. For instance, simulating two qubits requires a fusion space
of size four or larger. In the context of Fibonacci models, two practical
options arise. The first is to use five anyons prepared with an overall charge
of 1, resulting in a fusion space of five dimensions. The second is to
allocate three anyons to each qubit, resulting in five and eight fusion states
depending on whether the overall charge is 0 or 1, respectively. The former
option, termed dense encoding, is considered optimal in terms of resource
utilization. The latter, called sparse encoding, is compatible with the
picture of quantum circuits decomposed into separate qubits. A general scheme
for gate compilation within the dense encoding framework is explored in [29].
However, in this work, we opt for sparse encoding, aligning with the
mainstream [26, 37, 7]. Additionally, this study demonstrates several
advantages of this convention.

##  III Compiling Single-Qubit Gates

The algebraic group generated by Fibonacci braiding operations is densely
mapped to the $\text{SU}(2)$ group and is polynomially equivalent to a quantum
circuit [13]. In the context of three Fibonacci anyons, the matrix
representation of any Fibonacci braid sequence $\hat{B}$ assumes a general
form [37]:

$\displaystyle B$
$\displaystyle=\left(\begin{array}[]{cc|c}\lx@intercol\hfil\hbox{\multirowsetup$\pm\text{e}^{-iW(\hat{B})\pi/10}$
[\text{SU}(2)]}\hfil\lx@intercol\vrule\lx@intercol&\\\
\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\\\
\hline\cr&&\text{e}^{i3W(\hat{B})\pi/5}\end{array}\right),$ (7)

where $W(\hat{B})$, the winding number, is defined as the sum of the powers of
the braid sequence. Consequently, the global phase of $B$ in both blocks is
determined solely by the winding number, up to a $\pm$ sign. While the global
phase factor may not be of significant importance when compiling single-qubit
gates, it manifests measurable effects in the design of two and three-qubit
controlled gates. Furthermore, the phase difference between the two
independent sectors of fusion charges 1 and 0 gains relevance when executing
mixing operations between two qubits [37]. As a result, it is advisable to
employ a global phase-independent distance metric, given that the formula in
Eq. (7) precisely determines the global phase.

It has been established that the weave group, which is a subgroup of the braid
group, is also dense to $\text{SU}(2)$ [38]. Weaves are defined as sequences
of braids wherein a single anyon navigates around other stationary anyons. In
diagrammatic terms, there exists a solitary warp strand that weaves around the
remaining weft strands. The primary objective of this method is to simplify
the implementation of braiding circuits. In scenarios involving three anyons,
weaving sequences are restricted to braiding operations with even powers.

In scholarly discourse, numerous algorithms have been proposed to identify the
most accurate braid sequence of a certain length that approximates a targeted
unitary gate. The Solovay-Kitaev algorithm is a seminal method in this
context, as it demonstrates that an approximation of a given quantum gate with
a braid sequence up to the desired level of accuracy can be achieved
efficiently in polylogarithmic time [37]. As per the Solovay-Kitaev theorem,
the relationship between the error $\epsilon$ and the braiding length $L$ is
as follows:

$\displaystyle L=\text{poly}(\log(1/\epsilon)).$ (8)

Here, $\epsilon$ is proportional to any properly defined distance metric
between the braid unitary matrix and the target unitary matrix in the space of
unitary matrices. In this study, we utilize the spectral distance metric [26,
39] to quantify the error $\epsilon$:

$\displaystyle\mathcal{D}(U_{1},U_{2})=\sqrt{\text{maxEigenvalue}(AA^{\dagger})},$
(9)

such that $A$ is the difference between $U_{1}$ and $U_{2}$ after eliminating
the global phases.

Despite the fact that the brute-force algorithm imposes an exponential demand
on computational resources, it yields an optimal fundamental approximation.
This suffices to provide a proof of concept for the gates introduced in
subsequent sections. To render the brute-force algorithm more practical, its
performance has been augmented with a number of optimizations. Firstly, for
the optimization of linear calculations, it is advantageous to map the
$\text{SU}(2)$ components of the elementary braid matrices (7) to quaternions.
This approach is computationally more efficient as it reduces the number of
parameters and necessitates fewer real number multiplications. Secondly, the
cyclicity of the braid matrix powers should be taken into account.
Specifically, $\sigma_{1}^{10}=\sigma_{2}^{10}=\text{I}$. As a result, in the
context of weaves, it is adequate to span only sequences of the form
$\sigma_{i}^{p}$, where $i=1,2$ and $p=2,4,6,8$. Thirdly, the braid group
enforces algebraic relations that can effectively reduce the search space.
This is achieved by leveraging the similarity relation that associates each
braid sequence with its counterpart, wherein $\sigma_{1}$ and $\sigma_{2}$ are
permuted. Namely,

$\displaystyle\sigma_{1}^{p_{n}}\sigma_{2}^{q_{n}}\cdots\sigma_{1}^{p_{1}}\sigma_{2}^{q_{1}}=\Gamma^{\dagger}\sigma_{2}^{p_{n}}\sigma_{1}^{q_{n}}\cdots\sigma_{2}^{p_{1}}\sigma_{1}^{q_{1}}\Gamma,$
(10)

where $p_{i}$ and $q_{i}$ are integers and
$\Gamma=\sigma_{1}\sigma_{2}\sigma_{1}$. The physical interpretation of the
similarity relation, along with its proof, are elaborated in Appendix A.
Furthermore, we illustrate in Appendix B that if the target gate is Hermitian,
it suffices to span merely half of the search space. In essence, a Hermitian
target gate $H$ introduces a symmetry in the metric space since for each braid
matrix $B$:

$\displaystyle\mathcal{D}(B,H)=\mathcal{D}(B^{\dagger},H).$ (11)

From a computational perspective, the utilization of a high-performance
programming language can lead to substantial savings in terms of computational
time, energy, memory, and development duration 111In this study, the C++
language, recognized as one of the contemporary high-performance languages
[45], is employed.. Furthermore, a linear acceleration is achievable by
partitioning the search space across available processing units. Collectively,
all algebraic and numerical optimizations significantly reduced the search
time.

##  IV Compiling topological Conditional Three-Qubit Gates using controlled-
injection method

Figure 1: General scheme of the proposed three-qubit gate by the controlled-
injection method. The two upper qubits are the controlling qubits, while the
lower qubit is the controlled qubit. The $R$ operation prepares the
controlling qubits while the $S$ operation is the target operation which is
applied whenever the injected four anyons have a non-trivial overall
topological charge.

We introduce a class of quantum three-qubit gates $\mathbb{M}(R,S)$ that
perform controlled logic operations inherently compatible with the structure
of the Fibonacci model. We call this method as the controlled-injection
method. As shown in Fig. 1, the general form of this class of three-qubit
gates encodes qubits in four anyon groups. This convention is necessary for
this method and is the simplest encoding whose topological charge 0 sector has
two dimensions. The topological sector of charge 0 serves the purpose of the
exact implementation of the SWAP gate since the braiding of two 0 charges is
trivial. The latter is frequently needed to implement controlled gates between
non-adjacent qubits. In addition, the introduced $\mathbb{M}(R,S)$ three-qubit
gate involves three-anyon gates labeled $R,I$ and $S$ which act as
$\text{SU}(2)\oplus\text{U}(1)$ operations, where the $\text{SU}(2)$ part acts
on the topological charge 1 sector while U(1) part acts on the topological
sector of charge 0 as shown in Eq. (7). We compile these three-anyon gates by
weaving only one input anyon as explained in Section III. However, we are
allowed to weave pairs and groups of anyons keeping in mind only the fusion
outcomes of each group. In this case, the three-anyon gates will affect only
the fusion state of the fusion outcomes regardless to the constituent anyons.
As shown in Fig. 1, the gate $R$ takes pairs of anyons as inputs while weaving
only the upper pair, the gate $I$ injects the upper grouped two pairs in
yellow and red colors into the lower strand, and the $S$ gate weaves the upper
grouped two pairs returning them to the same strand position. In general, the
$\mathbb{M}(R,S)$ gate works in such a way that the $R$ gate prepares the
controlling qubits, so we call it the initialization gate, while the $S$ gate
is the target operation to be applied on the controlled qubit, then, we call
it the target gate. We call the intermediary identity gate the injection gate.
To understand precisely the mechanism of this gate, we will introduce two
variations of the controlled-injection three-qubit gates: $\mathbb{M}(I,S)$
and $\mathbb{M}(\text{NOT},S)$ whose initialization gates are the identity and
the NOT gate respectively.

###  IV.1 The $\mathbb{M}(I,S)$ controlled-injection gate

Let us consider the initialization gate to be the identity operation i.e
$R=I$. Notice that in the case of a four anyons qubit of overall topological
charge 0, the two constituent anyon pairs fuse simultaneously to 0 if the
state is $\ket{0}$:

$\displaystyle\ket{0}=\ket{((\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{0}}},(\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{0}}})_{\textbf{\scriptsize{0}}}},$
(12)

and fuse simultaneously to 1 if the state is $\ket{1}$:

$\displaystyle\ket{1}=\ket{((\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}},(\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}})_{\textbf{\scriptsize{0}}}}.$
(13)

Therefore, the identity initialization gate will have no effect but injecting
the lower pair of anyons from the upper qubit into the middle qubit. If one
and only one qubit of the two controlling qubits is in the state $\ket{0}$,
the middle two pairs will have different topological charges with overall
topological charge 1. In contrast, when the controlling qubits are both in the
state $\ket{0}$ or both in the state $\ket{1}$, the middle two pairs will have
the same topological charges, and since the initialization gate approximates
the identity in $\mathbb{M}(I,S)$, the overall topological charge of the
middle two pairs will remain 0. We summarize the truth table of this gate in
Tab. 2, where the target gate $S$ is applied if the controlling qubits have
opposite logic states. As a result, the exact matrix representation of
$\mathbb{M}(I,S)$ should be given by:

$\displaystyle\left(\begin{array}[]{cc|cc|cc|cc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ \hline\cr
0&0&\lx@intercol\hfil\hbox{\multirowsetup\huge$S$}\hfil\lx@intercol\vrule\lx@intercol&0&0&0&0\\\
0&0&&&0&0&0&0\\\ \hline\cr
0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup\huge$S$}\hfil\lx@intercol\vrule\lx@intercol&0&0\\\
0&0&0&0&&&0&0\\\ \hline\cr 0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&1\end{array}\right)$ (22)

In conclusion, the $\mathbb{M}(I,S)$ gate can be represented as the
application of three consecutive two-qubit controlled gates. Specifically, it
can be expressed as follows:

$\displaystyle\mathbb{M}(I,S)$ $\displaystyle=\left(\text{CNOT}\otimes
I\right)\left(I\otimes\text{CS}\right)\left(\text{CNOT}\otimes I\right).$ (23)

This representation provides a clear understanding of the operation of the
$\mathbb{M}(I,S)$ gate in terms of standard quantum gates.

Table 2:  Truth table of $\mathbb{M}(I,S)$ gate. Input | Output
---|---
qubit 1 | qubit 2 | qubit 3 | qubit 1 | qubit 2 | qubit 3
$\ket{0}$ | $\ket{0}$ | $\ket{\eta}$ | $\ket{0}$ | $\ket{0}$ | $\ket{\eta}$
$\ket{0}$ | $\ket{1}$ | $\ket{\eta}$ | $\ket{0}$ | $\ket{1}$ | $S\ket{\eta}$
$\ket{1}$ | $\ket{0}$ | $\ket{\eta}$ | $\ket{1}$ | $\ket{0}$ | $S\ket{\eta}$
$\ket{1}$ | $\ket{1}$ | $\ket{\eta}$ | $\ket{1}$ | $\ket{1}$ | $\ket{\eta}$

####  IV.1.1 Numerical simulation of $\mathbb{M}(I,\text{NOT})$

In practice, the initialization, injection, and target gates are approximated
to a predetermined braid length. This approximation process inevitably leads
to a degree of inaccuracy and leakage. To assess the efficacy of the
$\mathbb{M}(I,\text{NOT})$ gate, we conducted a numerical simulation with the
target gate chosen as the typical NOT gate, specifically $\pm iX$, the special
unitary version of the NOT gate. The initial step involves identifying the
weave approximation of the $\pm\text{I}$ gate for the initialization and
injection gates, under the stipulation that the weft strand’s initial tip
takes the upper rank while its end tip takes the lower rank. Concurrently, we
search for the weave approximation of the $S$ gate, which acts on three
strands, weaving the upper strand without changing its final rank. Through the
optimized brute-force approach, we derived relevant weaving sequences by
setting a fixed braid length of 48 braid operators and accounting for global
phases. It is noteworthy that there exists a multitude of weaving sequences
that exhibit optimal accuracy. In our numerical implementation, we select the
weaving sequences that combine to provide the best accuracy for the
$\mathbb{M}(I,\pm iX)$ gate. The sequences pertinent to this case are
delineated in Table 3.

Table 3:  Braid sequence and accuracy of the necessary three-strand gates to approximate the $\mathbb{M}(I,iX)$ gate with an overall error of $6.64\times 10^{-4}$ and leakage $\epsilon_{L}=3.26\times 10^{-6}$. $\mathbb{M}(R,S)$ Gates |  $\mathbb{M}(I,iX)$ Gates | Weave Sequence | Length | Error
---|---|---|---|---
R | I | $\sigma_{1}^{-1}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{-2}\sigma_{1}^{-4}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{4}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{2}^{-1}$ | 48 | $1.51\times 10^{-3}$
I | I | $\sigma_{1}^{1}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{4}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-1}$ | 48 | $1.51\times 10^{-3}$
S | $iX$ | $\sigma_{2}^{1}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{2}^{-1}$ | 48 | $8.55\times 10^{-4}$

The subsequent phase involves the conversion of these three-strand gates into
a six-strand braiding circuit, as depicted in Fig. 1. Ultimately, employing
the relevant matrix representations of the five elementary braid operations, a
systematic numerical method developed in [36] is utilized to obtain the
representation of the approximated $\mathbb{M}(I,iX)$ gate, as illustrated in
Fig. 2. The accuracy is quantified by calculating the error using the distance
metric defined in Eq. (9). The overall error of this $\mathbb{M}(I,iX)$
approximation is computed as $6.64\times 10^{-4}$. The error measured solely
on the controlled gate (the $S\equiv\pm iX$ block in the $\mathbb{M}(I,iX)$
matrix) is approximately $6.64\times 10^{-4}$. This value is commensurate with
the error computed on the $\pm iX$ approximation itself. The overall error is
comparable to the error on the target because the $\mathbb{M}(R,S)$ is always
accurate when the controlled gate is not applied since trivial braids are
involved, and the target gate contributes significantly to the inaccuracy.

The leakage error, denoted as $\epsilon_{L}$, can also be computed. This error
represents the quantity of information that the simulated gate inevitably
transmits to non-computational states. It is calculated using a modified form
of the spectral distance. Specifically, the leakage induced by a given gate
$U$ of $d$ dimensions is defined as:

$\displaystyle\epsilon_{L}=1-\sqrt{\text{minEigenvalue}(UU^{\dagger})}.$ (24)

In this equation, the second term yields the minimum factor by which the
matrix $U$ can alter the norm of a quantum state [7]. The leakage amount
measured on the simulated $\mathbb{M}(I,iX)$ gate is approximately $3.26\times
10^{-6}$. It is important to note that the initialization and injection gates
are anticipated to be primarily responsible for this leakage of information.

(a)

(b)

(c)

Figure 2: (a) Matrix representation of the approximated $\mathbb{M}(I,iX)$ in
the computational basis. The full braid matrix is $89\times 89$ in the sector
of charge 0 while the sector 1 is irrelevent because the overall charge cannot
be 1. (b) Exact $\mathbb{M}(I,iX)$ matrix representation. (c) The RGB color
map is designed to map each complex number, with an amplitude less than one,
to a specific color. This mapping process involves a linear transformation of
the radial component into spectral colors. Additionally, the amplitude is
mapped into luminosity in the vicinity of zero and into a saturation profile
near the identity, following a Gaussian distribution.

###  IV.2 The $\mathbb{M}(\text{NOT},S)$ controlled-injection gate

Let us now examine the scenario where the initialization gate assumes the role
of the NOT gate, i.e., $R=\pm iX$. The complex phase $\pm i$ is essential for
maintaining gates in the $\text{SU}(2)$ group and preventing complex phases in
the sector 1, as illustrated in Eq. 7. To comprehend the effect of the
initialization gate, one must revisit how the NOT gate operates on the fusion
states of three Fibonacci anyons [36]. When both controlling gates are in the
state $\ket{1}$, the initialization gate reverses the overall topological
charge of the middle two pairs from the charge 0 to the charge 1:

$\displaystyle\text{NOT}\ket{((\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{0}}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}}}=\ket{((\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}}},$
(25)

since the fusion states
$\ket{((\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{0}}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}}}$
and
$\ket{((\textbf{\scriptsize{1}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}},\textbf{\scriptsize{1}})_{\textbf{\scriptsize{1}}}}$
are the only logic states in case of three fibonacci anyons in the sector of
charge 1. In other cases, the action of the initialization gate is analogous
to the $R=I$ case and is trivial. Therefore, the target gate will only be
operational when neither of the controlling qubits are in the $\ket{0}$ state.
The truth table of $\mathbb{M}(iX,S)$ should align with that shown in Tab. 4.
Consequently, the relevant matrix representation of the $\mathbb{M}(NOT,S)$ is
as follows:

$\displaystyle\left(\begin{array}[]{cc|cc|cc|cc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ \hline\cr
0&0&\lx@intercol\hfil\hbox{\multirowsetup\huge$S$}\hfil\lx@intercol\vrule\lx@intercol&0&0&0&0\\\
0&0&&&0&0&0&0\\\ \hline\cr
0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup\huge$S$}\hfil\lx@intercol\vrule\lx@intercol&0&0\\\
0&0&0&0&&&0&0\\\ \hline\cr
0&0&0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup\huge$S$}\hfil\lx@intercol\\\
0&0&0&0&0&0&&\end{array}\right)$ (34)

Table 4:  Truth table of the $\mathbb{M}(iX,S)$ Gate. Input | Output
---|---
qubit 1 | qubit 2 | qubit 3 | qubit 1 | qubit 2 | qubit 3
$\ket{0}$ | $\ket{0}$ | $\ket{\eta}$ | $\ket{0}$ | $\ket{0}$ | $\ket{\eta}$
$\ket{0}$ | $\ket{1}$ | $\ket{\eta}$ | $\ket{0}$ | $\ket{1}$ | $S\ket{\eta}$
$\ket{1}$ | $\ket{0}$ | $\ket{\eta}$ | $\ket{1}$ | $\ket{0}$ | $S\ket{\eta}$
$\ket{1}$ | $\ket{1}$ | $\ket{\eta}$ | $\ket{1}$ | $\ket{1}$ | $S\ket{\eta}$

####  IV.2.1 Numerical simulation of $\mathbb{M}(\text{NOT},\text{NOT})$

As an illustrative instance, we consider $S=\pm iX$ and set the length of the
three-anyon braid gates to 48. The initial step involves identifying the
compilation of $\pm iX$ and $\pm I$, ensuring that the weaving process
commences from the upper strand and concludes at the lower strand.
Subsequently, we seek an approximate weave for the $S=\pm iX$ gate, which
operates on three strands, initiating the weaving process from the upper
strand and terminating at the same strand. Utilizing a an optimized brute-
force approach, we derive the necessary weave sequences, as presented in Table
5. These sequences are procured by spanning all combinations of the optimal
individual braid sequences to enhance the accuracy of the $\mathbb{M}(iX,iX)$
gate.

Table 5: Braid approximation of the necessary three-strand gates to compile the $\mathbb{M}(iX,iX)$ gate with an overall error of $6.64\times 10^{-4}$ and leakage $\epsilon_{L}=3.99\times 10^{-6}$. $\mathbb{M}(R,S)$ Gates |  $\mathbb{M}(iX,iX)$ Gates | Weave sequence | Length | Error
---|---|---|---|---
R | $iX$ | $\sigma_{1}^{-1}\sigma_{2}^{2}\sigma_{1}^{-4}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{1}$ | 48 | $8.55\times 10^{-4}$
I | I | $\sigma_{1}^{1}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{4}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-1}$ | 48 | $1.51\times 10^{-3}$
S | $iX$ | $\sigma_{2}^{1}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{2}^{-1}$ | 48 | $8.55\times 10^{-4}$

The matrix representation of the simulated $\mathbb{M}(iX,iX)$ gate is
depicted in Fig. 3. The computed error distance of the braid approximation is
found to be $6.64\times 10^{-4}$. However, the error of the controlled gate in
the level of $S$ blocks is approximately $6.644\times 10^{-4}$ if the
controlling qubits are either in the $\ket{01}$ state or in the $\ket{10}$
state, while it is computed to be $6.637\times 10^{-4}$ if the controlling
qubits are in the $\ket{11}$ state. The amount of leakage is around
$3.99\times 10^{-6}$.

(a)

(b)

Figure 3:  (a) Matrix representation of the approximated $\mathbb{M}(iX,iX)$
as expressed in the basis of the computational states. (b) Exact
$\mathbb{M}(iX,iX)$ matrix representation. The color map is depicted in Fig.
2(c).

###  IV.3 Controlled-Controlled-S gate with controlled-injection method

Figure 4: To compile the CCS gate with fewer braids, it is sufficient to add
NOT gates, replaced by $iX$ and its Hermitian conjugate, and $S$ gate to the
previously defined $\mathbb{M}(iX,S^{\dagger})$ three-qubit gate.

A direct application of the topological gates previously introduced is the
construction of the controlled-controlled-S (CCS) gate for all unitary $S$ up
to the global phase determined by Eq. (7). The operationality of the target
gate $S$ of this controlled-controlled gate is contingent upon both
controlling qubits being in the state $\ket{1}$. The implementation of the CCS
gate implies the execution of the Deutsch gates $\mathbb{D}(\theta)$, which
are intrinsically universal [41]. The Deutsch gate is characterized by its
target gate $S$, defined as:

$\displaystyle S(\theta)$
$\displaystyle=\begin{pmatrix}\cos{\theta}&-i\sin{\theta}\\\
-i\sin{\theta}&\cos{\theta}\end{pmatrix}$ (35)

for any angle $\theta$. A notable gate within this class is the Toffoli gate,
renowned for its ability to compute any arbitrary Boolean function, thereby
qualifying it as a universal reversible logic gate [42].
One way to compile the CCS gate is to combine $\mathbb{M}(I,S^{\dagger})$ and
$\mathbb{M}(iX,S)$ since:

CCS $\displaystyle=\mathbb{M}(I,S^{\dagger})\mathbb{M}(iX,S)$ (36)
$\displaystyle=\mathbb{M}(iX,S)\mathbb{M}(I,S^{\dagger}).$ (37)

However, it is possible to construct a CCS gate with a single implementation
of the $\mathbb{M}(R,S)$ gate. Initially, it should be noted that if we apply
the $S$ gate on the target qubit prior to applying the
$\mathbb{M}(iX,S^{\dagger})$ gate, we should obtain a controlled-controlled
gate which operates exclusively when both of the controlling qubits are in the
$\ket{0}$ state. Consequently, to produce the desired CCS gate, NOT gates
should be applied on the controlling qubits both before and after the
$\mathbb{M}(iX,S)$ is applied. Given that such NOT gates occur symmetrically,
we can set them to be $\pm iX$ then $\mp iX$ espectively, to eliminate the
additional phase factor on the controlling qubits. The illustration of the CCS
braid circuit is depicted in Fig. 4. The corresponding truth table is detailed
in Tab. 6. Therefore, the resulting matrix representation should be as
follows:

$\displaystyle\left(\begin{array}[]{cc|cc|cc|cc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ \hline\cr 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ \hline\cr
0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ \hline\cr
0&0&0&0&0&0&\lx@intercol\hfil\hbox{\multirowsetup\huge$S$}\hfil\lx@intercol\\\
0&0&0&0&0&0&&\end{array}\right)$ (46)

Table 6:  Truth table of $CCS$ Gate. Input | Output
---|---
qubit 1 | qubit 2 | qubit 3 | qubit 1 | qubit 2 | qubit 3
$\ket{0}$ | $\ket{0}$ | $\ket{\eta}$ | $\ket{0}$ | $\ket{0}$ | $\ket{\eta}$
$\ket{0}$ | $\ket{1}$ | $\ket{\eta}$ | $\ket{0}$ | $\ket{1}$ | $\ket{\eta}$
$\ket{1}$ | $\ket{0}$ | $\ket{\eta}$ | $\ket{1}$ | $\ket{0}$ | $\ket{\eta}$
$\ket{1}$ | $\ket{1}$ | $\ket{\eta}$ | $\ket{1}$ | $\ket{1}$ | $S\ket{\eta}$

####  IV.3.1 Numerical simulation of the controlled-controlled $\pm iX$ gate

To obtain a special unitary version of the Toffoli gate, let us consider
$S=\pm iX$. Initially, we should identify the compilation of $\pm\text{I}$ and
$\pm iX$ such that the weaving process commences from the upper strand and
concludes at the lower strand. We should also find the target $S=\pm iX$ gate
acting on three strands, initiating the weaving process from the upper strand
and terminating at the same strand. By employing a brute-force approach, we
derived the weaving sequences, as explicitly presented in Table 7. It is
important to note that these sequences are not the only possible optimal
sequences. However, their combination yields the most accurate $\pm i$Toffoli
gate.

Table 7:  Braid approximation of the necessary three-strand gates to build the
$\pm i$Toffoli gate with overall error of $2.07\times 10^{-3}$ and leakage
$\epsilon_{L}=1.62\times 10^{-6}$

.  CCS Gates $\pm i$Toffoli Gates Weave sequence Length Error R $iX$
$\sigma_{1}^{1}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{2}^{1}$
48 $8.55\times 10^{-4}$ I I
$\sigma_{1}^{1}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{4}\sigma_{2}^{4}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{2}\sigma_{2}^{-1}$
48 $1.51\times 10^{-3}$ S $iX$
$\sigma_{2}^{1}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{2}^{-1}$
48 $8.55\times 10^{-4}$ NOT $iX$
$\sigma_{2}^{1}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{2}^{-1}$
48 $8.55\times 10^{-4}$

Subsequently, the three-strand gates should be translated into a six-strand
braiding circuit, as depicted in Fig. 1. The relevant braid matrices are then
calculated, following the method employed in previous sections. Ultimately, we
obtain the representation of the approximated $\mathbb{M}(I,iX)$ gate, as
illustrated in Fig. 5. The overall distance of the approximated $i$Toffoli
gate is approximately $2.07\times 10^{-3}$. However, the error solely on the
controlled gate is about $8.54\times 10^{-4}$. Lastly, the amount of leakage
reaches $1.62\times 10^{-6}$.

(a)

(b)

Figure 5: (a) Matrix representation of the approximated $i$Toffoli matrix in
the basis of the computational states. It represents $i$Toffoli up to
$1.02\times 10^{-4}$ distance error. (c) Exact $i$Toffoli matrix
representation. The color map is depicted in Fig. 2(c).

##  V Discussion

A distinctive feature of the CCS gate, compiled using the controlled-injection
method, is its requirement for fewer than five two-qubit gates. Specifically,
only four two-qubit gates are required: the initialization gates $R$ and
$R^{\dagger}$, and the injection gates $I$ and $I^{\dagger}$. At first glance,
this seems to contradict the theorem etablished in [28] which asserts that
five two-qubit gates are necessary to implement the Toffoli gate.
Nevertheless, it is crucial to acknowledge that this theorem relies on the
Hilbert space of qubits, whereas the fusion space of anyons exhibits a
distinct structure with additional dimensions. The non-computational sub-space
assumes a pivotal role in the functioning of initialization gates,
facilitating the formation of $\mathbb{M}(R,S)$ gates. In summary, the
inclusion of non-computational states makes it feasible to diminish the number
of necessary two-qubit gates for the construction of a controlled three-qubit
gate. Furthermore, it is instructive to examine the disparities between the
controlled-injection method and the standard decomposition method of CCS
gates, detailed in Appendix C. Primarily, it is worth noting that the logic
tables of $\mathbb{M}(R,S)$ gates can additionally contribute to the diversity
of quantum logic gates. Additionally, the performance of these two methods are
compared in terms of four aspects: length (number of elementary braids),
accuracy, leakage, and ease of implementation. A numerical comparative study
of Toffoli gate compilation using both methods is depicted in Tab. 8. In our
study, we chose the braid length to be 48 to give a reasonably acceptable
approximation of the target gates up to an order of magnitude of $10^{-3}$,
without requiring unreasonable computational resources and long braid
sequences.

Table 8:  Comparative table of the controlled-injection method and the decomposition method applied on the Toffoli CCX gate as shown in Section IV.3.1 and Appendix C respectively. The parameters considered in this comparison include the number of required three-anyon gate approximations, the total number of braids needed to compile the gate (referred to as the required braid length), and the number of sequential braids in each gate (referred to as the depth). Additionally, numerical accuracy and leakage were also evaluated. In this context, $L$ denotes the braid length of single three-anyon gates. For the purpose of this numerical study, $L$ was set to 48. The error statistics in the target gate and leakage values were computed over all possible combinations of the optimal weaves. | Decomposition | Controlled-Injection
---|---|---
Two-qubit gates | 7 | 4
Three-anyon gates | 3 | 3
Length | 30L+32 | 25L
Depth | 30L+32 | 22L
Best error | $1.90\times 10^{-3}$ | $2.07\times 10^{-3}$
Leakage of the best | $3.96\times 10^{-6}$ | $1.62\times 10^{-6}$
Avg error in target | $2.32\times 10^{-3}$ | $2.35\times 10^{-3}$
Min error in target | $1.76\times 10^{-3}$ | $8.54\times 10^{-4}$
Max error in target | $3.16\times 10^{-3}$ | $3.34\times 10^{-3}$
Avg leakage | $1.75\times 10^{-6}$ | $2.031\times 10^{-6}$
Min leakage | $3.93\times 10^{-7}$ | $3.59\times 10^{-7}$
Max leakage | $4.09\times 10^{-6}$ | $3.99\times 10^{-6}$

Indeed, the controlled-injection method exhibits a shorter compilation length
compared to the standard decomposition method. Assuming three-anyon gates of
identical lengths, i.e., compiled with the same number $L$ of braids and
consuming the same amount of time to search for the best compilation, the
standard decomposition method would require $6L$ braids for each controlled
two-qubit gate and $16$ for each SWAP gate, resulting in a total of $(30L+32)$
braids. In contrast, the controlled-injection method requires $4L$ braids for
each three-anyon gate, i.e., $20L$ braids for the $\mathbb{M}(iX,S)$ gate, and
an additional $5L$ braids to compile the extra single-qubit gates, yielding a
total of $25L$ braids. When comparing them according to depth, which is the
number of sequential braids, we get $(30L+32)$ for decomposition and $22L$ for
the controlled-injection method. Generally, depth is more significant in terms
of compilation time. Therefore, the controlled-injection method provides gates
that are at least $27\%$ times shorter. Secondly, the controlled-injection
method is as accurate as the standard decomposition method, or at least within
the same order of magnitude, given that the composing gates are compiled with
similar accuracies. Asymptotically, both methods should exhibit behavior
identical to that of the $i$Toffoli gate. This demonstrates the efficacy of
the controlled-injection method in terms of accuracy, making it a viable
alternative to the standard decomposition method. Thirdly, both methods were
found to yield a numerically similar amount of leakage. Generally, the
injection and initialization gates are the primary sources of leakage. Given
that we employ injection braids of the same accuracy and leakage in both
methods, a significant difference in the amount of leakage between the two
methods is not anticipated. This observation underscores the comparable
performance of the controlled-injection method and the standard decomposition
method in terms of leakage. Lastly, the implementation of the controlled-
injection method may present certain challenges, as it involves braids of
multiple anyons (four anyons simultaneously), as opposed to weaving a single
particle through the entire circuit. The manipulation of numerous anyons could
pose significant technological hurdles [38]. Conversely, the standard
decomposition can be partially transformed at the level of CNOT gates into a
weaving sequence of a single particle. This transformation is feasible due to
the $FPF^{-1}$ braid circuit introduced in [37]. Thus, while the controlled-
injection method offers certain advantages, its practical implementation may
require overcoming additional complexities.

##  VI Conclusion

In this study, we present an efficient topological quantum circuit model for
compiling controlled-controlled gates within the Fibonacci anyon model, a
versatile framework for universal quantum computing. We elevate the
conventional numerical brute-force method by incorporating novel algebraic
relations, such as similar braid sequences, and leveraging the symmetry by
Hermitian target gates. Complemented by numerical methods like distributed
computing, we achieve optimal approximations of single-qubit gates with three
Fibonacci anyons.

In the concluding phase, we present a class of conditional three-qubit gates
designed for approximating controlled logic operations, including fundamental
gates like Deutsch and Toffoli. These gates are seamlessly compatible with the
inherent structure of the Fibonacci model. A comparative analysis between the
conventional decomposition method of three-qubit gates and the novel
controlled-injection approach reveals that the latter allows for a more
concise compilation, achieving shorter lengths and depths in implementing
controlled-controlled gates, while preserving a comparable degree of accuracy
and leakage.

Remarkably, the introduced controlled three-qubit gates are decomposed into
four, rather than the conventional five, two-qubit gates, as stipulated by the
decomposition theorem. This reduction is made possible by harnessing the non-
computational topological states within the fusion space. The controlled-
injection method leverages on the unique logical attributes of the Fibonacci
anyon model, presenting a potential template for analogous strategies across
diverse anyon models or with an increased number of qubits.

##  Appendix A Similar braid sequences

The similarity relation between the braid sequences of three anyons starting
with $\sigma_{1}$ and $\sigma_{2}$ has a practical use in optimizing the
search for the best approximation of a given target unitary gate. The main
observation that leads to similar braid sequences is the fact that applying
$\sigma_{1}$ and $\sigma_{2}$ braid operations on the state
$\ket{((a,b)_{i},c)_{j}}$ of three anyons $a$, $b$ and $c$ is equivalent to
applying $\sigma_{2}$ and $\sigma_{1}$, respectively, on the state
$\ket{(c,(b,a)_{i})_{j}}$, as another observer may choose to look at the same
set of anyons from the opposite side. In other words, $\sigma_{1}$ transforms
to $\sigma_{2}$ when rotating the surface $180^{\circ}$. Namely,

$\displaystyle\left|\bra{((a,b)_{i^{\prime}},c)_{j}}\sigma_{1}\ket{((a,b)_{i},c)_{j}}\right|^{2}$
$\displaystyle=\left|\bra{(c,(b,a)_{i^{\prime}})_{j}}\sigma_{2}\ket{(c,(b,a)_{i})_{j}}\right|^{2}$
(47)
$\displaystyle\left|\bra{((a,b)_{i^{\prime}},c)_{j}}\sigma_{2}\ket{((a,b)_{i},c)_{j}}\right|^{2}$
$\displaystyle=\left|\bra{(c,(b,a)_{i^{\prime}})_{j}}\sigma_{1}\ket{(c,(b,a)_{i})_{j}}\right|^{2}$
(48)

This is true because, topologically, the three following operations are
equivalent:

$\displaystyle\sigma_{1}\sigma_{2}\sigma_{1}\ket{((a,b)_{i},c)_{j}}$
$\displaystyle\equiv\sigma_{2}\sigma_{1}\sigma_{2}\ket{((a,b)_{i},c)_{j}},$
$\displaystyle\equiv R_{ab}R_{ic}\ket{((a,b)_{i},c)_{j}},$

and

$\displaystyle R_{ab}R_{ic}\ket{((a,b)_{i},c)_{j}}$
$\displaystyle=R_{ab}^{i}R_{ic}^{j}\ket{(c,(b,a)_{i})_{j}}.$ (49)

These three operations do nothing but rotate the frame of anyons
$180^{\circ}$. The first equivalence relation of the three is is one of the
Artin relations of the braid group.
Now, let’s define the operator $\Gamma$ such as:

$\displaystyle\Gamma$ $\displaystyle=\sigma_{1}\sigma_{2}\sigma_{1}$
$\displaystyle=\sigma_{2}\sigma_{1}\sigma_{2}.$

It is easy to see that

$\displaystyle\sigma_{1}$ $\displaystyle=\Gamma^{\dagger}\sigma_{2}\Gamma,$
(50) $\displaystyle\sigma_{2}$
$\displaystyle=\Gamma^{\dagger}\sigma_{1}\Gamma.$ (51)

In general, a braid sequence $\text{Braid}(\sigma_{1},\sigma_{2})$ applied on
three anyons takes the form:

$\displaystyle\text{Braid}(\sigma_{1},\sigma_{2})=\sigma_{1}^{p_{n}}\sigma_{2}^{p_{n-1}}\cdots\sigma_{1}^{p_{2}}\sigma_{2}^{p_{1}}$
(52)

where $p_{i}$ can be any integer and $n\geq 2$. Therefore, the permutation
between $\sigma_{1}$ and $\sigma_{2}$ yields the braid sequence
$\overline{\text{Braid}}(\sigma_{1},\sigma_{2})$ such that

$\displaystyle\overline{\text{Braid}}(\sigma_{1},\sigma_{2})$
$\displaystyle=\text{Braid}(\sigma_{2},\sigma_{1})=\sigma_{2}^{p_{n}}\sigma_{1}^{p_{n-1}}\cdots\sigma_{2}^{p_{2}}\sigma_{1}^{p_{1}}$
(53)

The previous relations Eq. (50) and Eq. (51) imply that Braid and
$\overline{\text{Braid}}$ are related by the similarity relation:

$\displaystyle\overline{\text{Braid}}(\sigma_{1},\sigma_{2})=\Gamma^{\dagger}\text{Braid}(\sigma_{1},\sigma_{2})\Gamma.$
(54)

This similarity relation can be deduced immediately by substitution.

##  Appendix B Distance symmetry induced by Hermitian target

######  Proposition.

Let $(G,\mathcal{D})$ be a metric space such that the set $G$ forms a group
and $\mathcal{D}$ is a bi-invariant metric. Then, for all $a,b\in G$, if $b$
is its own inverse (i.e., $b=b^{-1}$), it follows that
$\mathcal{D}(a,b)=\mathcal{D}(a^{-1},b)$.

A metric $\mathcal{D}$ is defined to be bi-invariant if, for all $a,b,c\in G$,
we have $\mathcal{D}(a,b)=\mathcal{D}(ac,bc)=\mathcal{D}(ca,cb)$ [43]. Given
that the set of special unitary matrices forms a group and the spectral
distance is bi-invariant, let $H$ be a target unitary that is a Hermitian
matrix. If $\mathcal{D}(B,H)$ is known for a given braid matrix $B$, then
$\mathcal{D}(B^{\dagger},H)$ holds the same value.

##  Appendix C Approximated Toffoli gate with decomposition method

The CCS gate can be decomposed to at least five two-qubit controlled
operations [44, 28] as shown in Fig. 6. Namely, for any unitary
$\text{U}\in\text{U(2)}$,

$\displaystyle\text{C}{\text{C}{\text{U}}}=$
$\displaystyle(\text{SWAP}\otimes\text{I})(\text{I}\otimes\text{C}{\sqrt{\text{U}}})(\text{SWAP}\otimes\text{I})$
$\displaystyle(\text{C}{\text{NOT}}\otimes\text{I})(\text{I}\otimes\text{C}{\sqrt{\text{U}}^{\dagger}})(\text{C}{\text{NOT}}\otimes\text{I})(\text{I}\otimes\text{C}{\sqrt{\text{U}}})$

& 1
1
S

=

& 1 1 2
1 iX 1 -iX
S S^† S

Figure 6: Controlled-Controlled-S (CCS) gate can be systematically decomposed
into a sequence of five two-qubit controlled gates. This sequence comprises
two Controlled-NOT (CNOT) gates and two controlled gates that implement the
square root of the S gate.  Figure 7: In the injection method, three-strand
gates can implement two-qubit gates. Notice that each lower pair of anyons in
the same qubit should yield the vacuum if the state is $\ket{0}$ and yield an
anyon if it is $\ket{1}$. Therefore, weaving a pair of anyons around the
anyons of the controlled qubit and then returning to the initial position does
not affect the state unless the state is $\ket{1}$. That is the spirit of a
controlled gate.

The SWAP gate is needed because the arrangement of anyons does not allow two-
qubit operation between non-neighboring three-anyons qubits without making
non-trivial exchanges with the intermediary anyons. Therefore, to compile the
CCS gate by decomposition, it is necessary to construct C$i$NOT, C$\sqrt{S}$
and SWAP gates with sufficient accuracy.

While the SWAP gate is trivial in the context of encoding qubits with groups
of four anyons whose overall topological charge is 0, the involved two-qubit
controlled gates C$i$NOT and C$\sqrt{S}$ can be approximated by the injection
method introduced in [26] as shown in Fig. 7.

####  C.0.1 Numerical simulation

Table 9: Braid approximation of the necessary three-strand gates to build the $i$Toffoli gate with the decomposition method. We use two different best braid sequences to approximate $I$ for CNOT and C$\sqrt{\text{NOT}}$ and optimize accuracy and leakage.  Gates | Target Gates | Weave sequence | Length | Error
---|---|---|---|---
CNOT Injection | I | $\sigma_{1}^{-1}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{2}^{-1}$ | 48 | $1.51\times 10^{-3}$
C$\sqrt{\text{NOT}}$ Injection | I | $\sigma_{1}^{1}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{4}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{4}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{-1}$ | 48 | $1.51\times 10^{-3}$
$\sqrt{\text{NOT}}$ | $-i\sqrt{iX}$ | $\sigma_{2}^{-1}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{1}^{4}\sigma_{2}^{4}\sigma_{1}^{2}\sigma_{2}^{-1}$ | 48 | $1.24\times 10^{-3}$
NOT | $iX$ | $\sigma_{2}^{1}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{-4}\sigma_{1}^{2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{4}\sigma_{1}^{-2}\sigma_{2}^{-4}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{2}^{-1}$ | 48 | $8.55\times 10^{-4}$

We should find first the compilation of $\pm\text{I}$ such that it starts
weaving from the upper strand and ends up in the lower strand. We need also to
approximate $\pm iX$ and $\sqrt{\pm iX}$ gate acting on three strands starting
weaving from the upper strand and ending up in the same strand. By brute-
forcing, we find the required weaving sequences as shown in Tab. 9. The
simulated $-i$Toffoli gate by decomposition is represented in Fig. 8. The
distance error of this approximated $-i$Toffoli gate is $1.90\times 10^{-3}$.
However, the error on the controlled gate only is about $1.77\times 10^{-3}$.
The leakage is computed as well and it is $3.96\times 10^{-6}$. These values
exhibit a similar magnitude of approximation as those obtained using the
controlled-injection method. However, the decomposition method generates
longer braid sequences, as discussed in Section V.

(a)

(b)

Figure 8: (a) Matrix representation of the approximated $-i$Toffoli gate,
expressed in the computational basis. (b) Exact matrix representation of the
$-i$Toffoli gate. The color map is depicted in Fig. 2(c).

## Acknowledgments

This document has been produced with the financial assistance of the European
Union (Grant no. DCI-PANAF/2020/420-028), through the African Research
Initiative for Scientific Excellence (ARISE), pilot programme. ARISE is
implemented by the African Academy of Sciences with support from the European
Commission and the African Union Commission. We are grateful to the Algerian
Ministry of Higher Education and Scientific Research and DGRST for the
financial support.

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Copy the proof of Claim 4.3, changing only the following steps:

(b) Set $N=On-O+1$. Get a collection $Q_{i}$ of $N$ pairwise-disjoint squares
whose value for agent $i$ is at least
$V_{i}(s_{i})\cdot\text{PropSame}\allowbreak(S,S,N)$.

(c) After allocating a smallest piece, at most $O$ pieces of other agents have
to be removed. ∎

Claims 4.3 and 4.4 can be attained as special cases of Claim D.7 by setting
$O=4$ or $O=8$, respectively. By calculating $\text{Overlap}\allowbreak(S)$
for different families of fat pieces, it is possible to prove analogous claims
with different constants.

For example, when $S$ is the family of $R$-fat rectangles,
$\text{Overlap}\allowbreak(S)=2\left\lceil R\right\rceil+2$. Substituting in
Claim D.7 gives:

$\text{RelProp}(C,\,R\,fat\,rectangles,\,n)\,\,\geq\,\,\frac{1}{(4\left\lceil
R\right\rceil+4)(n-1)+2}$

For example, when $S$ is the family of $R$-fat $d$-dimensional axis-parallel
boxes: $\text{Overlap}\allowbreak(S)=2^{d-1}\left\lceil
R\right\rceil+2^{d-1}$.

Finally, when all agents have the same value measure, we get:

######  Claim D.8.

For every cake $C$ which is a compact subset of $\mathbb{R}^{2}$ and for every
family $S$:

$\text{RelPropSame}(C,\,S,\,n)=\text{PropSame}\allowbreak(S,\,S,\,n)$

###### Proof.

Suppose the value measure of all $n$ agents is $V$. Let $q$ be a best
$S$-piece in $C$ \- an $S$-piece that maximizes $V$. By definition of the
utility function: $V(q)=V^{S}(C)$. Because$q\in S$, it is possible to allocate
$n$ $S$-pieces in $q$ with a value of at least
$V(q)\cdot\text{PropSame}\allowbreak(S,\,S,\,n)$. ∎

##  E Existence of Best pieces

This appendix shows how to prove the existence of a usable piece with a
maximum value (this is used in the proof of Claim 4.3). We start by defining a
metric space of pieces (recall that a _piece_ is a Borel subset of
$\mathbb{R}^{2}$ and $Area$ is its Lebesgue measure).

######  Definition E.1.

The _symmetric difference (SD) pseudo-metric_ is defined by:

$d_{SD}(X,Y)=Area[(X\setminus Y)\cup(Y\setminus X)]$

$d_{SD}$ is not a metric because there may be different pieces whose symmetric
difference has an area of 0, e.g, a square with an additional point and a
square with a missing point. To make SD a metric, we consider only pieces $X$
that are _regularly open_ , i.e, the interior of the closure of themselves:
$X=Int[Cl[X]]$.

######  Claim E.1.

SD is a metric on the set of all regularly-open pieces.

###### Proof.

181818We are thankful to Tony K., Phoemue X., Dafin Guzman, Henno Brandsma and
Ittay Weiss for contributing to this proof via discussions in the
math.stackexchange.com website
(http://math.stackexchange.com/a/1099461/29780).

Let $X$ and $Y$ be two regularly-open sets such that $d_{SD}(X,Y)=0$. We prove
that $X=Y$.

$d_{SD}(X,Y)=0$ implies $Area[X\setminus Y]=Area[Y\setminus X]=0$.

$Y\subseteq Cl[Y]$ so $X\setminus Y\supseteq X\setminus Cl[Y]$. Hence also
$Area[X\setminus Cl[Y]]=0$.

$X$ is open and $Cl[Y]$ is closed; hence $X\setminus Cl[Y]$ is open (it is an
intersection of two open sets).

The only open set with an area of 0 is the empty set (because any non-empty
open set contains a ball with a positive measure). Hence: $X\setminus
Cl[Y]=\emptyset$.

Equivalently: $X\,\subseteq\,Cl[Y]$.

By taking the Cl of both sides: $Cl[X]\subseteq Cl[Y]$

By a symmetric argument: $Cl[Y]\subseteq Cl[X]$

Hence: $Cl[Y]=Cl[X]$

By taking the Int of both sides and by the fact that they are regularly-open:
$Y=X$. ∎

Thus when we allocate a square we actually allocate only its interior. This
has no effect on the utility of the agents since the boundary has an area of 0
and so its value is 0 for all agents.

######  Claim E.2.

Let $D$ be the metric space defined by $d_{SD}$. Let $V$ be a measure
absolutely continuous with respect to area. Then $V$ is a uniformly continuous
function from $D$ to $\mathbb{R}$.

###### Proof.

The fact that $V$ is an absolutely continuous measure implies that, for every
$\epsilon>0$ there is a $\delta>0$ such that every piece $X$ with
$Area(X)<\delta$ has $V(X)<\epsilon$ [63, Proposition 15.5 on page 251].
Hence, for every two pieces $X$ and $Y$, if $d_{SD}(X,Y)<\delta$ then
$Area(X\setminus Y)<\delta$ and $Area(Y\setminus X)<\delta$, then
$V(X\setminus Y)<\epsilon$ and $V(Y\setminus X)<\epsilon$, then
$|V(X)-V(Y)|=|V(X\setminus Y)-V(Y\setminus X)|<\epsilon$.∎

######  Claim E.3.

Let $V$ be a measure absolutely continuous with respect to area and $Q$ a set
of pieces which is compact in the SD metric space. Then there exists a piece
$q\in Q$ for which $V$ is maximized.

###### Proof.

By the previous claim, $V$ is a uniformly continuous and hence a continuous
real-valued function. By the extreme value theorem, it attains a maximum in
every compact set. ∎

The value measures considered in this paper are always absolutely continuous
with respect to area. Hence, to prove that a certain set of pieces $Q$
contains a “best piece” it is sufficient to prove that $Q$ is compact. We do
this now for the special case in which $Q$ is the set of open squares
contained in a given cake (note that the same proof could be used for the set
of closed squares):

######  Claim E.4.

Let $C$ be a closed, bounded subset of $\mathbb{R}^{2}$. Let $Q$ be the set of
all open squares contained in $C$. Then $Q$ is compact in the SD metric space.

###### Proof.

It is sufficient to prove that $Q$ is sequentially compact, i.e. every
infinite sequence of open squares in $C$ has a subsequence converging to an
open square in $C$. Let $\\{q_{i}\\}_{i=1}^{\infty}$ be an infinite sequence
of open squares in $C$. For every $q_{i}$, let $(A_{i},B_{i})$ be a pair of
opposite corners. Because $C$ is compact, it contains $Cl[q]$ and hence
contains the points $A_{i}$ and $B_{i}$. Hence the infinite sequence of pairs
of points, $\\{(A_{i},B_{i})\\}_{i=1}^{\infty}$, is an infinite sequence in
$C\times C$. $C\times C$ is compact because it is a finite product of compact
sets. Hence, the sequence has a subsequence converging to a limit point
$(A^{*},B^{*})\in C$. From now on we assume that
$\\{(A_{i},B_{i})\\}_{i=1}^{\infty}$ is that converging subsequence. Let
$q^{*}$ be the open square having $A^{*}$ and $B^{*}$ as two opposite corners.
We show that: (a) $q^{*}$ is an open square in $C$; (b) The subsequence
$\\{q_{i}\\}_{i=1}^{\infty}$ converges to $q^{*}$.

(a) $q^{*}$ is a obviously an open square by definition. We have to show that
each point in $q^{*}$ is also a point of $C$. To every square $q_{i}$, attach
a local coordinate system in which corner $A_{i}$ has coordinates $0,0$ and
corner $B_{i}$ has coordinates $1,1$ and every other point in $Cl[q_{i}]$ has
coordinates in $[0,1]\times[0,1]$. For every coordinate
$(x,y)\in[0,1]\times[0,1]$, let $q_{i}(x,y)$ be the unique point with these
coordinates in $Cl[q_{i}]$ (e.g. $A_{i}=q_{i}(0,0)$ and $B_{i}=q_{i}(1,1)$).
The sequence $\\{q_{i}(x,y)\\}_{i=1}^{\infty}$ is a sequence of points which
are all in $C$, and they converge to $q^{*}(x,y)$. Since $C$ is closed,
$q^{*}(x,y)\in C$.

(b) For every $i$, the area of the symmetric difference between $q^{*}$ and
$q_{i}$ is bounded and satisfies the following inequality:

$d_{SD}(q^{*},q_{i})\leq
4\cdot\max(d(A^{*},A_{i}),d(B^{*},B_{i}))\cdot\max(d(A^{*},B^{*}),d(A^{*},B_{i}),d(A_{i},B^{*}),d(A_{i},B_{i}))$

Since all distances are bounded and $d(A^{*},A_{i})$, $d(B^{*},B_{i})$
converge to 0, the same is true for $d_{SD}(q^{*},q_{i})$. Hence, the
subsequence $\\{q_{i}\\}_{i=1}^{\infty}$ converges to $q$.

The previous paragraph proved that $Q$ is sequentially compact. Hence it is
compact. ∎

In a similar way it is possible to prove similar results for other families
$S$, such as the family of $R$-fat rectangles or cubes.

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