# New topological subsystem codes from semi-regular tessellations

Eduardo Brandani da Silva1 and Evandro Mazetto Brizola2
Maringa State University - Department of Mathematics
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###### Abstract

In this work, we present new constructions for topological subsystem codes
using semi-regular Euclidean and hyperbolic tessellations. They give us new
families of codes, and we also provide a new family of codes obtained through
an already existing construction, due to Sarvepalli and Brown. We also prove
new results that allow us to obtain the parameters of these new codes.

††footnotetext: _Key words and phrases_ : Quantum error-correcting codes,
subsystem codes, topological subsystem codes, semi-regular tessellations

##  1 Introduction

The first quantum computing model described in 1982 by Benioff [3], although
based on quantum kinematics and dynamics, was, in the computational sense
established in [11], effectively classical. Also in 1982, Feynman [13]
proposed the first computer model based on the principles of quantum
mechanics. This model was the closest thing to a universal quantum simulator.
As early as 1985, a major breakthrough occurred in this area when Deutsch [11]
described an entirely quantum model of a quantum version of the Turing
machine.

One of the biggest difficulties in performing quantum computing is quantum
decoherence, which is the issue of quantum decay, which was warned by Unruh
[28]. One way to get around this effect is to use quantum error-correcting
codes, which are mathematical tools used to encode the physical states of a
quantum system.

The first quantum error-correcting codes were inspired by the classical error-
correcting codes introduced in 1948 by Shannon [22]. In 1995, Shor [23]
introduced the first quantum error-correcting code. This code was a
combination of two other three-qubit codes, where one of them protects the
quantum state against bit flip errors and the other protects the quantum state
against phase-flip errors, where only the first of them has a classical
analog.

One of the biggest advances in quantum coding occurred in 1996 with the
construction of the CSS quantum codes by Calderbank and Shor [10] and Steane
[26]. From these codes, one of the most important classes of quantum codes
known to date was generated, the stabilizer quantum codes [14].

The following year, Kitaev [17] developed a new code called Kitaev’s toric
code, which is part of a class of codes known as topological codes. A great
advantage and novelty of this code was the fact that its stabilizer generators
acted on a small number of qubits in its neighborhood. Codes with this
property are said to be geometrically local. An advantage of stabilizing
generators acting on a small number of qubits in their neighborhood is that
this makes quantum words resistant to local noise.

Another important group of quantum codes that is also part of the stabilizer
codes are the subsystem codes [1], [2], [20]. These codes are the result of
applying the stabilizer formalism to quantum error correction of the operator
[18], [19], which generalizes standard quantum error correction theory and
provides a unified framework for active correction of errors and passive error
prevention techniques, making subsystem codes one of the most versatile class
of error-correcting quantum codes.

Over the years, numerous works have been carried out in this area. A great
step was given by the construction of topological subsystem codes done by
Bombín [5], which made use of trivalent and $3$-colorable tessellations.
Although in [2] tessellations have already been used to construct subsystem
codes, due to the stabilizer generators of these codes being non-local, they
cannot be considered topological.

Suchara, Bravyi, and Terhal [27] established some conditions for determining
topological subsystem codes and proved that topological subsystem codes can be
seen as a kind of generalization of Kitaev’s honeycomb model, for trivalent
hypergraphs. Also in [27], a necessary and sufficient condition was presented
about measuring the error syndrome of a subsystem code, which is a very
important result and is used to guarantee that a subsystem code is
topological.

Sarvepalli and Brown [21] presented a new construction of topological
subsystem codes inspired by [27], where they presented some new code families.
This is one of the few works where code parameters are provided, due to the
difficulty of determining certain standards within this class of codes.

The work is divided into the following sequence: In Section 2, we revise the
subsystem codes and present their main elements, such as the groups necessary
for their construction and definition. In Section 3, we will address
topological subsystem codes, where we will discuss some important advantages
of these codes in relation to topological quantum codes. We will also deal
with the main aspects of the construction of such codes done in [5] and their
generalization done in [27] and [21]. Finally, we will look at two families of
topological subsystem codes presented in [21]. In Section 4, we construct a
new hypergraph $\Gamma_{h}$ and prove that this hypergraph allows the
construction of subsystem codes. We also present, in general, some stabilizers
and other relevant data about this construction. In Section 5, we present four
families of topological subsystem codes, where two of these families come from
particular cases of the hypergraph $\Gamma_{h}$ constructed in Section 4. The
third family comes from the construction given in [21], but from a case that
was not addressed by the authors, and the fourth family comes from a
hypergraph built from tessellation $\\{p,3,4,3\\}$ in a similar way to that
used for previous families. Also in this section, we prove that these code
families are topological subsystem codes and provide some tables with
parameters.

##  2 Subsystem codes

In this section, we will cover subsystem codes, which are also stabilizer
codes. According to Bombín [5], these codes are the result of applying the
stabilizer formalism to the quantum error correction of the operator [18],
[19]. As can be seen in [20], Poulin claims that operator quantum error
correction generalizes the standard quantum error correction theory and
provides a unified framework for active error correction and passive error
prevention techniques, which, according to Aly and Klappenecker [1], made
subsystem codes the most versatile class of error-correcting quantum codes
known up to that time.

In subsystem codes, not all logical qubits that form the C code space are used
as logical qubits, that is, they are used to encode information. A portion of
these qubits are considered gauge qubits, which do not encode any information.
In addition to other consequences, these qubits absorb the effects of errors
that occur in them.

The definition of subsystem code is similar to stabilizer codes, as we also
have a stabilizer group S, and the code space C is also given in a similar
way.

We have a code space C where information is encoded. From this code space,
which is a subspace of a Hilbert space $H$, we can decompose $H$ into the
direct sum $H=\textbf{C}\oplus\textbf{C}^{\perp}$.

In subsystem codes, the code space C is no longer fully used. It admits a
subsystem structure, so we can write C as
$\textbf{C}=\mathbb{A}\otimes\mathbb{B}$, where $\mathbb{A}$ is the subsystem
of C  that is used to encode the information. The subsystem $\mathbb{B}$ does
not encode information, and errors that occur in it are ignored. Therefore, we
say that the subsystem $\mathbb{B}$ contributes with gauge degrees of freedom.
Because we use subsystems, the code is called subsystem code. To obtain the
subsystems $\mathbb{A}$ and $\mathbb{B}$, we need two groups, which are
fundamental for the construction of the subsystem codes.

The first of these is the gauge group, denoted by $\mathscr{G}$, which is a
subgroup of the Pauli group $P_{n}$. More precisely, this group will be a
normal subgroup of $N(\textbf{S})$ and will be generated by the stabilizer
group S, by $\langle i\rangle$, and by an arbitrary subset of $X_{j}^{\prime}$
and $Z_{j}^{\prime}$ with $j>s$, where $s$ is the number of independent
stabilizer generators.

The second fundamental group for the construction of subsystem codes is the
logical group $\mathscr{L}=N(\textbf{S})/\mathscr{G}$, which is a group
because $\mathscr{G}$ is a normal subgroup of $N(\textbf{S})$.

The elements of $\mathscr{G}$ are required to commute with the elements of
$\mathscr{L}$. According to Poulin [20], a consequence of their commuting is
that the generators $X_{j}^{\prime}$ and $Z_{j}^{\prime}$ of $\mathscr{G}$
must always appear in pairs. Therefore, we can assume, without loss of
generality, that

$\mathscr{G}=\langle
i,S_{1},\ldots,S_{s},X_{s+1}^{\prime},\ldots,X_{s+r}^{\prime},Z_{s+1}^{\prime},\ldots,Z_{s+r}^{\prime}\rangle,$
(1)

with $s+r\leq n$. We can ignored the phase factor, then the gauge group to be
considered is only

$\mathscr{G}=\langle\textbf{S},X_{s+1}^{\prime},\ldots,X_{s+r}^{\prime},Z_{s+1}^{\prime},\ldots,Z_{s+r}^{\prime}\rangle.$
(2)

Therefore, it follows that

$\mathscr{L}\simeq\langle
X_{s+r+1}^{\prime},Z_{s+r+1}^{\prime},\ldots,X_{n}^{\prime},Z_{n}^{\prime}\rangle.$
(3)

Since the elements of $\mathscr{G}$ commute with the elements of $\mathscr{L}$
and $\mathscr{G}\times\mathscr{L}\simeq N(\textbf{S})$, it can be seen from
[29] that it is possible to induce a subsystem structure in code space C, so
that we have $\textbf{C}=\mathbb{A}\otimes\mathbb{B}$, where the group
$\mathscr{G}$ acts trivially on $\mathbb{A}$ and $\mathscr{L}$ acts trivially
on $\mathbb{B}$. It also follows that
$\mathbb{A}\simeq({\mathbb{C}}^{2})^{\otimes k}$ and
$\mathbb{B}\simeq({\mathbb{C}}^{2})^{\otimes r}$. See that if we use
$\mathbb{B}$ as a one-dimensional space, then we will have exactly the usual
quantum error-correcting codes. Note that $\mathscr{L}$ acts as the logical
operators of the stabilizer codes. Thus, the biggest novelty so far in the
subsystem codes is the presence of the gauge group $\mathscr{G}$.

The action of the gauge operators in the code space C induces the following
equivalence relation in C. Let
$|\psi\rangle,|\psi\rangle^{\prime}\in\textbf{C}$ be, we have

$|\psi\rangle\sim|\psi\rangle^{\prime}\Leftrightarrow\exists g\in\mathscr{G}\
\ \textrm{tal que}\ \ |\psi\rangle^{\prime}=g|\psi\rangle\,.$

By definition, equivalent states
$|\psi_{\mathbb{A}}\rangle\otimes|\psi_{\mathbb{B}}\rangle$ and
$|\psi_{\mathbb{A}}\rangle\otimes|\psi^{\prime}_{\mathbb{B}}\ rangle$ carry
the same information, even if $|\psi_{\mathbb{B}}\rangle$ and
$|\psi^{\prime}_{\mathbb{B}}\rangle$ are different.

Another important fact about the gauge group is that the stabilizer group S of
the subsystem code can be described using $\mathscr{G}$ due to the identity
$\textbf{S}=\mathscr{G}\cap C(\mathscr{G})$, where $C(\mathscr{G})$ is the
centralizer of $\mathscr{G}$ in the Pauli group. Thus, to characterize the
subsystem codes, we only need to have their gauge group $\mathscr{G}$.

It is worth noting that in the stabilizer codes we had $n=k+s$. In the
subsystem codes, we divide the $n$ virtual qubits into three sets: $s$
stabilizer qubits, $r$ gauge qubits, and $k$ logical qubits, where the
physical qubits are now given by $n=k+r+s$. The operators related to the first
set are given by the independent stabilizer generators $S_{1},\ldots,S_{s}$
that fix the $2^{n-s=r+k}$-dimensional code space C. The second is generated
by $X_{s+j}^{\prime}$ and $Z_{s+j}^{\prime}$ with $j=1,\ldots,r$, which act on
the $r$ virtual qubits of subsystem $\mathbb{B}$, which do not encode useful
information and their only purpose is to absorb gauge transformations.
Therefore, the gauge group $\mathscr{G}$ has dimension $2r+s$. Finally, the
third set is given by the remaining operators, which are logical operators as
in stabilizer codes. Thus, we denote these operators by $\overline{X}_{j}$ and
$\overline{Z}_{j}$ with $j=1,\ldots,k$, which generate the set $\mathscr{L}$
and act only on $k$ logical qubits of subsystem $\mathbb{A}$.

Since the elements of $\mathscr{L}$ are the logical operators of the subsystem
code, they are elements of the normalizer of S, that is, they are elements of

$N(\textbf{S})=\langle\mathscr{G},\overline{X}_{1},\overline{Z}_{1},\ldots,\overline{X}_{k},\overline{Z}_{k}\rangle.$
(4)

Two other operators that appear frequently in the literature on subsystem
codes are bare logical operators and dressed logical operators. Bare logical
operators refer to the elements of $C(\mathscr{G})\setminus\mathscr{G}$. These
elements preserve the code space C and act trivially on the gauge qubits. The
dressed logical operators refer to the elements of
$N(\textbf{S})\setminus\mathscr{G}$. Since we have the identity
$N(\textbf{S})=C(\mathscr{G})\cdot\mathscr{G}$ hold, where $``\cdot"$ is the
product of the elements of $C(\mathscr{G})$ by the elements of $\mathscr{G}$,
we have that any dressed logical operator can be written as a product of a
bare logical operator and a gauge operator.

As can be seen in [7] the distance of subsystem codes is defined as follows.

######  Definition 1.

The distance $d$ of a subsystem code is the minimum weight of a Pauli operator
that commutes with all stabilizers and acts non-trivially on the logical
subsystem $\mathbb{A}$, that is, it will be the minimum weight of a dressed
logical non-trivial operator, that is,

$d=\min_{P\in
N(\textbf{S})\setminus\mathscr{G}}|P|=\min_{\begin{subarray}{c}P\in
C(\mathscr{G})\setminus\mathscr{G}\\\ G\in\mathscr{G}\end{subarray}}|PG|,$ (5)

where $|P|$ and $|PG|$ are the weights of the Pauli operators $P$ and $PG$,
respectively.

Thus, we have four parameters for the subsystem codes, which are denoted by
$\left[\left[n,k,r,d\right]\right]$, where $n$ is the number of physical
qubits, $k$ is the number of encoded qubits, $r$ is the number of gauge
qubits, and $d$ is the distance from the code.

For more information about the subsystem codes, their construction, the sets
presented here, and to see about error identification and correction, we
recommend [1], [7], [9], and [20].

##  3 Topological subsystem codes

In this section, we present the topological subsystem codes. These codes were
first developed by Bombín [5] in 2010, who made use of trivalent and
$3$-colorable tessellations. In the same year, they were generalized to
trivalent hypergraphs satisfying certain conditions by Suchara, Bravyi, and
Terhal [27]. Subsequently, several works on these codes emerged, also working
in the hyperbolic environment. In 2012, Sarvepalli and Brown [21] added one
more condition about hypergraphs that would be necessary in [27], but had not
been considered. They also presented two families of topological subsystem
codes and provided the parameters of these codes, which, according to them, is
not a trivial task for subsystem codes. These code families built by
Sarvepalli and Brown follow the construction model for obtaining trivalent
hypergraphs presented in [27]. These three works served as inspiration and
provided the necessary tools for building our families of topological
subsystem codes that we will present in the next section.

Topological subsystem codes have all the characteristic properties of
topological quantum codes, such as that we can visualize the code operators,
which according to Breuckmann [9] makes the work more easy and intuitive, and
also the property that the operators correspond to the interaction between
neighboring qubits, which means we have a more physical interpretation of the
codes. These codes also have some advantages that come from subsystem codes,
such as the fact that the codes are local. As we will see, one of the main
advantages compared to topological quantum codes is that these codes require
only two neighboring qubits at a time to perform the syndrome measurement for
error correction. To see other advantages of topological subsystem codes, we
recommend [5], [9], and [27]. The construction of topological subsystem codes
combines the following properties:

  * $(C_{1})$

The stabilizer group $\textbf{S}=\mathscr{G}\cap C(\mathscr{G})$ has spatially
local generators, that is, the number of elements different from the identity
(their weight) is within a disk of radius $O(1)$. The elements of S are
identified with the homologically trivial loops of the tessellation;

  * $(C_{2})$

Syndrome extraction can be done by measuring the eigenvalues of the link
operators, which are $2$-local (2-qubits);

  * $(C_{3})$

The code encodes one or more logical qubits. The bare logical operators (in
$C(\mathscr{G})\setminus\mathscr{G}$) can be identified with the homologously
non-trivial loops.

######  Definition 2.

A subsystem code that satisfies the properties $(C_{1})$, $(C_{2})$, and
$(C_{3})$ is called a topological subsystem code.

Property $(C_{1})$ requires that the stabilizer operators act on a small
number of qubits. Property $(C_{3})$ tells us that non-trivial undetectable
errors increase with the size of the tessellation and are non-local, which
according to Breuckmann [9] is an advantage, since the chance of occurring
non-local errors is much smaller than local errors. As can be seen in [8],
these two properties are often used to describe some properties of stabilizer
codes. Finally, property $(C_{2})$ is the main advantage of these codes
compared to topological quantum codes.

If only the property $(C_{3})$ is not satisfied, then we obtain a subsystem
code that does not encode logical qubits ($k=0$), that is, it does not encode
information. An example that satisfies $(C_{1})$ and $(C_{2})$ but does not
satisfy $(C_{3})$ is Kitaev’s Honeycomb model, which can be seen at [16]. In
[27], it is proved that the subsystem code built into this tessellation does
not encode any information. It is also proved that any tessellations or
trivalent graph do not encode information, seen as subsystem code.

Now we will address the main aspects of the construction of topological
subsystem codes carried out by Bombín in [5], by Suchara, Bravyi, and Terhal
in [27], and by Sarvepalli and Brown in [21]. But before that, we will define
some basic concepts about graph theory. To see more about graph theory, we
recommend [4], [12], and [15].

######  Definition 3.

A graph is a pair $\Gamma=(V,E)$, where $E$ is a subset of the set $V\times V$
of unordered pairs of $V$, that is, the elements of $V$ are the vertices of
the graph $\Gamma$ and the elements of $E$ are the edges of the graph
$\Gamma$.

######  Definition 4.

Let $\Gamma=(V,E)$ and $\gamma=(V^{\prime},E^{\prime})$ be two graphs. If
$V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$, then $\gamma$ is said to
be a subgraph of $\Gamma$.

######  Definition 5.

A hypergraph is a pair $\Gamma_{h}=(V,E)$, where the elements of $E$ are
subsets of the elements of $V$. If the subsets are all of size two, then
$\Gamma_{h}$ is a standard graph. And, a trivalent hypergraph is a hypergraph
$\Gamma_{h}$ such that every vertex belongs to exactly three distinct edges.
It is also said that such a hypergraph has a degree of $3$.

######  Definition 6.

Let $\Gamma_{h}=(V,E)$ be a hypergraph. Any element of $E$ whose size is
greater than $2$ is called a hyperedge, and its rank is its size. The rank of
a hypergraph is the maximum rank of its edges.

The construction made by Bombín starts from a trivalent and $3$-colorable
tessellation $\Gamma$ and takes the dual tessellation $\Gamma^{*}$, which is a
tessellation formed by triangles. Each edge of the dual tessellation is
transformed into a four-sided face, and each vertex is transformed into a face
with as many sides as the valence of the vertex, which is already done
automatically when generating the four-sided faces, as can be seen in Figure
1-($a$). This new tessellation will be denoted by $\overline{\Gamma}$.

In [5], edges are identified as being of three types: solid, dashed, and
dotted edges. However, to avoid confusion and to standardize with our
construction and the construction and language carried out in [21], we will
color the edges of this new tessellation as follows: We color the triangles
blue, and the edges of the faces obtained for each vertex alternate between
green and red colors, as can be seen in Figure 1-($b$).

Figure 1: ($a$) Tessellation $\\{6,6,6\\}$, its dual $\Gamma^{*}$ and the new
tessellation $\overline{\Gamma}$ given by black edges. ($b$) Colored edges of
tessellation $\overline{\Gamma}$.

The operators acting on each edge are called link operators (or edge
operators) and are given as follows: Each edge $e^{\prime}=(u,v)$ is
associated with an edge operator
$\overline{K}_{e^{\prime}}\in\\{X_{u}X_{v},Y_{u}Y_{v},Z_{u}Z_{v}\\}$, where
$\overline{K}_{e^{\prime}}=Z_{u}Z_{v}$ if $e^{\prime}$ is an edge of the
triangle (blue edge), $\overline{K}_{e^{\prime}}=X_{u}X_{v}$ if $e^{\prime}$
is a red edge, and $\overline{K}_{e^{\prime}}=Y_{u}Y_{v}$ if $e^{\prime}$ is a
green edge. Note that the edge operators are all Hermitian.

Using the edge operators, the gauge group $\mathscr{G}$ is defined as

$\mathscr{G}:=\langle\overline{K}_{e^{\prime}};\
e^{\prime}\in\overline{E}\rangle,$ (6)

where $\overline{K}_{e^{\prime}}$ are the edge operators and $\overline{E}$ is
the set of edges of the tessellation $\overline{\Gamma}$. Note that the gauge
group generators $\mathscr{G}$ are $2$-local.

As we saw in Section LABEL:codigosubsistem, subsystem codes can be
characterized through the gauge group $\mathscr{G}$ due to the identity
$\textbf{S}=\mathscr{G}\cap C(\mathscr{G})$.

Using string operators, in [5] is determined $C(\mathscr{G})$. The nature of
strings differs from the case of topological quantum codes.

Given any subgraph $\gamma\in\overline{\Gamma}$, which contains the three
edges of a triangle or none of them, this graph does not have “end points”, as
can be seen in Figure 2, that is, it is a closed string. This subgraph defines
a Pauli operator

$O_{\gamma}=\bigotimes_{v}P_{v},$ (7)

with $P_{v}=I,X,Y,Z$, according to the color of the edge of the subgraph to
which the vertex $v$ belongs.

Figure 2: Colored edges representing the $\gamma\in\overline{\Gamma}$
subgraph.

It follows that the operators $O_{\gamma}$ given in (7) belong to
$C(\mathscr{G})$, and there is a bijection between these subgraphs and
$C(\mathscr{G})$.

The triangles of these subgraphs are connected in pairs, and if we look at the
tessellation $\Gamma^{*}$, these triangles always connect vertices of the same
color. This allows you to classify the strings according to color. As with
color codes, strings of the same color commute, while strings of different
colors anti-commute.

Define $S_{v}^{c}=O_{\gamma}$, with $c=r,g,b$, then $\gamma$ is the smallest
colored string $c$ around the vertex $v\in V^{*}$ of the tessellation $\
Gamma^{*}$ (see Figure 3). These operators are the stabilizer generators of S,
that is, $\textbf{S}=\langle S_{v}^{c};\ v\in V^{*}\ \textrm{e}\
c=r,g,b\rangle$. However, these generators are not all independent, as the
following relationships apply:

$\prod_{c}S_{v}^{c}=I\ \ \textrm{e}\ \ \prod_{v}S_{v}^{c}=I,$ (8)

where in the first case we vary the color and the vertex is fixed, and in the
second we vary the vertex and the color is fixed. Therefore, the number of
independent stabilizer generators (i.e., the rank of S) will be $s=2V^{*}-2$,
where $V^{*}$ is the number of vertices of the dual tessellation $\Gamma^{*}$.
The number of physical qubits $n$ will be equal to $n=3F^{*}$, where $F^{*}$
is the number of faces of the tessellation $\Gamma^{*}$. The number of encoded
qubits is $k=2-\chi=2g$. As $n=k+r+s$, it follows that the number of gauge
qubits will be $r=2F^{*}-\chi$. Therefore, the gauge qubits depend on the
surface.

Figure 3: Colored edges of the operator $S_{v}^{g}$.

For the distance $d$, in [5] it is given that $d\leq d_{T}$, where $d_{T}$ is
the minimum length, in the number of triangles, between non-trivial closed
strings. This is justified, since given a string operator $O_{\gamma}\in
C(\mathscr{G})$, and defining the set $E^{\prime}$ as being the set of red,
green, and only one blue edge of each triangle of $\ gamma$, we have to

$G:=\prod_{e^{\prime}\in E^{\prime}}\overline{K}_{e^{\prime}}\in\mathscr{G},$

and the weight of the operator $O_{\gamma}G$ is equal to the number of
triangles in $O_{\gamma}$. It also provides a lower bound on the distance,
which is given by $d\geq d_{L}$, where $d_{L}$ is the minimum length in terms
of the number of edges, between the non-trivial homologically closed loops in
the tessellation $\Gamma^{*}$.

We also consider the topological subsystem codes built in [27]. To build these
codes, the authors used trivalent hypergraphs $\Gamma_{h}$ embedded in a plane
or a torus. The qubits are fixed at the vertices, and the set of all vertices
is denoted by $V$. Edges are of two types: edges that connect two vertices,
which are called link or rank-$2$ edges, and edges that connect three
vertices, which are called triangles or rank-$3$ edges. The sets of all
rank-$2$ and rank-$3$ edges are denoted by $E_{2}$ and $E_{3}$, respectively,
therefore, $E=E_{2}\cup E_{3}$.

######  Definition 7.

An edge $e^{\prime}$ is incident to a vertex $u$ if there exists $v\in V$ such
that $e^{\prime}=(u,v)$ or if there exist $v,w\in V$ such that
$e^{\prime}=(u,v,w)$.

There are four restrictions imposed in [27] for such hypergraphs and one more
imposed in [21]. They are:

  * ($H_{1}$)

$\Gamma_{h}$ has only rank-$2$ or rank-$3$ edges;

  * ($H_{2}$)

Each vertex has exactly three incident edges;

  * ($H_{3}$)

Two distinct edges intersect at at most one vertex;

  * ($H_{4}$)

Rank-$3$ edges are two by two disjoint;

  * ($H_{5}$)

The edges of the hypergraph $\Gamma_{h}$ must be $3$-colorable.

As in [5], in [27] edge operators are also defined, but unlike there, there
are two types here. The link operators $K_{e^{\prime}}$ where $e^{\prime}\in
E_{2}$, and the triangle operators $K_{e^{\prime}}$ where $e^{\prime}\in
E_{3}$. By convention, given an edge $e^{\prime}=(u,v)\in E_{2}$, we have that
$K_{e^{\prime}}:=X_{u}X_{v}$ or $K_{e^{\prime}}:=Y_{u}Y_{v}$. If
$e^{\prime}=(u,v,w)\in E_{3}$, we have that $K_{e^{\prime}}:=Z_{u}Z_{v}Z_{w}$.
This choice does not change the code in any way, as a depolarizing channel is
being assumed, i.e, errors of the types $X$, $Y$, and $Z$ occur with the same
probability.

The next result is from [9].

######  Theorem 8.

Let $e^{\prime},e^{\prime\prime}\in E$ be any two edges of $\Gamma_{h}$. So,
for their respective operators, $K_{e^{\prime}}$ and $K_{e^{\prime\prime}}$,
we have

$K_{e^{\prime}}K_{e^{\prime\prime}}=(-1)^{\eta(e^{\prime},e^{\prime\prime})}K_{e^{\prime\prime}}K_{e^{\prime}},$
(9)

with $\eta(e^{\prime},e^{\prime\prime})=0$, if $e^{\prime}$ and
$e^{\prime\prime}$ share an even number of vertices or
$e^{\prime},e^{\prime\prime}\in E_{3}$, and otherwise
$\eta(e^{\prime},e^{\prime\prime})=1$.

######  Remark 9.

To obtain a subsystem code from the hypergraph $\Gamma_{h}$, it is essential
that the commutation rule be satisfied for all edge operators of the
hypergraph $\Gamma_{h}$. Therefore, in [21], the authors added the restriction
$(H_{5})$ to the hypergraph $\Gamma_{h}$. Only $(H_{1}),\ldots,(H_{4})$ does
not guarantee the existence of subsystem codes. For instance, as can be seen
in [21], the Petersen graph satisfies $(H_{1}),\ldots,(H_{4})$ but does not
admit defining a subsystem code.

We will now define a closed hypercycle in a hypergraph. Roughly speaking,
restricting to the cases covered in [5], these closed hypercycles are the
strings seen previously.

######  Definition 10.

We call a subset of edges $M\subset E$ a closed hypercycle, or just a
hypercycle, if every vertex of the hypergraph has an even number of edges
incident to $M$.

Using these hypercycles, we define an operator that will be very important,
called the loop operator.

######  Definition 11.

Let $M\subset E$ be any closed hypercycle. The operator

$W(M)=\prod_{e^{\prime}\in M}K_{e^{\prime}}$ (10)

is the tensor product between the link operators and the triangle operators
acting on the edges of $M$. We call $W(M)$ the loop operator.

The next results tell us when an edge operator (link or triangle) commutes
with a loop operator, and when two loop operators commute. Their proof can be
seen at [9] and [27].

######  Theorem 12.

Let $e^{\prime}\in E$ be any edge and $M\subset E$ be any closed hypercycle.
Their corresponding operators commute if and only if $e^{\prime}$ is not a
triangle contained in $M$, that is,

$[W(M),K_{e^{\prime}}]=0\Leftrightarrow e^{\prime}\notin M\cap E_{3}\,.$

######  Corollary 13.

Let $M,M^{\prime}\subset E$ be any two closed hypercycles, then

$W(M)W(M^{\prime})=(-1)^{\eta(M,M^{\prime})}W(M^{\prime})W(M)\,,$

where $\eta(M,M^{\prime}):=|M\cap M^{\prime}\cap E_{3}|$ is the number of
triangles shared by $M$ and $M^{\prime}$.

######  Remark 14.

It follows from the Corollary 13 that two loop operators will commute if they
have an even number of triangles in common, and anticommute if they have an
odd number of triangles in common. Thus, the number of triangles in any closed
hypercycle must be even for $W(M)$ to commute with itself.

We also define the group

$\mathscr{G}_{loop}:=\\{W(M);M\subseteq E\ \textrm{is a closed
hypercycle}\\}.$ (11)

Through this group, we can obtain the gauge group $\mathscr{G}$ and the
stabilizer group $\textbf{S}=\mathscr{G}\cap C(\mathscr{G})$ using the
identities

$\mathscr{G}=C(\mathscr{G}_{loop})\ \ \textrm{e}\ \
\textbf{S}=\mathscr{G}_{loop}\cap C(\mathscr{G}_{loop}).$ (12)

Thus, it follows that $C(\mathscr{G})=\mathscr{G}_{loop}$.

The stabilizer operators will be the loop operators that commute with all the
other loop operators. The bare logical operators
$\overline{X}_{i},\overline{Z}_{i}\in C(\mathscr{G})$ can be chosen as a pair
of loop operators satisfying the standard commutation rules of Pauli
operators.

To prove that $C(\mathscr{G})=\mathscr{G}_{loop}$ and the other results, it is
necessary to transform the hypergraph $\Gamma_{h}$ into an ordinary graph,
denoted by $\overline{\Gamma}_{h}$. Firstly, the set of vertices
$\overline{V}$, that is, of qubits of $\overline{\Gamma}_{h}$, will be the
same as that of $\Gamma_{h}$. The edges of $\overline{\Gamma}_{h}$ will all be
rank-$2$, so the links, that is, the edges of $E_{2}$, will continue to be the
same links in $\overline{\Gamma}_{h}$. Triangles will be transformed into
links as follows: Given a triangle $(u,v,w)$, we will consider $(u,v)$,
$(u,w)$ and $(v,w)$ as links, that is, each triangle contributes three new
links in $\overline{\Gamma}_{h}$. Thus, $\overline{\Gamma}_{h}$ is obtained
from $\Gamma_{h}$ just by transforming each triangle into three links.

In this way, the triangle operator $K_{e^{\prime}}=Z_{u}Z_{v}Z_{w}$ will be
divided into three operators: $\overline{K}_{u,v}=Z_{u}Z_{v}$,
$\overline{K}_{u,w}=Z_{u}Z_{w}$, and $\overline{K}_{v,w}=Z_{v}Z_{w}$. Note
that these three operators are not independent, as
$\overline{K}_{u,v}\overline{K}_{u,w}\overline{K}_{v,w}=(Z_{u}Z_{v})(Z_{u}Z_{w})(Z_{v}Z_{w})=I$.
Thus, denoting the link operators $K_{e^{\prime}}$ also by
$\overline{K}_{e^{\prime}}$, we have that the gauge group $\mathscr{G}$
follows as in (6), that is, $\mathscr{G}=\langle\overline{K}_{e^{\prime}};\
e^{\prime}\in\overline{E}\rangle$, where $\overline{E}$ are the edges of
$\overline{\Gamma}_{h}$.

The proof of the next result can be seen in [27].

######  Lemma 15.

Let $\mathscr{G}$ be the group generated by the link operators
$\overline{K}_{e^{\prime}}$, where $e^{\prime}\in\overline{E}$ is an edge of
$\overline{\Gamma}_{h}$. So $\mathscr{G}_{loop}=C(\mathscr{G})$.

The next result, which can also be seen in [9, 27], is one of the most
important results, as it tells us when ($C_{2}$) is satisfied. Remembering
that the item ($C_{2}$) says that the extraction of the syndrome can be done
by measuring the eigenvalues of the link operators, which are $2$-local, that
is, measuring only the elements of the gauge group $\mathscr{G}$.

######  Theorem 16.

The eigenstate of a stabilizer $S\in\textbf{S}$ can be measured by a set of
link operators if and only if $S$ can be written as a product of these link
operators, i.e.

$S=\overline{K}_{m}\cdots\overline{K}_{1}$ (13)

and $\overline{K}_{j}$ commutes with the ordered product of all preceding link
operators, that is,

$[\overline{K}_{j},\overline{K}_{j-1}\cdots\overline{K}_{1}]=0\ \forall
j\in\\{2,\ldots,m\\}.$ (14)

The way of writing the stabilizer $S$ does not matter and is not necessarily
unique, but it must satisfy (14).

Although this construction of subsystem codes using the group
$\mathscr{G}_{loop}$ made in [27] provides subsystem codes, these codes are
not necessarily topological. In [21], the authors give a counterexample, where
a stabilizer is obtained that is associated with a homological non-trivial
closed hypercycle, that is, it will not satisfy the condition $(C_{1})$ of the
definition 2. Even satisfying $(C_{2})$ and $(C_{3})$, it will not be a
topological subsystem code.

Fortunately, this is not a restriction to be subsystem code, it just won’t be
topological. Thus, to use this construction, we will need more properties to
guarantee that they will actually give topological subsystem codes.

Obtaining the parameters of topological subsystem codes is not an easy task,
as they depend on the type of hypergraph and the surface where the hypergraph
is embedded.

We will now consider the topological subsystem codes built in [21]. In this
construction, the authors start from a trivalent and 3-colorable graph,
denoted by $\Gamma_{2}$, imposing the restriction that there is a non-empty
set of faces whose number of edges is a multiple of and greater than $4$.

From this graph, a trivalent hypergraph $\Gamma_{h}$ is constructed,
satisfying the conditions $(H_{1}),\ldots,(H_{5})$ seen previously, thus, it
is guaranteed that the codes obtained are subsystem codes, which are described
by $\mathscr{G}_{loop}$ as in [27].

We will now describe how the process of constructing the hypergraph
$\Gamma_{h}$ is carried out. Consider a trivalent and $3$-colorable graph
$\Gamma_{2}$ with the requirement that there be a non-empty set of faces as
previously stated. Let $F_{R}$, $F_{B}$, and $F_{G}$ be the sets of all red,
blue, and green faces, respectively. Suppose without loss of generality that
the faces $f\in F$, where $F\subseteq F_{R}$, are such that the number of
sides of $f$ is congruent to $4(\textrm{mod}\ 0)$ and greater than $4$. For
each $f\in F$, a face $f^{\prime}$ is added inside $f$, so that $f^{\prime}$
has half the number of edges of $f$.

Coloring the edges of $f$ alternately using the colors blue and green, there
are two ways to construct the edges of rank-$3$ (the triangles) using the
vertices of the face $f^{\prime}$ inside $f$. The first way is to consider
triangles using a vertex of $f^{\prime}$ and a blue edge of $f$, as in Figure
4-($a$), so that these triangles do not intersect, that is, they are disjoint.
The second way is analogous to the first, but now the green edges are
considered instead of the blue ones, as in Figure 4-($b$).

Figure 4: Rank-$3$ edges inserted into face $f\in F$ from the vertices of face
$f^{\prime}$. ($a$) blue rank-$3$ edges and ($b$) green rank-$3$ edges.

The rank-$3$ edges have the same color as the edge $f$ used in its
construction. The edges of $f^{\prime}$ are of different colors from the color
of the edges of rank-$3$, as in Figures 4-($a$) and 4-($b$). The resulting
graph is the desired $\Gamma_{h}$ hypergraph.

In [21], the authors proved that the graph $\Gamma_{h}$ satisfies
$(H_{1}),\ldots,(H_{5})$. Therefore, it gives rise to subsystem codes, whose
group of loop operators is given as in (11) and the gauge group is given by
$\mathscr{G}=C(\mathscr{G}_{loop})$.

Because $F\subseteq F_{R}$ is a subset, the stabilizers of the code vary
depending on $F$. However, given any face $f\in F$, there are two independent
hypercycles in $\Gamma_{h}$ that we can associate with this face, and
consequently, two independent stabilizing generators. The first of them,
denoted by $f_{\sigma_{1}}$, is composed only of the edges of $f^{\prime}$, as
in Figure 5-($a$). The second, denoted by $f_{\sigma_{2}}$, is formed by the
triangles inserted in $f$, together with the rank-$2$ edges on the border of
$f$ and the edges of $f^{\prime}$ of the same color as the edges of $f$, as in
Figure 5-($b$). In [21] it is proved that the operators $W(f_{\sigma_{1}})$
and $W(f_{\sigma_{2}})$ associated with these hypercycles are independent
stabilizer generators, and satisfy the Theorem 16.

It is possible to obtain a third hypercycle $f_{\sigma_{3}}$ (see Figure
5-($c$)) and, consequently, a third stabilizer generator $W(f_{\sigma_{3}})$,
doing the modulo 2 sum of $f_{\sigma_{1}}$ and $f_{\sigma_{2}}$, and
multiplying $W(f_{\sigma_{1}})$ by $W(f_{\sigma_{2}})$, respectively.

Figure 5: Hypercycles corresponding to face $f\in F\subseteq F_{R}$
represented by colored edges. ($a$) Hypercycle $f_{\sigma_{1}}$, ($b$)
hypercycle $f_{\sigma_{2}}$ and ($c$) Hypercycle $f_{\sigma_{3}}$.

Note that the operators $W(f_{\sigma_{1}})$ and $W(f_{\sigma_{2}})$, when
restricted to the cases covered in [5], coincide with the operators
$S_{v}^{c}$ seen previously.

For distance, the authors in [21] proved that $d$ will be upper bound by the
smallest number of triangles in a homological non-trivial closed hypercycle,
as done in [5], that is, it will be the smallest weight among the operators of
$\mathscr{G}_{loop}\setminus\textbf{S}$, that is,
$C(\mathscr{G})\setminus\textbf{S}$.

Before presenting the new families of codes, where it will be possible to
obtain the parameters, we need to present a way to construct a trivalent and
$3$-colorable graph $\Gamma_{2}$ from another graph embedded in a surface.
This construction is due to [6], but it can also be seen in [21] and is used
to build one of the families that we will present below.

Given a graph $\Gamma$ embedded in a surface, one of the colors is chosen to
color all its faces. Each edge of the graph is transformed into two edges, and
the new faces obtained are now all colored with the same color but a different
color from the color already chosen, as can be seen in Figure 6. Finally, each
vertex of valence $j$ is transformed into a face of $j$ edges. These new faces
are all colored with the color that has not yet been used, and, therefore, a
trivalent and $3$-colorable graph is obtained, which is denoted by
$\Gamma_{2}$.

The next result from [21] tells us how to construct the first family of codes
made by the authors, along with the parameters of these codes.

Figure 6: Transforming any graph $\Gamma$ into a trivalent and $3$-colorable
graph $\Gamma_{2}$.

We need the definition of a bipartite graph.

######  Definition 17.

A graph $\Gamma=(V,E)$ is called bipartite, if $V$ admits a partition into two
sets such that every edge has its endpoints in different sets. Vertices in the
same set cannot be adjacent.

######  Theorem 18.

Let $\Gamma$ be a graph such that every vertex has an even degree (valence)
greater than two. Construct a trivalent, $3$-colorable graph $\Gamma_{2}$
using the construction described previously. Then, using $\Gamma_{2}$ to
construct the hypergraph $\Gamma_{h}$, where $F$ will be the set of $v$ faces
of $\Gamma_{2}$, and the edges of rank-$3$ are the boundaries of $e$-faces of
$\Gamma_{2}$. If $l$ is the number of edges of rank-$3$ of a homologically
non-trivial hypercycle, then we obtain a subsystem code with parameters

$[[6e,1+\delta_{\Gamma^{*},\textrm{bipartido}}-\chi,4e-\chi,d\leq l]],$ (15)

where $e=|O(\Gamma)|$ and $\delta{\Gamma^{*},\textrm{bipartite}}=1$ if
$\Gamma^{*}$ is bipartite and $0$ otherwise.

Before presenting the next result that we will use to construct our second
family of codes, we need the definition of a medial graph. The proof of the
result can also be found in [21].

######  Definition 19.

We call the medial graph of a graph $\Gamma$ the graph that is obtained by
placing a vertex on each edge of $\Gamma$ and an edge connecting two of these
vertices, if the edges associated with these vertices in $\Gamma$ are
adjacent. We denote the medial graph of $\Gamma$ by $\Gamma_{m}$.

######  Theorem 20.

Let $\Gamma$ be a graph whose vertices have even valences greater than $2$ and
$\Gamma_{m}$ be its medial graph. Construct the trivalent and $3$-colorable
graph $\Gamma_{2}$ from $\Gamma_{m}^{*}$ (dual of $\Gamma_{m})$ using the
construction presented previously. If $\Gamma_{m}^{*}$ is bipartite, the set
of $v$-faces of $\Gamma_{2}$, denoted by $F_{v}$, forms a bipartition
$F_{v}\cup F_{f}$, where $|F_{v}|=|V(\ Gamma)|$. Making the construction of
$\Gamma_{h}$ with $F_{v}\subset F$ such that the edges of rank-$3$ are not on
the borders of the $e$-faces of $\Gamma_{2}$. Let $l$ be the number of edges
of rank-$3$ in a homologically non-trivial hypercycle, then we obtain a
subsystem code with parameters

$[[10e,1+\delta_{\Gamma^{*},\textrm{bipartido}}-\chi,6e-\chi,d\leq l]],$ (16)

where $e=|O(\Gamma)|$ and $\delta{\Gamma^{*},\textrm{bipartite}}=1$ if
$\Gamma^{*}$ is bipartite and $0$ otherwise.

In [21], the authors proved that these codes cannot be obtained by the
construction made in [5]. In particular, all the codes of Theorem 20 are
different from those obtained in [5], and the codes of Theorem 18 are
different when the graph $\Gamma^{*}$ is non-bipartite. They also proved that
these codes satisfy the $(C_{2})$ condition and that they are topological
subsystem codes.

##  4 A new construction to obtain topological subsystem codes

Inspired by the constructions of topological subsystem codes in the last
section, in this section we present a generalization of the construction given
in [21], where we will construct new subsystem codes and present four new
families of topological subsystem codes. We will prove that these codes are
topological subsystem codes.

In our construction, instead of starting from a trivalent and $3$-colorable
graph, we will start from a trivalent and $3$-colorable tessellation. Although
the graph approach is more generic, using tessellations, we will be able to
obtain greater control over the parameters of the codes obtained. We also draw
attention to the fact that in [24] all trivalent and $3$-colorable
tessellations existing on the surfaces were determined, as well as the
quantities of faces, edges, and vertices of these tessellations were
calculated, the which we will use here.

Consider a trivalent and $3$-colorable tessellation
$\\{2p_{1},2p_{2},2p_{3}\\}$. We denote the set of red, green, and blue faces
by $F_{R}$, $F_{G}$, and $F_{B}$, respectively. In order to make our
construction easier, we will fix the colors of the tessellation polygons as
follows: Suppose without loss of generality that the $2p_{1}$-gons are red,
the $2p_{2}$-gons are green, and the $2p_{3}$-gons are blue.

Consider $F\subseteq F_{R}$ such that $F\neq\emptyset$ and assume that
$2p_{1}>4$ with $p_{1}>2$ odd. If we make the construction proposed in [21],
only for trivalent and 3-colorable tessellations $\\{2p_{1},2p_{2},2p_{3}\\}$,
we will have the restriction that $2p_{1}>4$ with $p_{1}>2$ even, thus we will
only work with the case $p_{1}>2$ odd.

To construct the hypergraph $\Gamma_{h}$, we insert a face $f^{\prime}$ with
$p_{1}$ sides inside each face $f\in F$ and construct the triangles from the
vertices of $f^{\prime}$, so that one of its edges is also an edge of a blue
$2p_{3}$-gon, and these triangles do not intersect each other, as in Figure 7.

Figure 7: Tessellation $\\{6,6,6\\}$, where inside the hexagonal face $f\in
F_{R}$ we insert the triangular face $f^{\prime}$, and from the vertices of
$f^{\prime}$ we construct three triangles.

By fixing this choice in relation to the construction of triangles together
with fixing the colors, we are also fixing an order for the tessellations, as
tessellation $\\{6,12,4\\}$ will provide a different hypergraph than
tessellation $\\{6,4,12\\}$, as can be seen in Figure 8.

######  Remark 21.

If $p_{1}$ is even, to obtain the construction done in [21],just color the
edges as we saw in Section 3. However, it is worth highlighting that in our
approach, we are starting from a tessellation and not from a graph, as in
[21]. Furthermore, if $p_{1}$ is even, $p_{3}=2$, and $F=F_{R}$, then we are
in the conditions of Theorem 18, as we have $\\{2p_{1},2p_{2},4\\}$. This is
the only family of tessellations to which this theorem applies.

Figure 8: Triangles inserted inside two faces of $F_{R}$ in the tessellations
(a) $\\{6,12,4\\}$ and (b) $\\{6,4,12\\}$.

For this construction with the odd case $p_{1}$, the resulting hypergraph does
not satisfy the restriction $(H_{5})$, therefore, it does not satisfy the
switching rule and, therefore, does not admit the construction of subsystem
codes. To correct this problem, we choose any three distinct edges of
$f^{\prime}$ and take a point on each of them, different from the extremes.
Then we connect these points to obtain a triangle, as in Figure 9. The edges
where we chose the points give rise to two new edges, and these points are new
vertices. Therefore, the face $f^{\prime}$ will have $p_{1}+3$ edges.

Figure 9: Face $f\in F_{R}$ of the hyperbolic tessellation $\\{10,10,4\\}$,
where we construct a triangle inside the face $f^{\prime}$. Thus, $f^{\prime}$
now has 8 edges.

We color all the edges of the triangles blue, and the other edges of $f$,
where the triangles were inserted, we color red. For possible faces $f\in
F_{R}$ such as $f\not\in F$, we alternate between red and blue, where the blue
edges are those connected to the $2p_{3}$-gons. For the $p_{1}+3$ edges of
$f^{\prime}$, we alternate between the colors red and green, and the remaining
edges, which belong to the $2p_{2}$-gons and $2p_{3}$-gons, we color green, as
in Figure 10.

Figure 10: Hypergraph $\Gamma_{h}$ obtained from tessellation $\\{6,12,4\\}$.

Note that when we construct the triangle from points taken on the edges of
$f^{\prime}$, we correct the problem of the hypergraph not satisfying
$(H_{5})$. We denote the resulting hypergraph by $\Gamma_{h}$ and use it to
construct the subsystem codes in the same way as in [27].

To see that the hypergraph $\Gamma_{h}$ given in our construction can be used
to construct subsystem codes as in [27], we will prove that $\Gamma_{h}$
satisfies $(H_{1}),\ldots,(H_{5})$.

######  Theorem 22.

The hypergraph $\Gamma_{h}$ obtained by the previous construction, starting
from a tessellation $\\{2p_{1},2p_{2},2p_{3}\\}$ with odd $p_{1}>2$, satisfies
the constraints $(H_{1}),\ldots,(H_{5})$, and, therefore, it gives rise to
subsystem codes whose group formed by loop operators is given by

$\mathscr{G}_{loop}=\\{W(M);\ M\subseteq E\ \ \textrm{\'{e} um hiperciclo
fechado}\\},$ (17)

where $E$ is the set formed by the edges of rank-$2$ and rank-$3$, and the
gauge group $\mathscr{G}$ is given by $\mathscr{G}=C(\mathscr{G}_{loop})$.

###### Proof.

In the tessellation $\\{2p_{1},2p_{2},2p_{3}\\}$, we only have rank-$2$ edges.
When we insert $f^{\prime}$ into $f\in F\subseteq F_{R}$ and create triangles
from the vertices of $f^{\prime}$, we are producing only rank-$2$ and rank-$3$
edges. In the same way, when we create the triangle inside $f^{\prime}$ from
points taken on three of its edges, we are producing one edge of rank-$3$ and
three of rank-$2$. Therefore, $\Gamma_{h}$ satisfies $(H_{1})$.

It also follows from the way we insert the faces $f^{\prime}$ and create the
triangles that the original vertices of $\\{2p_{1},2p_{2},2p_{3}\\}$ will
continue to be trivalent. The vertices of $f^{\prime}$ are also trivalent
because of the triangles created from their vertices. Likewise, the three new
vertices will be trivalent, as they divide an edge into two and have the new
triangle as the third edge. Thus, $\Gamma_{h}$ satisfies $(H_{2})$.

The restriction $(H_{3})$ is satisfied because, despite creating rank-$2$ and
rank-$3$ edges, it follows from the way they were created that any two
distinct edges intersect at most one vertex.

The restriction $(H_{4})$ is also satisfied, because when we create the
triangles from $f^{\prime}$, we already require that they do not intersect.
And the last triangle is also created without intersecting with the others.
Therefore, the rank-$3$ edges of $\Gamma_{h}$ are two-by-two disjoint.

Finally, $(H_{5})$ is satisfied, as we constructed the triangle inside
$f^{\prime}$ so that the commutation rule is valid, that is, so that the edges
of $\Gamma_{h}$ were $3$-colorable. Therefore, $\Gamma_{h}$ gives rise to
subsystem codes as constructed in [27]. ∎

######  Remark 23.

The choice of the three edges of $f^{\prime}$ that will have the vertices of
the triangle is made, in most cases, randomly, that is, we have no
restrictions. However, we will have a small restriction for two families of
tessellations. The first of them is for tessellation $\\{2p_{1},2p_{2},4\\}$
with $F=F_{R}$, and the second of them is tessellation $\\{2p_{1},4,6\\}$ also
with $F=F_{R}$. This restriction is due to the fact that the Theorem 16 has to
be satisfied to guarantee that $(C_{2})$ is valid. We will study these two
cases in particular in the next section. For any other family of
tessellations, we can take the three edges of $f^{\prime}$ at random.

As we saw in Section 2, we fixed the triangle operators
$K_{e^{\prime}}=Z_{u}Z_{v}Z_{w}$, where $e^{\prime}=(u,v,w)$. When we
transform $\Gamma_{h}$ into the ordinary graph $\overline{\Gamma}_{h}$, we
will have $\overline{K}_{u,v}=Z_{u}Z_{v}$, $\overline{K}_{u,w}=Z_{u}Z_{w}$ and
$\overline{K}_{v,w}=Z_{v}Z_{w}$. Thus, in $\overline{\Gamma}_{h}$ given
$e^{\prime}=(u,v)\in\overline{E}$, we have that
$\overline{K}_{e^{\prime}}=Z_{u}Z_{v}$ if $e^{\prime}\in\overline{E}_{B}$,
$\overline{K}_{e^{\prime}}=X_{u}X_{v}$ if $e^{\prime}\in\overline{E}_{R}$ and
$\overline{K}_{e^{\prime}}=Y_{u}Y_{v}$ if $e^{\prime}\in\overline{E}_{G}$,
where $\overline{E}_{B}$, $\overline{E}_{R}$ and $\overline{E}_{G}$ are the
sets with all blue, red and green edges, respectively, of the graph
$\overline{\Gamma}_{h}$.

As the non-empty set $F\subseteq F_{R}$ is any set, we cannot generally
determine all the stabilizers in the code. However, for each $f\in F$, we can
always determine three stabilizer generators, where only two are independent,
and for each $2p_{2}$-gon, we can determine one more independent stabilizer
generator. We show this in the following lemma.

######  Lemma 24.

Let $f_{1}\in F$ and $f_{2}\in F_{G}$, and let $\Gamma_{h}$ be given by our
construction. Consider the three hypercycles (two independent) that we can
associate with the face $f_{1}$ and one independent hypercycle that we can
associate with the face $f_{2}$. Then, there are three stabilizer generators
(two independent) associated with $f_{1}$ and one independent stabilizer
generator associated with $f_{2}$.

###### Proof.

Let $f_{1}^{\prime}$ be the face inserted into $f\in F\subseteq F_{R}$, with
the three extra edges coming from the insertion of the triangle inside
$f_{1}^{\prime}$. Define $f_{1\sigma_{1}}$ as the hypercycle formed by the
$p_{1}+3$ rank-$2$ edges on the border of $f_{1}^{\prime}$. Therefore, it
follows that

$W(f_{1\sigma_{1}})=\prod_{e^{\prime}\in\partial(f_{1}^{\prime})\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f_{1}^{\prime})\cap\overline{E}_{G}}\overline{K}_{e^{\prime}}$
(18)

belongs to the group $\mathscr{G}_{loop}$, and by the Corollary 13 we have
that $W(f_{1\sigma_{1}})\in C(\mathscr{G}_{loop})$. Thus,
$W(f_{1\sigma_{1}})\in\mathscr{G}_{loop}\cap
C(\mathscr{G}_{loop})=\textbf{S}$.

Now, define $f_{1\sigma_{2}}$ to be the hypercycle consisting of all rank-$3$
edges embedded within $f_{1}$ and within $f_{1}^{\prime}$, together with the
rank-$2$ edges on the boundary of $f_{1}$ and the rank-$2$ red edges on the
border of $f_{1}^{\prime}$. Therefore, it follows that

$W(f_{1\sigma_{2}})=\prod_{e^{\prime}\in\partial(f_{1})\cap\overline{E}_{B}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f_{1})\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f_{1}^{\prime})\cap\overline{E}_{G}}\overline{K}_{e^{\prime}}$
(19)

belongs to the group $\mathscr{G}_{loop}$. We also know that every hypercycle
contains none or an even number of $W(f_{1\sigma_{2}})$ triangles. Thus, by
the Corollary 13 we have that $W(f_{1\sigma_{2}})\in C(\mathscr{G}_{loop})$.
Thus, $W(f_{1\sigma_{2}})\in\mathscr{G}_{loop}\cap
C(\mathscr{G}_{loop})=\textbf{S}$.

The third and final hypercycle associated with $f_{1}$, which we denote by
$f_{1\sigma_{3}}$, is the modulo $2$ sum of the hypercycles $f_{1\sigma_{1}}$
and $f_{1\sigma_{2}}$. The loop operator $W(f_{1\sigma_{3}})$ will be the
product of $W(f_{1\sigma_{1}})$ by $W(f_{1\sigma_{2}})$.

Finally, for the face $f_{2}\in F_{G}$, define $f_{2\sigma_{1}}$ as the
hypercycle formed by the $2p_{2}$ rank-$2$ edges on the border of $f_{2}$.
Therefore, by the same reasoning used for $W(f_{1\sigma_{1}})$, we have that

$W(f_{2\sigma_{1}})=\prod_{e^{\prime}\in\partial(f_{2})\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f_{2})\cap\overline{E}_{G}}\overline{K}_{e^{\prime}}$
(20)

is an independent stabilizer generator. ∎

######  Lemma 25.

The stabilizer operators $W(f_{1\sigma_{i}})$ with $i=1,2,3$ and
$W(f_{2\sigma_{1}})$ from Lemma 24 satisfy the condition $(C_{2})$ of the
definition of topological subsystem code.

###### Proof.

It follows from the equations (18), (19), and (20). ∎

######  Example 26.

Let $f_{1}\in F\subseteq F_{R}$ and $f_{2}\in F_{G}$ and consider the
construction of $\Gamma_{h}$ from tessellation $\\{6,6,6\\}$, as in Figure 11.

Figure 11: Face $f_{1}\in F\subseteq FR$ and face $f_{2}\in F_{G}$ of the
hypergraph $\Gamma_{h}$ obtained from tessellation $\\{6,6,6\\}$.

See that $f_{1\sigma_{1}}$ is given by the $6$ edges of $f_{1}^{\prime}$
inserted inside $f_{1}$. Therefore, the loop operator is given as
$(\ref{hiperciclo1})$, that is,
$W(f_{1\sigma_{1}})=(X_{3}X_{12})(X_{6}X_{10})(X_{9}X_{11})(Y_{3}Y_{10})(Y_{6}Y_{11})(Y_{9}Y_{12})$.
Thus, $W(f_{1\sigma_{1}})$ satisfies the Theorem 16 and the condition
$(C_{2})$.

On the other hand, $f_{1\sigma_{2}}$ is formed by the three triangles inserted
in $f_{1}$, by the triangle inserted in $f_{1}^{\prime}$, by the red edges on
the border of $f_{1}$, and by the red edges on the border of $f_{1}^{\prime}$.
It follows from $(\ref{hiperciclo2})$ that the loop operator is given by

$\displaystyle W(f_{1\sigma_{2}})$ $\displaystyle=$
$\displaystyle(Z_{1}Z_{2}Z_{3})(Z_{4}Z_{5}Z_{6})(Z_{7}Z_{8}Z_{9})(Z_{10}Z_{11}Z_{12})(X_{1}X_{8})(X_{2}X_{4})(X_{5}X_{7})(X_{3}X_{12})$
$\displaystyle(X_{9}X_{11})(X_{6}X_{10})$ $\displaystyle=$
$\displaystyle(Z_{1}Z_{2})(Z_{4}Z_{5})(Z_{7}Z_{8})(X_{1}X_{8})(X_{2}X_{4})(X_{5}X_{7})(Y_{3}Y_{10})(Y_{6}Y_{11})(Y_{9}Y_{12}),$

which also satisfies the Theorem 16 and the condition $(C_{2})$.

Finally, we have that $f_{2\sigma_{1}}$ is formed by the $6$ rank-$2$ edges of
$f_{2}$. Therefore, the loop operator is given as in $(\ref{hiperciclo3})$,
that is,
$W(f_{2\sigma_{1}})=(X_{2}X_{4})(X_{13}X_{14})(X_{15}X_{16})(Y_{4}Y_{16})(Y_{2}Y_{13})(Y_{14}Y_{15})$.
Thus, $W(f_{2\sigma_{1}})$ satisfies the Theorem 16 and the condition
$(C_{2})$.

To conclude this section, let’s remember the number of faces $N_{f}$, edges
$N_{e}$, and vertices $N_{v}$ of the tessellation $\\{2p_{1},2p_{2},2p_{3}\\}$
on a compact orientable surface of genus $g\geq 2$, as well as the number of
red $F_{R}$, green $F_{G}$, and blue $F_{B}$ faces. These values will be
needed so that we can determine the parameters of three of the four families
of topological subsystem codes that we will present in the next section.

As we can see in [25], the number of faces, edges, and vertices of this
tessellation is given by

$\displaystyle N_{f}$ $\displaystyle=$
$\displaystyle\frac{2(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})(g-1)}{p_{1}p_{2}p_{3}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}},$
(21) $\displaystyle N_{e}$ $\displaystyle=$
$\displaystyle\frac{6p_{1}p_{2}p_{3}(g-1)}{p_{1}p_{2}p_{3}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}},$
(22) $\displaystyle N_{v}$ $\displaystyle=$
$\displaystyle\frac{4p_{1}p_{2}p_{3}(g-1)}{p_{1}p_{2}p_{3}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}}.$
(23)

The number of red, green, and blue faces is given by

$\displaystyle F_{R}$ $\displaystyle=$
$\displaystyle\frac{2p_{2}p_{3}(g-1)}{p_{1}p_{2}p_{3}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}},$
$\displaystyle F_{G}$ $\displaystyle=$
$\displaystyle\frac{2p_{1}p_{3}(g-1)}{p_{1}p_{2}p_{3}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}},$
$\displaystyle F_{B}$ $\displaystyle=$
$\displaystyle\frac{2p_{1}p_{2}(g-1)}{p_{1}p_{2}p_{3}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}}.$

##  5 New topological subsystem codes

In this section, we present four families of subsystem codes and prove that
they are, in fact, topological. We also display the $n$, $k$, $r$, and $d$
parameters of these codes.

Two of these families are particular cases of our construction, seen in the
last section. The first family refers to the case in which we start from
tessellation $\\{2p_{1},2p_{2},4\\}$ and consider $F=F_{R}$. The second and
third families refer to the case in which we start from the tessellation
$\\{2p_{1},4,6\\}$ with $p_{1}>4$ even and with $p_{1}>6$ odd, respectively,
and also consider $F=F_{R}$. As the case $p_{1}>4$ even was not analyzed in
[21], we will address it here. Finally, the fourth and final family that we
will present is related to tessellation $\\{p,4,3,4\\}$ with $p$ odd, where we
solve the problem of $p$ being odd by introducing a triangle inside the
$p$-gon, just as we did in our construction in the last section.

These codes will be built both on the torus and on other compact orientable
surfaces with genus $g\geq 2$. However, in the case of the torus, we only have
one example for the first family, which is starting from the tessellation
$\\{6,12,4\\}$, one example for the second family, which is starting from the
tessellation $\\{12,4.6\\}$, and no examples for the third and fourth
families.

###  5.1 Topological subsystem codes from tessellations
$\\{2p_{1},2p_{2},4\\}$

Consider the trivalent, $3$-colorable tessellation $\\{2p_{1},2p_{2},4\\}$
with odd $p_{1}>2$ on a compact orientable surface $\mathbb{M}$. Using
$F=F_{R}$, construct the hypergraph $\Gamma_{h}$ as seen in Section 4.
However, here, the way we choose the edges of $f^{\prime}$, where we will take
the vertices of the triangle that we insert inside it, cannot be done
randomly.

As we saw in Section 3, when constructing the hypergraph $\Gamma_{h}$ for
$p_{1}>2$ even and $F=F_{R}$, we have that each $f\in F_{R}$ determines three
stabilizer generators (2 independent). Furthermore, because we consider
$F=F_{R}$, we also have that each $f\in F_{G}$ determines three stabilizer
generators (2 independent), as can be seen in Figure 12. In the proof of the
Theorem 18, which can be seen in [21], it is shown that the latter are
stabilizing generators.

Figure 12: Hypercycles of the face $f\in F_{G}$ over the hypergraph
$\Gamma_{h}$ obtained from tessellation $\\{8,8,4\\}$. In black, we have the
hypercycles: $(a)$ $f_{\sigma_{1}}$ given by the rank-$2$ edges of $f$; $(b)$
$f_{\sigma_{2}}$ given by the rank-$3$ edges around $f$, by the rank-2 red
edges of $f$ and other red and green edges around $f$, necessary to complete
the hypercycle; $(c)$ $f_{\sigma_{3}}$ sum modulo $2$ of $f_{\sigma_{1}}$ and
$f_{\sigma_{2}}$.

For our construction, we also want that each $f\in F_{G}$ provide us with
three stabilizer generators ($2$ independent). But, when we add the triangles
inside $f_{i}^{\prime}$, we are creating an additional edge to the hypercycle
involving the triangles around some faces $f\in F_{G}$. This will be a problem
because, in this way, the stabilizer coming from this hypercycle will not
satisfy the Theorem 16; therefore, it does not satisfy $(C_{2})$, and
consequently, we will not be able to obtain topological subsystem codes. To
solve this, we impose the restriction that these triangles be constructed in
$f_{i}^{\prime}$, so that it does not increase new vertices or increase an
even number of vertices in each hypercycle involving the triangles around
$f\in F_{G}$, as can be seen in Figure LABEL:10-8-4-hyperbolico, where we
added four new vertices to the hypercycle involving the triangles around $f$.

Figure 13: Part of a hypergraph $\Gamma_{h}$ obtained from hyperbolic
tessellation $\\{10,8,4\\}$, where we added $4$ new vertices (highlighted in
white), that is, $4$ new edges for the hypercycle involving the triangles
around $f\in F_{G}$.

If no vertex is increased in the hypercycle $f_{\sigma_{2}}$ with $f\in
F_{G}$, we have that the loop operator will be given by

$W(f_{\sigma_{2}})=\prod_{e^{\prime}\in\partial(\overline{f})\cap\overline{E}_{B}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(\overline{f})\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(\overline{f})\cap\overline{E}_{G}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f)\cap\overline{E}_{R}}\overline{K}_{e^{\prime}},$
(24)

where $\overline{f}$ is the “face” composed of the rank-$2$ edges of
$\overline{\Gamma}_{h}$ that encompass the face $f\in F_{G}$ (see Figure 14),
and are used in the hypercycles $f_{\sigma_{2}}$ and $f_{\sigma_{3}}$,

Figure 14: Representation of the “face” $\overline{f}$, which is composed of
the black edges of $\overline{\Gamma}_{h}$ that encompass the face $f\in
F_{G}$.

If the number of vertices increases by an even amount, we will not be able to
write the loop operator as in the equation (24), separating by the colors of
the edges, but this will not prevent the loop operator from satisfying the
Theorem 16.

######  Example 27.

Consider the face $f$ given in Figure 15-(a), which represents a face of
$F_{G}$ arising from the construction of the hypergraph $\Gamma_{h}$ over the
hyperbolic tessellation $\\{10,8,4\\}$. If we have no new vertices on the
rank-$2$ edges of $\overline{f}$, it follows that the hypercycle
$f_{\sigma_{2}}$ surrounding the face $f\in F_{G}$ gives the loop operator

$\displaystyle W(f_{\sigma_{2}})$ $\displaystyle=$
$\displaystyle(Z_{1}Z_{2})(Z_{4}Z_{5})(Z_{7}Z_{8})(Z_{10}Z_{11})(Z_{13}Z_{14})(Z_{16}Z_{17})(Z_{19}Z_{20})(Z_{22}Z_{23})$
$\displaystyle(X_{1}X_{23})(X_{5}X_{7})(X_{11}X_{13})(X_{17}X_{19})(Y_{2}Y_{4})(Y_{8}Y_{10})(Y_{14}Y_{16})(Y_{20}Y_{22})$
$\displaystyle(X_{3}X_{24})(X_{6}X_{9})(X_{12}X_{15})(X_{18}X_{21}).$

Considering that only one vertex increases (qubit $25$ in Figure 15-(b)), it
will not be possible to write the $21$ edges in such a way that they satisfy
the Theorem 16, because we have an odd amount of the rank-$2$ edges of
$\overline{f}$, so they will never commute. Therefore, we require that the
number of vertices, if it increases, be even.

On the other hand, in Figure LABEL:example10-8-4-(c) increased two vertices
(qubits $25$ and $26$). Therefore, the loop operator will be given by

$\displaystyle W(f_{\sigma_{2}})$ $\displaystyle=$
$\displaystyle(Z_{1}Z_{2})(Z_{4}Z_{5})(X_{7}X_{26})(Y_{8}Y_{10})(X_{11}X_{13})(Y_{14}Y_{16})(X_{17}X_{19})(Y_{20}Y_{22})(X_{23}X_{25})$
$\displaystyle(Y_{1}Y_{25})(Y_{2}Y_{4})(Y_{5}Y_{26})(Z_{7}Z_{8})(Z_{10}Z_{11})(Z_{13}Z_{14})(Z_{16}Z_{17})(Z_{19}Z_{20})(Z_{22}Z_{23})$
$\displaystyle(X_{3}X_{24})(X_{6}X_{9})(X_{12}X_{15})(X_{18}X_{21}).$

Note that we cannot separate the link operators by colors as in the
$(\ref{equa1})$ equation, but even so, this operator satisfies Theorem 16.

Figure 15: Representation of a face $f\in F_{G}$ resulting from the
construction of the hypergraph $\Gamma_{h}$ over tessellation $\\{10,8,4\\}$.
$(a)$ No new vertices, $(b)$ one new vertex, and $(c)$ two new vertices on the
rank-$2$ edges of $\overline{f}$.

This guarantees that we can make this restriction regarding the way we fix the
triangles inside $f_{i}^{\prime}$ so that the hypercycles of $f\in F_{G}$
provide stabilizers satisfying the Theorem 16, given the fact that we have a
finite amount of $2p_{1}$-gons and $2p_{2}$-gons. Therefore, we have that each
$f\in F_{G}$ will provide three stabilizer generators ($2$ independent).

Thus, the stabilizer generators of this code will be given by emma 24, by the
loop operators $W(f_{\sigma_{2}})$ and by $W(f_{\sigma_{3}})$ with $f\in
F_{G}$, where $W(f_{\sigma_{2}})$ is the loop operator just described, and
$W(f_{\sigma_{3}})$ is the product of $W(f_{\sigma_{1}})$ by
$W(f_{\sigma_{2}})$. Therefore, this subsystem code satisfies the condition
$(C_{2})$.

Let us determine the number of independent stabilizer generators. It follows
from what we have just seen and the Lemma 24 that each $f\in F_{R}$ determines
$2$ independent stabilizer generators, and each $f\in F_{G}$ also determines
$2$ independent stabilizer generators. However, when we add all these
stabilizers together, not all of them will be independent, as we have the
following valid independence relationship

$\prod_{f\in F_{G}}W(f_{\sigma_{2}})=\prod_{f\in F_{R}}W(f_{\sigma_{1}}).$
(25)

For the case of $p_{1}$ even, that is, for the Theorem 18, other independence
relations may exist, as can be seen in the proof of this theorem in [21].
However, for our construction, the only valid relation in this case is the
relation (25). In this way, the number of independent stabilizing generators
will be

$s=2F_{R}+2F_{G}-1.$ (26)

Let’s now determine the parameters $n$, $k$, $r$, and $d$ of our subsystem
code. As we have only one Euclidean case, which is given by $\\{6,12,4\\}$,
the determination of the parameters is made in relation to tessellations
embedded in compact orientable surfaces of genus $g\geq 2$. However, the same
construction applies to the case $\\{6,12,4\\}$, with the difference that we
need to count the number of faces manually. As we know the values of $N_{v}$
and $F_{R}$, we can determine the quantity $n$ of physical qubits, which is
given by the number of vertices of the hypergraph $\Gamma_{h}$, that is,

$n=N_{v}+p_{1}F_{R}+3F_{R}=\frac{12(p_{1}p_{2}+2p_{2})(g-1)}{p_{1}p_{2}-2p_{1}-2p_{2}}.$
(27)

We also know that the quantity of independent stabilizing generators $s$ is
given in (26) and that $\dim\mathscr{G}=2r+s$, so we will determine the
quantity $r$ of gauge qubits. Remembering that the gauge group $\mathscr{G}$
is given by $\mathscr{G}=\langle\overline{K}_{e^{\prime}};\
e^{\prime}\in\overline{E}\rangle$.

To determine $\dim\mathscr{G}$, we need to determine the number of independent
generators, that is, the number of independent link operators. As quoted in
Section 2, the rank-$3$ edges, that is, the $\Gamma_{h}$ triangles, when
transformed into three rank-$2$ edges, provide us with only two independent
link operators. Combining these two independent link operators with all the
link operators of $\Gamma_{h}$, we also have that they will not all be
independent, as they obey the independence relations obtained for $s$, which
in this case is given only by the equation (25). Thus, we have that the number
of independent generators of $\mathscr{G}$ will be the number of rank-$2$
edges of $\Gamma_{h}$, plus twice the number of rank-$3$ edges, minus the
number of independence relations. We have that the rank-$2$ and rank-$3$ edges
for the hypergraph $\Gamma_{h}$, obtained by our construction on the
tessellation $\\{2p_{1},2p_{2},4\\}$, are given by

$E_{2}=2p_{2}F_{G}+(p_{1}+3)F_{R}\ \ \textrm{e}\ \ E_{3}=(p_{1}+1)F_{R}.$ (28)

Thus,

$\displaystyle\dim\mathscr{G}$ $\displaystyle=$ $\displaystyle
2p_{2}F_{G}+(p_{1}+3)F_{R}+2(p_{1}+1)F_{R}-1$ $\displaystyle=$ $\displaystyle
2p_{2}F_{G}+3p_{1}F_{R}+5F_{R}-1$ $\displaystyle=$ $\displaystyle
N_{v}+N_{e}+5F_{r}-1,$

since $2p_{2}F_{G}=N_{v}$ e $3p_{1}F_{R}=N_{e}$. As $\dim\mathscr{G}=2r+s$ e
$s=2F_{G}+2F_{R}-1$, we have

$\displaystyle 2r+2F_{G}+2F_{R}-1$ $\displaystyle=$ $\displaystyle
N_{v}+N_{e}+5F_{R}-1\Leftrightarrow$ $\displaystyle 2r$ $\displaystyle=$
$\displaystyle N_{v}+N_{e}+5F_{R}-2F_{G}-2F_{R}-2F_{B}+2F_{B}+2N_{v}-2N_{v}$
$\displaystyle=$ $\displaystyle-2(N_{v}-N_{e}+N_{f})+2N_{v}+5F_{R}$
$\displaystyle=$ $\displaystyle-2\chi+2N_{v}+5F_{R},$

since $F_{G}+F_{R}+F_{B}=N_{f}$ e $2F_{B}+N_{v}=N_{e}$. Therefore,

$r=-\chi+N_{v}+\frac{5F_{R}}{2},$ (29)

which will only make sense if $r$ is a positive integer. As we already know
the values of $n$, $r$, and $s$, it is easy to determine the value of $k$, as
it is given by $k=n-r-s$. So,

$\displaystyle k$ $\displaystyle=$ $\displaystyle
N_{v}+p_{1}F_{R}+3F_{R}+\chi-N_{v}-\frac{5F_{R}}{2}-2F_{R}-2F_{G}+1$
$\displaystyle=$ $\displaystyle
N_{e}-N_{v}-2F_{R}-2F_{G}-2F_{B}+2F_{B}+3F_{R}+\chi-\frac{5F_{R}}{2}+1$
$\displaystyle=$
$\displaystyle\frac{-2N_{f}+4F_{B}+6F_{R}-5F_{R}+2+2N_{v}-2N_{v}+2N_{e}-2N_{e}}{2}$
$\displaystyle=$
$\displaystyle\frac{-2(N_{v}-N_{e}+N_{f})+3N_{v}-2N_{e}+F_{R}+2}{2}$
$\displaystyle=$ $\displaystyle-\chi+1+\frac{F_{R}}{2},$

since $N_{v}+p_{1}F_{R}=N_{e}$, $4F_{B}=N_{v}$ e $3N_{v}-2N_{e}=0$. Thus,

$k=-\chi+1+\frac{F_{r}}{2},$ (30)

which will only make sense if $k$ is a positive integer. Finally, the distance
$d$ follows in a similar way to what was done in Section 2, that is, we will
limit it by the smallest number of triangles in a homological non-trivial
closed hypercycle, that is, it will be the smallest weight among the operators
of $C(\mathscr{G})\setminus\textbf{S}$.

The next two results are used to prove that $\Gamma_{h}$ satisfies the
conditions $(C_{1})$ and $(C_{3})$ of the topological subsystem code
definition. These results can be seen in [21], but here we will prove that
they are valid for our construction.

######  Lemma 28.

Let $\sigma$ be a homologically non-trivial closed hypercycle of $\Gamma_{h}$.
Then $\sigma$ must contain edges of rank-$3$.

###### Proof.

Suppose the hypercycle $\sigma$ does not contain rank-$3$ edges. It follows
from our construction that every vertex of the hypergraph $\Gamma_{h}$ is
trivalent and has a rank-$3$ edge incident on it. We also have that
$\Gamma_{h}$ satisfies $(H_{5})$, and the rank-$3$ edges are all blue. Thus,
$\sigma$ is formed by rank-$2$ edges that have alternating colors between red
and green.

If $\sigma$ contains some vertex $v\in f_{1}^{\prime}$ for $f_{1}^{\prime}$
inserted in $f_{1}\in F=F_{R}$, then all $p_{1}+3$ vertices of
$f_{1}^{\prime}$ are in $\sigma$, that is, the hypercycle $f_{1\sigma_{1}}$,
which is homologically trivial, is part of $\sigma$. We can make the sum
modulo $2$ of $\sigma$ with $f_{1\sigma_{1}}$, thus discarding
$f_{1\sigma_{1}}$ of $\sigma$. This way, only hypercycles that do not contain
vertices of $f_{1}^{\prime}$ will remain in $\sigma$. We can do this for all
faces $f^{\prime}$ that have vertices in $\sigma$. Therefore, the remaining
vertices of $\sigma$ will be vertices of faces of $F_{G}$. Again, due to the
fact that $\sigma$ does not have rank-$3$ edges, it follows that it will have
all $2p_{2}$ vertices of some face $f_{2}\in F_{G}$, that is, it has
$f_{2\sigma_{1}}$. Then, $\sigma$ is a homologically trivial hypercycle, which
is absurd. Therefore, $\sigma$ must contain rank-$3$ edges. ∎

######  Lemma 29.

Let $\sigma$ be a homologically non-trivial closed hypercycle of $\Gamma_{h}$.
Then, $W(\sigma)$ is not in the gauge group $\mathscr{G}$.

###### Proof.

Suppose $W(\sigma)\in\mathscr{G}$. Then

$W(\sigma)=\prod_{e^{\prime}\in
E_{2}\cap\sigma}K_{e^{\prime}}\prod_{e^{\prime}\in
E_{3}\cap\sigma}K_{e^{\prime}}\in\mathscr{G},$

where $E_{2}$ and $E_{3}$ are the sets of rank-$2$ and rank-$3$ edges of
$\Gamma_{h}$, respectively. We know that the edges of $E_{2}\cap\sigma$ are
edges of $\overline{\Gamma}_{h}$, and the link operators $K_{e^{\prime}}$ and
$\overline{K}_{e^{\prime}}$ are the same as $e^{\prime}\in E_{2}\cap\sigma$.
Thus, taking the product of $W(\sigma)\in\mathscr{G}$ and
$\prod_{e^{\prime}\in
E_{2}\cap\sigma}\overline{K}_{e^{\prime}}\in\mathscr{G}$, it follows that the
operator of the type $Z$

$O_{\sigma}=\prod_{e^{\prime}\in
E_{3}\cap\sigma}K_{e^{\prime}}=W(\sigma)\prod_{e^{\prime}\in
E_{2}\cap\sigma}\overline{K}_{e^{\prime}}\in\mathscr{G},$

that is, it will be generated by the link operators of the form $\\{X\otimes
X,Y\otimes Y,Z\otimes Z\\}$.

Suppose $O_{\sigma}$ is generated by $O_{\sigma}=K^{(X,Y)}K^{(Z)}$, where
$K^{(X,Z)}$ consists only of operators of the form $X\otimes X$ and $Y\otimes
Y$. $K^{(Z)}$ only consists of operators of the form $Z\otimes Z$. Therefore,
$O_{\sigma}K^{(Z)}=K^{(X,Y)}$ and the edges in the support of $K^{(X,Y)}$ must
form a closed hypercycle consisting only of rank-$2$ edges. According to the
previous lemma, this closed hypercycle is homologically trivial and consists
of the union of hypercycles that are boundaries of a collection of faces $f\in
F_{G}$ and $f_{i}^{\prime}$ with $f_{i}\in F_{R}$. We denote this collection
by $\mathscr{F}$. Thus, the rank-$3$ edges of $\sigma$ are incident on the
vertices of this hypercycle obtained through $K^{(X,Y)}$.

Consider the operator $O^{\prime}=\prod_{f\in\mathscr{F}}W(f_{\sigma_{i}})$,
where $f_{\sigma_{i}}$ is a hypercycle associated with $f$, which contains all
edges of rank-$3$ of $\sigma$.

Claim: The rank-$3$ edges, which are incident on the vertices in the support
of $K^{(X,Y)}$ but are not in $\sigma$, are also not in $O^{\prime}$.

Indeed, suppose that $v$ is not a vertex in the support of $O_{\sigma}$ but is
a vertex in the support of $K^{(X,Y)}$. Then, the three gauge operators
$X_{v}X_{i}$, $Y_{v}Y_{j}$, and $Z_{v}Z_{u}$ act on $v$. Therefore, suppose,
without loss of generality, that $u$ is a vertex in the support of $K^{(Z)}$.
Thus, $u$ is not a vertex in the support of $O_{\sigma}$, otherwise, as $u$
belongs to the support of $K^{(Z)}$, then $v$ would belong to the support of
$O_{\sigma}$, which is absurd.

Now, let $e^{\prime}\in E_{3}\setminus\sigma$ be such that $e^{\prime}$ has
exactly two vertices in the support of $O_{\sigma}K^{(Z)}=K^{(X,Y)}$.
Therefore, there are two faces $f_{a}$ and $f_{b}$ associated with these two
vertices. Thus, there is a hypercycle $f_{a\sigma_{2}}$ that encompasses
$f_{a}$ and contains $e^{\prime}$, and there is a hypercycle $f_{b\sigma_{2}}$
that encompasses $f_{b}$ and contains $e^{\prime}$. So the operator
$W(f_{a\sigma_{2}})W(f_{b\sigma_{2}})$ is not supported in $e^{\prime}$.
Therefore, the vertices of $e^{\prime}$ are not in the support of
$O^{\prime}$.

Continuing this procedure, we obtain stabilizers that are supported on the
same set of edges as $O_{\sigma}$, which implies that $\sigma$ is
homologically trivial, as it will be generated by a combination of trivial
hypercycles, which is absurd. Therefore, $W(\sigma)\not\in\mathscr{G}$. ∎

######  Remark 30.

Lemma 28 guarantees that the homologically non-trivial closed hypercycles
$\sigma$ have edges of rank-$3$. This is very important, because otherwise
$W(\sigma)$ would be a stabilizer and $(C_{1})$ and $(C_{3})$ would not be
satisfied. The Lemma 29 guarantees that
$W(\sigma)\not\in\mathscr{G}=C(\mathscr{G}_{loop})$, thus guaranteeing that
logical operators can be identified with homologically closed hypercycles that
are not trivial, and stabilizer operators can be identified with homologically
trivial closed hypercycles, that is, $(C_{1})$ and $(C_{3})$ are satisfied.

######  Theorem 31.

Consider a trivalent, $3$-colorable tessellation $\\{2p_{1},2p_{2},4\\}$ with
odd $p_{1}>2$. Applying the construction of $\Gamma_{h}$ with $F=F_{R}$ seen
in the section 4, we obtain topological subsystem codes with parameters

$\displaystyle[[N_{e}+3F_{R},-\chi+1+\frac{F_{R}}{2},-\chi+N_{v}+\frac{5F_{R}}{2},d\leq
l]]$ (31)

where $l$ is the number of triangles in a homologically non-trivial
hypercycle.

###### Proof.

It follows from Theorem 22 that $\Gamma_{h}$ provides subsystem codes, and it
follows from Lemma 25 and the equation (24) that this subsystem code satisfies
$(C_{2})$. By the Lemmas 28 and 29, we have that it will satisfy $(C_{1})$ and
$(C_{3})$. Therefore, the code obtained is a topological subsystem code.
Finally, by the equations (27), (29), and (30) we have that the parameters
$n$, $r$, and $k$ are given as in (31). The fact that the distance is limited
by the number of triangles in a homologically non-trivial hypercycle follows
from the discussion in this section. ∎

Figure 16: Hypergraph $\Gamma_{h}$ arising from tessellation $\\{6,12,4\\}$.

######  Example 32.

Consider the hypergraph $\Gamma_{h}$ arising from tessellation $\\{6,12,4\\}$
given in Figure 16. By the Theorem 31, we obtain a topological subsystem code
through $\Gamma_{h}$. Let’s determine the parameters of this code.

This hypergraph has $96$ vertices, that is, $96$ qubits. Therefore, it follows
that $n=96$. To determine $r$, we need to determine $s$, $E_{2}$, $E_{3}$, and
$\dim\mathscr{G}$. Remembering that the number of independent stabilizer
generators is given by $s=2F_{R}+2F_{G}-1$ and, in this case, $F_{R}=8$ and
$F_{G}=4$, therefore, $s=23$. $E_{2}$ and $E_{3}$ are given by
$E_{2}=2p_{2}F_{G}+(p_{1}+3)F_{R}$ and $E_{3}=(p_{1}+1)F_{R}$. Then, in this
case, $E_{2}=96$ and $E_{3}=32$. Since $\dim\mathscr{G}=E_{2}+2E_{3}-1$, it
follows that $\dim\mathscr{G}=159$. But $\dim\mathscr{G}=2r+s$, so it follows
that $159=2r+23\Longleftrightarrow r=68$. Now, since $k=n-r-s$, it follows
that $k=5$.

Finally, searching for homologically non-trivial hypercycles, the one with the
smallest number of triangles found has $l=4$ triangles. Therefore, $d\leq 4$.
Therefore, we obtain a topological subsystem code with parameters
$[[96,5,68,4]]$.

To conclude the study on this family of codes, we present in Tables 1 and 2
the parameters $n$, $k$ and $r$, obtained from the hypergraph $\Gamma_{h}$
originating from the tessellations $\\{2p_{1},2p_{2},4\\}$ on a compact
orientable surface $\mathbb{M}$ with genus $g=2,3,4$ and $5$. We also present
in each case the quantity $s$ of independent stabilizer generators together
with the tessellations $\\{2p_{1},2p_{2},4\\}$ used to construct $\Gamma_{h}$.
We do not present the value of $d$ in any of the tables, because despite
having an upper bound for the distance, we still cannot determine a value in
general.

$g$ |  Tesselação |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
2 |  $\\{6,14,4\\}$ |  79 |  $[[336,17,240,d]]$
2 |  $\\{14,6,4\\}$ |  79 |  $[[288,9,200,d]]$
2 |  $\\{6,16,4\\}$ |  43 |  $[[192,11,138,d]]$
2 |  $\\{6,18,4\\}$ |  31 |  $[[144,9,104,d]]$
2 |  $\\{10,8,4\\}$ |  35 |  $[[144,7,102,d]]$
2 |  $\\{6,20,4\\}$ |  25 |  $[[120,8,87,d]]$
2 |  $\\{18,6,4\\}$ |  31 |  $[[120,5,84,d]]$
2 |  $\\{6,24,4\\}$ |  19 |  $[[96,7,70,d]]$
2 |  $\\{6,36,4\\}$ |  13 |  $[[72,6,53,d]]$
2 |  $\\{10,10,4\\}$ |  15 |  $[[72,5,52,d]]$
2 |  $\\{10,20,4\\}$ |  5 |  $[[36,4,27,d]]$
3 |  $\\{6,14,4\\}$ |  159 |  $[[672,33,480,d]]$
3 |  $\\{14,6,4\\}$ |  159 |  $[[576,17,400,d]]$
3 |  $\\{6,16,4\\}$ |  87 |  $[[384,21,276,d]]$
3 |  $\\{6,18,4\\}$ |  63 |  $[[288,17,208,d]]$

$g$ |  Tesselação |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
3 |  $\\{10,8,4\\}$ |  71 |  $[[288,13,204,d]]$
3 |  $\\{6,20,4\\}$ |  51 |  $[[240,15,174,d]]$
3 |  $\\{18,6,4\\}$ |  63 |  $[[240,9,168,d]]$
3 |  $\\{6,24,4\\}$ |  39 |  $[[192,13,140,d]]$
3 |  $\\{6,28,4\\}$ |  33 |  $[[168,12,123,d]]$
3 |  $\\{6,36,4\\}$ |  27 |  $[[144,11,106,d]]$
3 |  $\\{10,10,4\\}$ |  31 |  $[[144,9,104,d]]$
3 |  $\\{6,60,4\\}$ |  21 |  $[[120,10,89,d]]$
3 |  $\\{10,12,4\\}$ |  21 |  $[[108,8,79,d]]$
3 |  $\\{10,20,4\\}$ |  11 |  $[[72,7,54,d]]$
3 |  $\\{18,12,4\\}$ |  9 |  $[[60,6,45,d]]$
3 |  $\\{14,28,4\\}$ |  5 |  $[[48,6,37,d]]$

Table 1: Parameters of topological subsystem codes from tessellations
$\\{2p_{1},2p_{2},4\\}$ with $p_{1}>2$ odd, $g=2$ and $3$.

$g$ |  Tesselação |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
4 |  $\\{6,14,4\\}$ |  239 |  $[[1008,49,720,d]]$
4 |  $\\{14,6,4\\}$ |  239 |  $[[864,25,600,d]]$
4 |  $\\{6,16,4\\}$ |  131 |  $[[576,31,414,d]]$
4 |  $\\{6,18,4\\}$ |  95 |  $[[432,25,312,d]]$
4 |  $\\{10,8,4\\}$ |  107 |  $[[432,19,306,d]]$
4 |  $\\{6,20,4\\}$ |  77 |  $[[360,22,261,d]]$
4 |  $\\{18,6,4\\}$ |  95 |  $[[360,13,252,d]]$
4 |  $\\{6,24,4\\}$ |  59 |  $[[288,19,210,d]]$
4 |  $\\{6,30,4\\}$ |  47 |  $[[240,17,176,d]]$
4 |  $\\{6,36,4\\}$ |  41 |  $[[216,16,159,d]]$
4 |  $\\{10,10,4\\}$ |  47 |  $[[216,13,156,d]]$
4 |  $\\{6,48,4\\}$ |  35 |  $[[192,15,142,d]]$
4 |  $\\{14,8,4\\}$ |  43 |  $[[192,11,138,d]]$
4 |  $\\{30,6,4\\}$ |  47 |  $[[192,9,136,d]]$
4 |  $\\{6,84,4\\}$ |  29 |  $[[168,14,125,d]]$
4 |  $\\{10,20,4\\}$ |  17 |  $[[108,10,81,d]]$
4 |  $\\{14,14,4\\}$ |  15 |  $[[96,9,72,d]]$
4 |  $\\{18,36,4\\}$ |  5 |  $[[60,8,47,d]]$
5 |  $\\{6,14,4\\}$ |  319 |  $[[1344,65,960,d]]$
5 |  $\\{14,6,4\\}$ |  319 |  $[[1152,33,800,d]]$

$g$ |  Tesselação |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
5 |  $\\{6,16,4\\}$ |  175 |  $[[768,41,552,d]]$
5 |  $\\{6,18,4\\}$ |  127 |  $[[576,33,416,d]]$
5 |  $\\{10,8,4\\}$ |  143 |  $[[576,25,408,d]]$
5 |  $\\{6,20,4\\}$ |  103 |  $[[480,29,348,d]]$
5 |  $\\{18,6,4\\}$ |  127 |  $[[480,17,336,d]]$
5 |  $\\{6,24,4\\}$ |  79 |  $[[384,25,280,d]]$
5 |  $\\{6,28,4\\}$ |  67 |  $[[336,23,246,d]]$
5 |  $\\{6,36,4\\}$ |  55 |  $[[288,21,212,d]]$
5 |  $\\{10,10,4\\}$ |  63 |  $[[288,17,208,d]]$
5 |  $\\{6,44,4\\}$ |  49 |  $[[264,20,195,d]]$
5 |  $\\{6,60,4\\}$ |  43 |  $[[240,19,178,d]]$
5 |  $\\{6,108,4\\}$ |  37 |  $[[216,18,161,d]]$
5 |  $\\{10,12,4\\}$ |  43 |  $[[216,15,158,d]]$
5 |  $\\{10,20,4\\}$ |  23 |  $[[144,13,108,d]]$
5 |  $\\{14,12,4\\}$ |  25 |  $[[144,12,107,d]]$
5 |  $\\{18,12,4\\}$ |  19 |  $[[120,11,90,d]]$
5 |  $\\{10,60,4\\}$ |  13 |  $[[108,12,83,d]]$
5 |  $\\{14,28,4\\}$ |  11 |  $[[96,11,74,d]]$
5 |  $\\{30,12,4\\}$ |  13 |  $[[96,10,73,d]]$
5 |  $\\{22,44,4\\}$ |  5 |  $[[72,10,57,d]]$

Table 2: Parameters of topological subsystem codes from tesellations
$\\{2p_{1},2p_{2},4\\}$ with $p_{1}>2$ odd, $g=4$ and $5$.

###  5.2 Topological subsystem codes From tessellation $\\{2p_{1},4,6\\}$

Consider the trivalent and $3$-colorable tessellation $\\{2p_{1},4,6\\}$ on a
compact orientable surface $\mathbb{M}$. We will construct two families of
topological subsystem codes. The first one is for even $p_{1}>4$. This family
is not considered in [21]. In this way, we will provide a family of codes
following the construction given in [21], as we saw in Section 3. The second
family will be for odd $p_{1}>6$, where we make our construction of
$\Gamma_{h}$ given in Section 4. For both cases, we keep fixing the
$2p_{1}$-gons as red faces, the $4$-gons as green faces, and the $6$-gons as
blue faces.

### Codes From tessellation $\\{2p_{1},4,6\\}$ with even $p_{1}>4$

Using $F=F_{R}$, construct the hypergraph $\Gamma_{h}$ as seen in Section 3,
so that the triangles border the $6$-gons as in Figure 17. Note that the only
Euclidean example of this case is $\\{12,4,6\\}$.

Figure 17: Hypergraph $\Gamma_{h}$ arising from tessellation $\\{12,4,6\\}$.

As in Section 3, each $f\in F_{R}$ provides $3$ stabilizer generators ($2$
independent), and, just as in the Lemma 24, we will have a stabilizer for each
face $f\in F_{G}$, which is given by

$W(f_{\sigma_{1}})=\prod_{e^{\prime}\in\partial(f)\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f)\cap\overline{E}_{G}}\overline{K}_{e^{\prime}}.$
(32)

However, in this case, even with $F=F_{R}$, there is no other stabilizer for
$f\in F_{G}$. But there will be a third independent stabilizer generator for
each face $f\in F_{R}$, which comes from a hypercycle, denoted by
$\overline{f}_{\sigma}$, as in Figure 18-($a$), where $\overline{f}$ is the
“face” given in Figure 18-($b$). See that the stabilizer coming from the
hypercycle $\overline{f}_{\sigma}$ satisfies the Theorem 16, since it can be
written as

$\displaystyle
W(\overline{f}_{\sigma})=\prod_{e^{\prime}\in\partial(\overline{f})\cap\overline{E}_{B}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(\overline{f})\cap\overline{E}_{G}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(\overline{f})\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}\prod_{\begin{subarray}{c}e^{\prime}\in\partial(f_{i})\cap\overline{E}_{G}\\\
f_{i}\in F_{G};\
f_{i}\subset\overline{f}\end{subarray}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f)\cap\overline{E}_{B}}\overline{K}_{e^{\prime}}\prod_{e^{\prime}\in\partial(f)\cap\overline{E}_{R}}\overline{K}_{e^{\prime}}$
(33)

Figure 18: ($a$) Hypercycle $\overline{f}_{\sigma}$ with colored edges, which
corresponds to face $f\in F_{R}$. ($b$) Colored edges highlighting the “face”
$\overline{f}$.

Thus, the stabilizer generators will be given as we saw in Section 3, that is,
they are given by $W(f_{\sigma_{1}})$, $W(f_{\sigma_{2}})$, and
$W(f_{\sigma_{3}})$, where
$W(f_{\sigma_{3}})=W(f_{\sigma_{1}})W(f_{\sigma_{2}})$. They are also given by
(32) and (33). Therefore, this subsystem code satisfies the condition
$(C_{2})$.

We need to determine the number of independent stabilizer generators. We have
that each $f\in F_{R}$ determines three independent stabilizer generators, and
each $f\in F_{G}$ determines one single independent stabilizer generator.
However, when we add all these stabilizers together, not all of them will be
independent, as we have the following valid independence relationship

$\prod_{f\in F_{R}}W(\overline{f}_{\sigma})=\prod_{f\in
F_{R}}W(f_{\sigma_{1}})\prod_{f\in F_{G}}W(f_{\sigma_{1}}).$ (34)

Now, let $\Gamma_{R}$ be the red reduced tessellation obtained from
$\\{2p_{1},4,6\\}$, that is, we reduce the red faces to a point and connect
these points, so that we obtain triangles as in Figure 19. If $\Gamma_{R}$ is
tripartite, that is, if the set of vertices admits a partition into three
sets, such that every edge has its end points in different sets, then we
divide the faces of $F_{R}$ into three sets, according to the tripartition,
which we denote by $F_{1}$, $F_{2}$, and $F_{3}$. Define $F^{F_{i}}_{G}$ to be
the set of green faces that are adjacent to the hypercycles
$\overline{f}_{\sigma}$ with $f\in F_{i}$ and $i=1,2,3$. Therefore, there are
three independence relations

$\displaystyle\prod_{f\in F_{1}}W(\overline{f}_{\sigma})\prod_{f\in
F^{F_{1}}_{G}}W(f_{\sigma_{1}})$ $\displaystyle=$ $\displaystyle\prod_{f\in
F_{2}}W(f_{\sigma_{2}})\prod_{f\in F_{3}}W(f_{\sigma_{2}}),$ (35)
$\displaystyle\prod_{f\in F_{2}}W(\overline{f}_{\sigma})\prod_{f\in
F^{F_{2}}_{G}}W(f_{\sigma_{1}})$ $\displaystyle=$ $\displaystyle\prod_{f\in
F_{1}}W(f_{\sigma_{2}})\prod_{f\in F_{3}}W(f_{\sigma_{2}}),$ (36)
$\displaystyle\prod_{f\in F_{3}}W(\overline{f}_{\sigma})\prod_{f\in
F^{F_{3}}_{G}}W(f_{\sigma_{1}})$ $\displaystyle=$ $\displaystyle\prod_{f\in
F_{1}}W(f_{\sigma_{2}})\prod_{f\in F_{2}}W(f_{\sigma_{2}}),$ (37)

of which only two are independent. Together with (34), we will have three
independence relations, and, therefore, the number of independent stabilizer
generators will be

$s=3F_{R}+F_{G}-1-2\delta_{\Gamma_{R}},$ (38)

onde $\delta_{\Gamma_{R}}=1$ se $\Gamma_{R}$ for tripartida e 0, caso
contrário.

Figure 19: Red dashed edges correspond to reduced tessellation $\Gamma_{R}$ of
the hypergraph $\Gamma_{h}$. See that $\Gamma_{R}$ is tripartite.

Let’s now determine the parameters $n$, $k$, $r$, and $d$ of our subsystem
code. As we have only one Euclidean case, $\\{12,4,6\\}$, the determination of
the parameters is made in relation to the hyperbolic tessellations embedded in
compact orientable surfaces of genus $g\geq 2$. However, the same construction
applies to the case $\\{12,4,6\\}$, with the difference that we will have to
count the number of faces manually. As we know the values of $N_{v}$ and
$F_{R}$, we can determine the quantity $n$ of physical qubits, which is given
by the number of vertices of the hypergraph $\Gamma_{h}$, that is,

$n=N_{v}+p_{1}F_{R}=\frac{36p_{1}(g-1)}{p_{1}-6}=N_{e}.$ (39)

We also know that the number of independent stabilizer generators $s$ is given
in (38) and that $\dim\mathscr{G}=2r+s$. But, to determine $\dim\mathscr{G}$,
we need to determine the number of independent generators of $\mathscr{G}$,
that is, we need to determine the number of independent link operators. We
have that the number of rank-2 and rank-3 edges for the hypergraph
$\Gamma_{h}$ is given by

$E_{2}=4F_{G}+p_{1}F_{R}\ \ \textrm{e}\ \ E_{3}=p_{1}F_{R}.$ (40)

Thus,

$\displaystyle\dim\mathscr{G}$ $\displaystyle=$ $\displaystyle
4F_{G}+p_{1}F_{R}+2p_{1}F_{R}-1-2\delta_{\Gamma_{R}}$ $\displaystyle=$
$\displaystyle N_{v}+N_{e}-1-2\delta_{\Gamma_{R}},$

since $4F_{G}=N_{v}$ and $3p_{1}F_{R}=N_{e}$. As $\dim\mathscr{G}2r+s$ and
$s=3FR+FG-1-2\delta_{\Gamma_{R}}$, making the appropriate substitutions, we
determine that the quantity $r$ of gauge qubits is given by

$r=-\chi+2N_{v}-\frac{N_{f}+N_{e}}{2},$ (41)

where $r$ must be a positive integer.

As we already know the values of $n$, $r$, and $s$ it is easy to determine the
value of $k$, as it is given by $k=n-r-s$. Thus, making the appropriate
substitutions, we have that the quantity $k$ of logical qubits is given by

$k=\chi-3F_{R}+\frac{F}{2}+1+2\delta_{\Gamma_{R}},$ (42)

where $k$ must be a positive integer.

Finally, the distance $d$ will be the same as for the family constructed
previously, that is, we get an upper bound to the code distance by the
smallest number of triangles in a homologically non-trivial closed hypercycle,
that is, it will be the smallest weight among the
$C(\mathscr{G})\setminus\textbf{S}$ operators.

Finally, the distance $d$ is calculated in the same way as in the previous
case, thus we obtain an upper limit for the code distance considering the
smallest number of triangles in a non-trivial homologically closed hypercycle,
that is, it will be the smallest weight among the
$C(\mathscr{G})\setminus\textbf{S}$ operators.

The Lemmas 28 and 29 are also valid here, and their proofs are done in a
similar way. From this, it follows that this code satisfies the conditions
$(C_{1})$ and $(C_{3})$. We have already seen that this code satisfies
$(C_{2})$ and it was shown in [21] that the graph $\Gamma_{h}$ built on this
type of tessellation satisfies $(H_{1}),\ldots,(H_{5})$. Therefore, this
demonstrates the following theorem.

######  Theorem 33.

Consider a trivalent and $3$-colorable tessellation $\\{2p_{1},4,6\\}$ with
even $p_{1}>4$. Applying the construction of $\Gamma_{h}$ given in [21] with
$F=F_{R}$ such that the edges of rank-$3$ are the border of the $6$-gons, we
obtain topological subsystem codes with parameters

$[[N_{e},\chi-3F_{R}+\frac{F}{2}+1+2\delta_{\Gamma_{R}},-\chi+2N_{v}-\frac{N_{f}+N_{e}}{2},d\leq
l]],$ (43)

where $l$ is the number of triangles in a homologically non-trivial
hypercycle.

######  Example 34.

Consider the hypergraph $\Gamma_{h}$ arising from tessellation $\\{6,12,4\\}$
illustrated in Figure 17. By the Theorem 33, we obtain a topological subsystem
code through $\Gamma_{h}$. Let’s determine the parameters of this code.

This hypergraph has $72$ vertices, that is, $72$ qubits. Then, $n=72$. To
determine $r$, we need to determine $s$, $E_{2}$, $E_{3}$, and
$\dim\mathscr{G}$. Remembering that the number of independent stabilizer
generators is given by $s=3F_{R}+F_{G}-1-2\delta_{\Gamma_{R}}$ and, in this
case, $F_{R}=4$ and $F_{G}=12$. As the reduced tessellation $\Gamma_{R}$ is
not tripartite, it follows that $\delta_{\Gamma_{R}}=0$, therefore, $s=23$.
$E_{2}$ and $E_{3}$ are given by $E_{2}=4F_{G}+p_{1}F_{R}$ and
$E_{3}=p_{1}F_{R}$. Therefore, in this case, $E_{2}=72$ and $E_{3}=24$. Since
$\dim\mathscr{G}=E_{2}+2E_{3}-1$, it follows that $\dim\mathscr{G}=119$. But
$\dim\mathscr{G}=2r+s$, so it follows that $119=2r+23\Longleftrightarrow
r=48$. Now, since $k=n-r-s$, we have that $k=1$.

Finally, searching for homologically non-trivial hypercycles, the one with the
smallest number of triangles found has $l=4$ triangles. Therefore, $d\leq 4$.
Therefore, we obtain a topological subsystem code with parameters
$[[72,1,48,4]]$.

To conclude the study on this family of codes, we present in the table 3 the
parameters obtained from the hypergraph $\Gamma_{h}$ arising from the
tessellations $\\{2p_{1},4,6\\}$ on a compact orientable surface $\mathbb{M}$
with genders $g=2$, $3$, $4$, and $5$. As we cannot determine whether or not
the graph will be tripartite, we will determine the two possible values for
the encoded qubits, which we call $k_{1}$ and $k_{2}$, respectively. We also
present in each case the tessellations $\\{2p_{1},4,6\\}$ used to construct
$\Gamma_{h}$, together with the quantity $s_{1}$ and $s_{2}$ of independent
stabilizer generators, if the graph is tripartite and if not, respectively.

$g$ |  Tessellation |  $s_{1}$ |  $[[n,k_{1},r,d]]$ |  $s_{2}$ |  $[[n,k_{2},r,d]]$
---|---|---|---|---|---
2 |  $\\{16,4,6\\}$ |  39 |  $[[144,6,99,d]]$ |  41 |  $[[144,4,99,d]]$
2 |  $\\{20,4,6\\}$ |  21 |  $[[90,6,63,d]]$ |  23 |  $[[90,4,63,d]]$
2 |  $\\{24,4,6\\}$ |  15 |  $[[72,6,51,d]]$ |  17 |  $[[72,4,51,d]]$
2 |  $\\{36,4,6\\}$ |  9 |  $[[54,6,39,d]]$ |  11 |  $[[54,4,39,d]]$
3 |  $\\{16,4,6\\}$ |  81 |  $[[288,9,198,d]]$ |  83 |  $[[288,7,198,d]]$
3 |  $\\{20,4,6\\}$ |  45 |  $[[180,9,126,d]]$ |  47 |  $[[180,7,126,d]]$
3 |  $\\{24,4,6\\}$ |  33 |  $[[144,9,102,d]]$ |  35 |  $[[144,7,102,d]]$
3 |  $\\{28,4,6\\}$ |  27 |  $[[126,9,90,d]]$ |  29 |  $[[126,7,90,d]]$
3 |  $\\{36,4,6\\}$ |  21 |  $[[108,9,78,d]]$ |  23 |  $[[108,7,78,d]]$
3 |  $\\{60,4,6\\}$ |  15 |  $[[90,9,66,d]]$ |  17 |  $[[90,7,66,d]]$
4 |  $\\{16,4,6\\}$ |  123 |  $[[432,12,297,d]]$ |  125 |  $[[432,10,297,d]]$
4 |  $\\{20,4,6\\}$ |  69 |  $[[270,12,189,d]]$ |  71 |  $[[270,10,189,d]]$
4 |  $\\{24,4,6\\}$ |  51 |  $[[216,12,153,d]]$ |  53 |  $[[216,10,153,d]]$
4 |  $\\{36,4,6\\}$ |  33 |  $[[162,12,117,d]]$ |  35 |  $[[162,10,117,d]]$
4 |  $\\{48,4,6\\}$ |  27 |  $[[144,12,105,d]]$ |  29 |  $[[144,10,105,d]]$
4 |  $\\{84,4,6\\}$ |  21 |  $[[126,12,93,d]]$ |  23 |  $[[126,10,93,d]]$
5 |  $\\{16,4,6\\}$ |  165 |  $[[576,15,396,d]]$ |  167 |  $[[576,13,396,d]]$
5 |  $\\{20,4,6\\}$ |  93 |  $[[360,15,252,d]]$ |  95 |  $[[360,13,252,d]]$
5 |  $\\{24,4,6\\}$ |  69 |  $[[288,15,204,d]]$ |  71 |  $[[288,13,204,d]]$
5 |  $\\{28,4,6\\}$ |  57 |  $[[252,15,180,d]]$ |  59 |  $[[252,13,180,d]]$
5 |  $\\{36,4,6\\}$ |  45 |  $[[216,15,156,d]]$ |  47 |  $[[216,13,156,d]]$
5 |  $\\{44,4,6\\}$ |  39 |  $[[198,15,144,d]]$ |  41 |  $[[198,13,144,d]]$
5 |  $\\{60,4,6\\}$ |  33 |  $[[180,15,132,d]]$ |  35 |  $[[180,13,132,d]]$
Table 3: Parameters of topological subsystem codes from tessellation
$\\{2p_{1},4,6\\}$ with even $p_{1}>4$, $g=2$, $3$, $4$, and $5$, where
$s_{1}$ and $k_{1}$ are the values when $\Gamma_{R}$ is tripartite and $s_{2}$
and $k_{2}$ are the values when $\Gamma_{R}$ is not tripartite.

### Codes from tessellation $\\{2p_{1},4,6\\}$ with odd $p_{1}>6$

Using $F=F_{R}$, construct the hypergraph $\Gamma_{h}$ as in Section 4. Lemma
24 gives that each $f\in F_{R}$ provides $3$ stabilizer generators ($2$
independent) and each $f\in F_{G}$ provides $1$ independent stabilizer
generator. We want the stabilizer generators $W(\overline{f}_{\sigma})$ with
$f\in F_{R}$, seen for the case $p_{1}>4$ even, to continue being stabilizers.
But when we add the triangles inside $f^{\prime}$, we are creating an extra
vertex in some of the $\overline{f}_{\sigma}$ hypercycles, just as it happened
for the $\overline{f}_{\sigma_{2}}hypercycles$ with $f\in F_{G}$ from the
first family of codes we built. This will be a problem, because, in this way,
the stabilizer coming from the hypercycle $\overline{f}_{\sigma}$ will not
satisfy the Theorem 16, therefore, it does not satisfy $(C_{2})$, and
consequently, we will not obtain topological subsystem codes.

Again, to solve this, we impose the restriction that these triangles be
constructed in $f^{\prime}$, so that it does not increase new vertices or
increase an even number of vertices in each hypercycle of
$\overline{f}_{\sigma}$.

If we have no new vertices in $\overline{f}_{\sigma}$ hypercycle, we have that
the loop operator will be given by (33). If an even number of vertices
increases, we will not be able to write the loop operator as in the equation
(33), separating by the colors of the edges, but this will not prevent the
loop operator from satisfying the 16 Theorem.

Again, which guarantees that we can make this restriction on the way we fix
the triangles inside $f^{\prime}$, so that the hypercycles of
$\overline{f}_{\sigma}$ provide stabilizers satisfying the Theorem 16, is the
fact that we have a finite amount of $2p_{1}$-gons. Therefore, it follows that
each $f\in F_{R}$ will provide us with four stabilizing generators ($3$
independent), and each $f\in F_{G}$ will provide an independent stabilizer
generator. Thus, the stabilizer generators of this code will be given by the
Lemma 24 and the loop operators $W(\overline{f}_{\sigma})$. Therefore, this
subsystem code satisfies the condition $(C_{2})$.

Let us determine the number of independent stabilizer generators. For odd
$p_{1}>6$, we have that $\Gamma_{R}$ will never be tripartite. Thus, the only
independence relation that will be valid is (34). Therefore, the number of
independent stabilizing generators will be

$s=3F_{R}+F_{G}-1.$ (44)

Now we determine the parameters $n$, $k$, $r$, and $d$ of our subsystem code.
Note that there are no Euclidean tessellations for this case. Therefore, the
determination of the parameters will be made in relation to the tessellations
embedded in compact orientable surfaces of genus $g\geq 2$. The quantity of
physical qubits, $n$, is given by the quantity of vertices of the hypergraph
$\Gamma_{h}$,

$n=N_{v}+p_{1}F_{R}+3F_{R}=N_{e}+3F_{R}.$ (45)

The quantity of gauge qubits, $r$, is determined in a similar way to the other
cases. Here, the rank-$2$ and rank-$3$ edges are given by

$E_{2}=4F_{G}+p_{1}F_{R}+3F_{R}\ \ \textrm{e}\ \ E_{3}=p_{1}F_{R}+F_{R}.$ (46)

Then,

$\displaystyle\dim\mathscr{G}$ $\displaystyle=$ $\displaystyle
4F_{G}+p_{1}F_{R}+3F_{R}+2p_{1}F_{R}+2F_{R}-1$ $\displaystyle=$ $\displaystyle
N_{v}+N_{e}+5F_{R}-1.$

As $\dim\mathscr{G}=2r+s$ and $s=3F_{R}+F_{G}-1$, we have

$r=-\chi+2N_{v}-\frac{N_{f}+N_{e}}{2}+\frac{5F_{R}}{2},$ (47)

where $r$ is a positive integer. Now, for the parameter $k$, we know that
$k=n-r-s$. Then, we have

$k=\chi+\frac{N_{f}}{2}-\frac{5F_{R}}{2}+1,$ (48)

where $k$ is a positive integer.

Finally, the distance $d$ follows as in the other cases studied, that is, it
will be upper bound by the smallest number of triangles in a homologically
non-trivial closed hypercycle, that is, it will be the smallest weight among
the operators of $C(\mathscr{G})\setminus\textbf{S}$.

Lemmas 28 and 29 are also valid in this case, and their proofs are done in a
similar way. From this, it follows that this code satisfies the conditions
$(C_{1})$ and $(C_{3})$. We have already seen that this code satisfies
$(C_{2})$, and it is given in the Theorem 22 that the graph $\Gamma_{h}$
satisfies $(H_{1}),\ldots,(H_{5})$. Therefore, this proves the following
theorem.

######  Theorem 35.

Consider a trivalent, $3$-colorable tessellation $\\{2p_{1},4,6\\}$ with odd
$p_{1}>6$. Applying the construction of $\Gamma_{h}$ with $F=F_{R}$ seen in
Section 4, we obtain topological subsystem codes with parameters

$[[N_{e}+3F_{R},\chi+\frac{N_{f}}{2}-\frac{5F_{R}}{2}+1,-\chi+2N_{v}-\frac{N_{f}+N_{e}}{2}+\frac{5F_{R}}{2},d\leq
l]],$ (49)

where $l$ is the number of triangles in a homologically non-trivial
hypercycle.

To conclude, we present in Table 4 the parameters $n$, $k$, and $r$, obtained
from the hypergraph $\Gamma_{h}$ originating from the tessellations
$\\{2p_{1},4,6\\}$ on a compact orientable surface $\mathbb{M}$ with genus
$g=2,3,4$ and $5$. We also present in each case the quantity $s$ of
independent stabilizer generators together with the tessellations
$\\{2p_{1},4,6\\}$ used to construct $\Gamma_{h}$.

$g$ |  Tessellation |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
2 |  $\\{14,4,6\\}$ |  77 |  $[[288,10,201,d]]$
2 |  $\\{18,4,6\\}$ |  29 |  $[[120,6,85,d]]$
3 |  $\\{14,4,6\\}$ |  155 |  $[[576,19,402,d]]$
3 |  $\\{18,4,6\\}$ |  59 |  $[[240,11,170,d]]$
4 |  $\\{14,4,6\\}$ |  233 |  $[[864,28,603,d]]$
4 |  $\\{18,4,6\\}$ |  89 |  $[[360,16,255,d]]$
4 |  $\\{30,4,6\\}$ |  41 |  $[[192,12,139,d]]$
5 |  $\\{14,4,6\\}$ |  311 |  $[[1152,37,804,d]]$
5 |  $\\{18,4,6\\}$ |  119 |  $[[480,21,340,d]]$
Table 4: Parameters of topological subsystem codes coming from tessellations
$\\{2p_{1},4,6\\}$ with odd $p_{1}>6$ for $g=2,3,4$ and $5$.

### Codes from tessellation $\\{p,4,3,4\\}$

Here, we construct topological subsystem codes based on the tessellation
$\\{p,4,3,4\\}$ for odd $p\geq 7$, which is obtained from the tessellation
$\\{p,3\\}$ as follows: Given a tessellation $\\{p,3\\}$ on a compact surface
$\mathbb{M}$ of genus $g\geq 2$, take the dual tessellation $\\{3,p\\}$ and
keep the two tessellations $\\{p,3\\}$ and $\\{3,p\\}$ overlapping. In this
way, we form quadrilaterals with angles $\frac{2\pi}{p}$, $\frac{\pi}{2}$,
$\frac{2\pi}{3}$, and $\frac{\pi}{2}$. As each quadrilateral has an incenter,
we take the incenter of each of these quadrilaterals. Connecting these
incenters, we obtain the tessellation $\\{p,4,3,4\\}$, as can be seen in the
figure 1-($a$).

We saw the construction of the topological subsystem codes from [5] in Section
3: it begins from a trivalent and $3$-colorable tessellation
$\\{2p_{1},2p_{2},2p_{3}\\}$. When this tessellation is regular, that is, when
it is of the form $\\{p,3\\}$, after carrying out the constructions of
triangles and squares, the tessellation $\\{p,4,3,4\\}$ is obtained with even
$p$; that is, it is the same as carrying out the process described above,
starting from the tessellation $\\{p,3\\}$ with even $p$.

If $p$ is odd, the tessellation $\\{p,3\\}$ is not $3$-colorable, so the
construction given in [5] does not apply in this case. That’s why we are
considering an alternative approach to working with the odd $p$ case.

The process of building this family of topological subsystem codes is very
simple and similar to what we did in Section 4. First, we take a tessellation
$\\{p,3\\}$ with odd $p$ and carry out the process described previously, thus
obtaining the tessellation $\\{p,4,3,4\\}$. As $p$ is odd, considering
$\\{p,4,3,4\\}$ being the resulting hypergraph, it does not satisfy the
restriction $(H_{5})$, therefore, it does not satisfy the commutation rule,
and, thus, it does not support the construction of subsystem codes. To fix
this, we place a triangle inside each $p$-gon. These triangles cannot be
placed randomly, but we will explain how to place them later. Finally, we
color the triangles blue and alternate the other edges between red and green.

We denote the resulting hypergraph by $\Gamma_{h}$ and use it to construct the
topological subsystem code. To prove that our construction of $\Gamma_{h}$ can
be used to construct subsystem codes using the construction of [27], we will
prove that $\Gamma_{h}$ satisfies the constraints $(H_{1}),\ldots,(H_{5})$.

######  Theorem 36.

The hypergraph $\Gamma_{h}$ obtained from the tessellation $\\{p,4,3,4\\}$
with $p\geq 7$ odd satisfies the constraints $(H_{1}),\ldots,(H_{5})$ and,
therefore, it gives rise to subsystem codes whose group formed by loop
operators is given by

$\mathscr{G}_{loop}=\\{W(M);\ M\subseteq E\ \ \textrm{\'{e} um hiperciclo
fechado}\\},$ (50)

where $E$ is the set given by the edges of rank-$2$ and rank-$3$, and the
gauge group $\mathscr{G}$ is given by $\mathscr{G}=C(\mathscr{G}_{loop})$.

###### Proof.

Considering the triangles of tessellation $\\{p,4,3,4\\}$ as being rank-$3$
edges, we will only have rank-$2$ and rank-$3$ edges. Likewise, when we create
the triangle inside the $p$-gon from points taken on three of its edges, we
are producing one edge of rank-$3$, and three of rank-$2$. Therefore,
$\Gamma_{h}$ satisfies $(H_{1})$.

When we consider triangles to be rank-$3$ edges, it follows that every vertex
of $\\{p,4,3,4\\}$ will be trivalent. Likewise, the three new vertices will be
trivalent, as they divide an edge into two and have the new triangle as the
third edge. Thus, $\Gamma_{h}$ satisfies $(H_{2})$.

It follows from the construction of the tessellation $\\{p,4,3,4\\}$ that the
constraint $(H_{3})$ is satisfied. Even introducing the triangle inside each
$p$-gon does not change this condition.

The constraint $(H_{4})$ is also satisfied since the rank-$3$ edges of
$\\{p,4,3,4\\}$ do not intersect, and when we create the triangle inside the
$p$-gon, we take the points on the edges so that there is no intersection with
the other triangles. Therefore, the rank-$3$ edges of $\Gamma_{h}$ are two by
two disjoint.

Finally, with the introduction of the triangle inside each $p$-gon, the
restriction $(H_{5})$ is satisfied. We just color all the triangles blue and
alternate each $p+3$ edge between red and green. Therefore, $\Gamma_{h}$ gives
rise to subsystem codes as constructed in [27]. ∎

The operators $\overline{K}_{e^{\prime}}$ will be given in the same way as in
the previous cases. When we consider the ordinary graph
$\overline{\Gamma}_{h}$ given by $\Gamma_{h}$, we will have for
$e^{\prime}=(u,v)\in\overline{E}$ that $\overline{K}_{e^{\prime}}=Z_{u}Z_{v}$
if $e^{\prime}\in\overline{E}_{B}$, $\overline{K}_{e^{\prime}}=X_{u}X_{v}$ if
$e^{\prime}\in\overline{E}_{R}$, and $\overline{K}_{e^{\prime}}=Y_{u}Y_{v}$ if
$e^{\prime}\in\overline{E}_{G}$, where $\overline{E}_{B}$, $\overline{E}_{R}$
and $\overline{E}_{G}$ are the sets with all blue, red and green edges,
respectively, of the graph $\overline{\Gamma}_{h}$.

In the case of tessellation $\\{p,4,3,4\\}$ with even $p$, there are three
stabilizers for each $p$-gon, where only two are independent. One is formed by
the border of the $p$-gon, and the other by the triangles involving the
$p$-gon, and the edges necessary to close the hypercycle. As in our case $p$
is odd, when we introduce the triangle inside the $p$-gon, we create three
more edges, making the $p$-gon, now with three more edges, a stabilizer
satisfying the Theorem 16. However, when we introduce the triangle, this will
increase a vertex in the hypercycles involving the triangles of the $p$-gons
“adjacent” to the one where the triangle was introduced. For these hypercycles
to provide stabilizers, we require that these triangles be fixed within the
$p$-gons in such a way that an even number of vertices increases for each
hypercycle $f_{\sigma_{2}}$, or no new vertices increase. In this way, we
obtain three stabilizer generators (two independent) per $p$-gon.

######  Lemma 37.

Each $p$-gon of our construction on the tessellation $\\{p,4,3,4\\}$ provides
three hypercycles (two independent) and, consequently, three stabilizer
generators (two independent).

###### Proof.

The proof of this result follows in a similar way to that seen in Section 4.
The first hypercycle $f_{\sigma_{1}}$ is given by the $p+3$ edges of rank-$2$
created in $p$-gon, and the loop operator $W(f_{\sigma_{1}})$ is given by
(18). The second hypercycle $f_{\sigma_{2}}$ is given by the rank-$3$ edges
around the $p$-gon, together with the rank-$3$ edge created inside the $p$-gon
and the edges necessary to close the hypercycle. If no new edges have been
increased in this hypercycle, the loop operator $W(f_{\sigma_{2}})$ will be
given as (19). If an even number of new edges have increased in this
hypercycle, we will not be able to write the loop operator as in (19),
separating by colors, but it will continue as we saw in the case
$\\{2p_{1},2p_{2},4\\}$, that $W(f_{\sigma_{2}})$ will be a stabilizer. The
third and final hypercycle is given by the modulo $2$ sum of $f_{\sigma_{1}}$
and $f_{\sigma_{2}}$. The stabilizer will be given by the product of
$W(f_{\sigma_{1}})$ by $W(f_{\sigma_{2}})$. ∎

Then, it follows that these loop operators satisfy Theorem 16. Therefore, they
satisfy the condition $(C_{2})$ of the topological subsystem code definition.

Thus, we will have three stabilizer generators, where two are independent for
each $p$-gon. In the case of $p$ even, we had two independence relations,
which are given in (8). In the case of odd $p$, as we are not starting from a
$3$-colorable tessellation, and by introducing the triangle inside each
$p$-gon, there will be no independence relationship. Therefore, the quantity
$s$ of independent stabilizing generators will be twice the quantity of
$p$-gons.

To determine the value of $s$ and the other code parameters, we need the
number of faces, vertices, and edges of the tessellation $\\{p,3\\}$, which
are given in (LABEL:fevcolors), or it is,

$\displaystyle n_{f}=\frac{12(g-1)}{p-6},\ \ n_{e}=\frac{6p(g-1)}{p-6}\ \
\textrm{e}\ \ n_{v}=\frac{4p(g-1)}{p-6}.$

Consider $F_{\mathbb{P}}$, $F_{\mathbb{T}}$, and $F_{{\mathbb{Q}}}$ the amount
of $p$-gons, triangles, and squares of $\\{p,4,3,4\\}$, respectively. It
follows from the construction process of $\\{p,4,3,4\\}$ that the number of
$p$-gons is equal to the number of tessellation faces of $\\{p,3\\}$, the
number of triangles is equal to the number of tessellation vertices of
$\\{p,3\\}$, and the number of squares is equal to the number of tessellation
edges of $\\{p,3\\}$, that is,

$\displaystyle F_{\mathbb{P}}=\frac{12(g-1)}{p-6},\ \
F_{\mathbb{T}}=\frac{4p(g-1)}{p-6}\ \ \textbf{e}\ \
F_{{\mathbb{Q}}}=\frac{6p(g-1)}{p-6}.$

So, we have

$s=2F_{\mathbb{P}}=\frac{24(g-1)}{p-6}.$ (51)

Introducing the triangle inside each $p$-gon, we obtain 3 more vertices for
each $p$-gon. Thus, the quantity $n$ of physical qubits will be

$n=(p+3)F_{\mathbb{P}}.$ (52)

To determine the quantity of gauge qubits, $r$, we will proceed as in the
previous cases. To do this, we need the quantities of rank-$2$ and rank-$3$
edges of $\Gamma_{h}$, which are given by $E_{2}=(p+3)F_{\mathbb{P}}$ and
$E_{3}=F_{\mathbb{T}}+F_{\mathbb{P}}$, respectively. Therefore,
$\dim\mathscr{G}=E_{2}+2E_{3}=(p+3)F_{\mathbb{P}}+2F_{\mathbb{T}}+2F_{\mathbb{P}}$.
As $s=2F_{\mathbb{P}}$ and $\dim\mathscr{G}=2r+s$, it follows that

$\displaystyle 2r+2F_{\mathbb{P}}$ $\displaystyle=$
$\displaystyle(p+3)F_{\mathbb{P}}+2F_{\mathbb{T}}+2F_{\mathbb{P}}\Longleftrightarrow$
$\displaystyle 2r$ $\displaystyle=$
$\displaystyle(p+3)F_{\mathbb{P}}+2F_{\mathbb{T}}$ $\displaystyle=$
$\displaystyle 5F_{\mathbb{T}}+3F_{\mathbb{P}},$

since $pF_{\mathbb{P}}=3F_{\mathbb{T}}$. Thus,

$r=\frac{5F_{\mathbb{T}}+3F_{\mathbb{P}}}{2},$ (53)

where $r$ is a positive integer.

As we already know the values of $n$, $r$, and $s$, it is easy to determine
the value of $k$, as it is given by $k=n-r-s$. So, we have to

$\displaystyle k$ $\displaystyle=$
$\displaystyle(p+3)F_{\mathbb{P}}-\frac{5F_{\mathbb{T}}+3F_{\mathbb{P}}}{2}-2F_{\mathbb{P}}$
(54) $\displaystyle=$ $\displaystyle\frac{F_{\mathbb{T}}-F_{\mathbb{P}}}{2},$

where $k$ is a positive integer.

Finally, the distance $d$ follows as in the other cases, that is, it will be
upper bound by the smallest number of triangles in a homologically non-trivial
closed hypercycle, that is, it will be the smallest weight among the operators
of $C(\mathscr{G})\setminus\textbf{S}$.

Lemmas 28 and 29 are also valid here, and their proofs are done in a similar
way. From this, it follows that this code satisfies the conditions $(C_{1})$
and $(C_{3})$. We have already seen that this code satisfies $(C_{2})$, and it
was shown in Theorem 36 that the graph $\Gamma_{h}$ satisfies
$(H_{1}),\ldots,(H_{5})$. Therefore, this proves the following theorem.

######  Theorem 38.

Consider a tessellation $\\{p,3\\}$ with odd $p$ embedded in a compact
orientable surface $\mathbb{M}$ of genus $g\geq 2$. By carrying out the
process to obtain the tessellation $\\{p,4,3,4\\}$ and carrying out the
construction described previously, we obtain a topological subsystem code with
parameters

$[[(p+3)F_{\mathbb{P}},\frac{F_{\mathbb{T}}-F_{\mathbb{P}}}{2},\frac{5F_{\mathbb{T}}+3F_{\mathbb{P}}}{2},d\leq
l]],$ (55)

where $F_{\mathbb{P}}$ and $F_{\mathbb{T}}$ are the amount of $p$-gons and
triangles of the tessellation $\\{p,4,3,4\\}$, respectively, and $l$ is the
number of triangles in a homologously non-trivial hypercycle.

To conclude, we present in Table 5 the parameters $n$, $k$, and $r$, obtained
from the hypergraph $\Gamma_{h}$ originating from the tessellations
$\\{p,4,3,4\\}$ on a compact orientable surface $\mathbb{M}$ with genus $g=2$
to $g=7$. We also present in each case the quantity $s$ of independent
stabilizing generators together with the tessellations $\\{p,4,3,4\\}$ used to
construct $\Gamma_{h}$.

$g$ |  Tessellation |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
2 |  $\\{7,4,3,4\\}$ |  24 |  $[[120,8,88,d]]$
2 |  $\\{9,4,3,4\\}$ |  8 |  $[[48,4,36,d]]$
3 |  $\\{7,4,3,4\\}$ |  48 |  $[[240,16,176,d]]$
3 |  $\\{9,4,3,4\\}$ |  16 |  $[[96,8,72,d]]$
4 |  $\\{7,4,3,4\\}$ |  72 |  $[[360,24,264,d]]$
4 |  $\\{9,4,3,4\\}$ |  24 |  $[[144,12,108,d]]$
4 |  $\\{15,4,3,4\\}$ |  8 |  $[[72,8,56,d]]$
5 |  $\\{7,4,3,4\\}$ |  96 |  $[[480,32,352,d]]$

$g$ |  Tesselação |  $s$ |  $[[n,k,r,d]]$
---|---|---|---
5 |  $\\{9,4,3,4\\}$ |  32 |  $[[192,16,144,d]]$
6 |  $\\{7,4,3,4\\}$ |  120 |  $[[600,40,440,d]]$
6 |  $\\{9,4,3,4\\}$ |  40 |  $[[240,20,180,d]]$
6 |  $\\{11,4,3,4\\}$ |  24 |  $[[168,16,128,d]]$
6 |  $\\{21,4,3,4\\}$ |  8 |  $[[96,12,76,d]]$
7 |  $\\{7,4,3,4\\}$ |  144 |  $[[720,48,528,d]]$
7 |  $\\{9,4,3,4\\}$ |  48 |  $[[288,24,216,d]]$
7 |  $\\{15,4,3,4\\}$ |  16 |  $[[144,16,112,d]]$

Table 5: Parameters of topological subsystem codes coming from tessellation
$\\{p,4,3,4\\}$ with $p\geq 7$ odd for $g=2$ to $g=7$.

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