# Magic Boundaries of 3D Color Codes

Zijian Song  Department of Physics and Astronomy, University of California,
Davis, California 95616, USA  Guanyu Zhu  IBM Quantum, T.J. Watson Research
Center, Yorktown Heights, NY 10598 USA

###### Abstract

We investigate boundaries of 3D color codes and provide a systematic
classification into 101 distinct boundary types. The elementary types of
boundaries are codimension-1 (2D) boundaries that condense electric particle
($Z$-type) or magnetic flux ($X$-type) excitations in the 3D color code,
including the $Z$-boundary condensing only electric particles, the
$X$-boundary condensing only the magnetic flux, and other boundaries
condensing both electric and magnetic excitations. Two novel types of
boundaries can be generated based on certain elementary types. The first type
is generated by applying transversal-$T$ gate on the entire code in the
presence of the $X$-boundary, which effectively sweeps the codimension-1 (2D)
$T$-domain wall across the system and attaches it to the $X$-boundary. Since
the $T$-domain wall cannot condense on the $X$-boundary, a new magic boundary
is produced, where the boundary stabilizers contain $XS$-stabilizers going
beyond the conventional Pauli stabilizer formalism and hence contains ‘magic’.
Neither electric nor magnetic excitations can condense on such a magic
boundary, and only the composite of the magnetic flux and codimension-2 (1D)
$S$-domain wall can condense on it, which makes the magic boundary going
beyond the classification of the Lagrangian subgroup. The second type is
generated by applying transversal-$S$ gate on a codimension-1 (2D) submanifold
in the presence of certain codimension-1 (2D) boundaries, which effectively
sweeps the $S$-domain wall across this submanifold and attaches it onto the
boundary. This generates a codimension-2 (1D) nested boundary at the
intersection. We also connect these novel boundaries to their previously
discovered counterpart in the $\mathbb{Z}_{2}^{3}$ gauge theory equivalent to
three copies of 3D toric codes, where the $S$ and $T$ domain walls correspond
to gauged symmetry-protected topological (SPT) defects. New boundaries are
produced whenever the corresponding symmetry of the SPT defect remains
unbroken on the boundary. Applications of the magic boundaries include
implementing fault-tolerant non-Clifford logical gates, e.g., in the context
of fractal topological codes.

###### Contents

  1. I Introduction
    1. I.1 Main results and ideas
    2. I.2 Outline
  2. II Backgrounds
    1. II.1 The 3D toric code
    2. II.2 The 3D color code
    3. II.3 Gapped boundaries of topological phases
    4. II.4 Unfolding the 3D Color Code without Boundaries
  3. III Unfolding the 3D color code with boundaries
    1. III.1 The $Z$-boundary
    2. III.2 The $X$-Boundary
    3. III.3 Other boundaries
      1. III.3.1 The $\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries
      2. III.3.2 The $\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries
      3. III.3.3 The folded boundaries
      4. III.3.4 The $pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$-boundary
      5. III.3.5 The $y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary
      6. III.3.6 Other boundaries with composite condensations
      7. III.3.7 Summary of the elementary boundary types
  4. IV Boundaries from attaching domain walls
    1. IV.1 Domain-wall sweeping
    2. IV.2 An overview of domain walls and boundaries using the TQFT description
      1. IV.2.1 The $(e_{i},m_{j},m_{k})$-boundary
      2. IV.2.2 The $(e_{i},e_{j},m_{k})$\- and $(e_{1},e_{2},e_{3})$-boundaries
      3. IV.2.3 The folded boundaries
      4. IV.2.4 The $m_{1}m_{2}m_{3}$\- and $e_{1}e_{2}e_{3}$-boundaries
      5. IV.2.5 The $(m_{1},m_{2},m_{3})$-boundary
      6. IV.2.6 Other boundaries classified by the Lagrangian subgroup
    3. IV.3 Transversal gates on the 3D color code
    4. IV.4 The magic boundary
    5. IV.5 The other elementary boundaries
      1. IV.5.1 The $\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries
      2. IV.5.2 The $\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundary
      3. IV.5.3 The $\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$-boundary
      4. IV.5.4 The folded boundaries
      5. IV.5.5 The $pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$\- and the $y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundaries
    6. IV.6 The nested boundaries
      1. IV.6.1 The $\mathbf{c}_{i}$-color boundary
      2. IV.6.2 The $X$-boundary
      3. IV.6.3 The fold$|m$-boundary
      4. IV.6.4 The $y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary
      5. IV.6.5 The other boundaries
    7. IV.7 Summary of boundary types
  5. V Summary and discussion
  6. A Detailed description of the unfolding unitary transformation
  7. B 3D toric code from gauging 3D Ising model
    1. B.1 Gauging without boundary
    2. B.2 Gauging in the presence of Boundaries
      1. B.2.1 The smooth boundary
      2. B.2.2 The rough boundary
      3. B.2.3 The folded boundary
      4. B.2.4 The $m_{1}m_{2}m_{3}$-boundary
      5. B.2.5 The $e_{1}e_{2}e_{3}$-boundary
  8. C A Minimal Model Example

##  I Introduction

In recent decades, there has been significant progress on the development of
quantum error-correcting codes and fault-tolerant quantum computation. A well-
known class of error-correcting codes is the topological stabilizer codes
(TSCs). The error syndrome that occurs in TSCs can usually be detected by
measuring the stabilizers, and the code distance can be made arbitrarily large
by increasing the size of the lattice [1]. The family of TSCs includes the
surface codes [2, 3], the color codes [4], and their generalizations with non-
trivial topology, boundary conditions, and topological defects, etc. [5, 6, 7,
8]. From the perspective of quantum matter, a topological stabilizer code also
has a corresponding Abelian topological order, which can be described by an
underlying topological quantum field theory (TQFT) or equivalently a $G$-gauge
theory with a discrete Abelian gauge group $G$.

Among a large family of topological stabilizer codes, the color code has
obtained significant interest due to its capability for fault-tolerant quantum
operations, particularly transversal logical gates [4]. Logical gates from the
Clifford group can be implemented transversally in the 2D color code by
applying a unitary operation to each physical qubit. Additionally, it is worth
noting that the 2D color code boundary can be used to distill magic states
through code deformation [9], which can be utilized for universal quantum
computing [10]. More generally, a $D$-dimensional color code admits
transversal logical gates in the $D^{\text{th}}$ level of the Clifford
hierarchy [11]. Furthermore, an equivalence is established between
$D$-dimensional color codes and $D$ copies of $D$-dimensional toric codes
[11]. From the TQFT perspective, a $D$-dimensional toric code corresponds to a
($D$+1)-dimensional $\mathbb{Z}_{2}$ gauge theory, while a $D$-dimensional
color code corresponds to a ($D$+1)-dimensional $\mathbb{Z}_{2}^{D}$ gauge
theory.

It was demonstrated in Ref. [12] that for (2+1)D Abelian topological order,
each gapped boundary can be associated with a corresponding Lagrangian
subgroup. This subgroup constitutes the abelian subset of the topological
excitations in the bulk that can condense (annihilate) on the boundary.
However, an exotic boundary in three copies of 3D toric codes equivalent to a
(3+1)D $\mathbb{Z}_{2}^{3}$ gauge theory, which will be dubbed as the ‘magic
boundary’, was constructed in Ref. [13] that goes beyond the classification of
the Lagrangian subgroup even though the corresponding (3+1)$D$ topological
order is still Abelian. Given the equivalence between a 3D color code and
three copies of 3D toric codes ($\mathbb{Z}_{2}^{3}$ gauge theory), a natural
question arises: what are the microscopic descriptions and properties of this
boundary in the context of the 3D color code?

###  I.1 Main results and ideas

In this paper, we explicitly construct an exact microscopic lattice model of
this exotic boundary based on the 3D color code and further scrutinize its
properties. We refer to this boundary as the magic boundary since it goes
beyond the conventional Pauli stabilizer formalism, or refer to it as the
$XS$-boundary since it has $XS$-stabilizers on it. This boundary is realized
by first creating an $X$-boundary of the 3D color code, followed by applying a
transversal-$T$ gate, which is effectively sweeping the corresponding
$T$-domain wall across the system. Given the duality between the $T$-gate in
the 3D color code and the $\mathrm{CCZ}$-gate in three copies of 3D toric
codes, the magic boundary realized in our work is also dual to the boundary
explored in Ref. [13].

This magic boundary displays exotic properties that extend beyond the
conventional classification by the Lagrangian subgroup. In this context, none
of the topological charges or fluxes from the bulk can condense on this
boundary. This is attributed to the fact that the $T$-domain wall can not
condense on the $X$-boundary; instead, it attaches to it. The interplay
between the $X$-boundary and the $T$-domain wall imparts this boundary with
its exotic character. We further demonstrate that altering the condensations
on this exotic boundary causes its unique properties to vanish, rendering it
equivalent to an elementary boundary found in the 3D color code.

To complete the classification of boundaries, we also consider the boundaries
attached with codimension-2 domain walls, namely, the $S$-domain walls. This
boundary is dubbed as the ‘nested boundary’. There are three independent
$S$-domain walls in the 3D color code. We explicitly show that, for the
$X$-boundary, no $S$-domain wall can condense on it; instead, it attaches to
it. Consequently, the condensation set at the intersection differs from other
parts of the boundary which leads to a nested boundary. We further demonstrate
that for the boundaries preserving $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetries, the corresponding $S$-domain walls cannot condense on them.
Conversely, for the boundaries where the corresponding
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetries are spontaneously broken, the
corresponding $S$-domain walls can condense on them.

From the TQFT perspective, the transversal $T$ and $S$ gates correspond to two
types of emergent symmetries of the $\mathbb{Z}_{2}^{3}$ gauge theory.
Applying these symmetries transversally within a specific region $\mathcal{R}$
generates gapped domain walls along the boundaries. As pointed out in Ref.
[14, 15, 16, 17, 18, 19], those two types of domain walls are gauged symmetry-
protected topological (SPT) defects, which can be generated by decorating a
(3+1)D invertible phases with a (2+1)D $\mathbb{Z}_{2}^{3}$ SPT (3-cocycle of
type-III) or (1+1)D $\mathbb{Z}_{2}^{2}$ SPT (2-cocycle of type-II) [20, 21],
respectively, and then gauging the global $\mathbb{Z}_{2}^{3}$ symmetry.
Applying transversal $T$ or $S$ gate is effectively sweeping the corresponding
domain walls across the entire system or codimension-1 submanifold.

In the presence of boundaries, the $e$-boundary (boundary with $Z$-type
condensations) spontaneously breaks the $\mathbb{Z}_{2}$ symmetry, whereas the
$m$-boundary (boundary with $X$-type condensations) preserves the symmetry.
Consequently, when the gauged SPT defects, such as $T$ or $S$ domain walls,
touch the $e$-boundary, the corresponding global symmetry is also
spontaneously broken, which trivializes the gauged SPT defects, and hence
allows them to condense on the $e$-boundary. In contrast, when the gauged SPT
defects touch the $m$-boundary, the global symmetry is preserved. As a result,
they cannot condense on the $m$-boundary but instead attach to it and produce
a new boundary.

Based on these insights, we systematically classify and explicitly construct
101 distinct types of boundaries for the 3D color code, including 1 magic
boundary and 70 nested boundaries. While permuting all the color labels could
yield additional boundary types, these permutations result in boundaries that
are physically equivalent. This work also serves as a classification of the
gapped boundaries of the 3D color code based on the current understanding of
emergent symmetries, symmetry defects, and gapped boundaries in the
corresponding $\mathbb{Z}^{3}_{2}$ gauge theory.

###  I.2 Outline

The paper is organized as follows. In Section II.1 and II.2, we review the
definition of the 3D toric code, the 3D color code, and their corresponding
topological excitations. In Section II.3, we review the definition of the
Lagrangian subgroup, and In Section II.4, we introduce the unfolding unitary
transformation that maps between the toric code and the color code without
boundaries. In Section III, we introduce the unfolding unitaries on the
systems with boundaries, and further define the $X$-boundary, the $Z$-boundary
and the other boundaries with a mixture of $X$ and $Z$ types condensations. In
Section IV, we construct the magic boundary from the 3D color code with
$X$-boundaries, and show the exotic properties of this boundary explicitly. We
further discuss the nested boundaries, which are boundaries with codimension-2
(1D) $S$-domain wall defects. In Section V, we draw the conclusion and have a
discussion of our work, followed by an outlook towards future directions. A
detailed description of the unfolding unitary transformation is discussed in
Appx. A. Gauging the 3D Ising model in the presence of boundaries is discussed
in Appx. B. A minimal model that has two $Z$-boundaries and four
$X$-boundaries is constructed explicitly in Appx. C.

##  II Backgrounds

In this section, we briefly review the construction of $3$-dimensional toric
codes [2] and $3$-dimensional topological color codes [4]. Then we introduce
the concept of the Lagrangian subgroup and its application in the
classification of gapped boundaries [22, 12]. In the last subsection, we
briefly review the unfolding technique to transform the 3D color code to three
copies of the 3D toric code without the presence of boundaries [11].

###  II.1 The 3D toric code

Consider a $3$-dimensional square lattice
$\mathcal{L}(\mathcal{V},\mathcal{E},\mathcal{F},\mathcal{C})$. $\mathcal{V}$
is the set of vertices $v\in\mathcal{V}$, $\mathcal{E}$ is the set of edges
$e\in\mathcal{E}$, $\mathcal{F}$ is the set of faces $f\in\mathcal{F}$, and
$\mathcal{C}$ is the set of cells $c\in\mathcal{C}$.

The Hamiltonian of the $3$-dimensional toric code is given by

$\displaystyle H_{TC}=-\sum_{v}\prod_{v\in\partial
e}X(e)-\sum_{f}\prod_{e\in\partial f}Z(e),$ (1)

where $X(e)$ and $Z(e)$ represent for the Pauli $X$\- and $Z$-stabilizers
applying on the edges labelled by $e$. Pictorially, these two kinds of
stabilizers can be drawn as follows:

<EMAIL_ADDRESS>(2)

One can check that all the Hamiltonian terms are mutually commuting.

In the 3D toric code, there exist two types of topological excitations:
electric excitations, referred to as $e$-particles, and magnetic excitations,
referred to as $m$-flux loops. These excitations arise from violations of the
$X$-stabilizers and $Z$-stabilizers, respectively. We provide examples of
these excitations below.

<EMAIL_ADDRESS>(3)

Electric charges always appear at the end points of strings of Pauli
$Z$-operators and flux loops always appear at the boundaries of Pauli
$X$-membranes.

In this paper we consider two types of elementary boundaries of the 3D toric
code: the rough (e) boundary and the smooth (m) boundary. $Z$-strings can
terminate on the rough boundary without generating any electric excitations at
the intersections, while $X$-membranes can terminate on the smooth boundary
without generating any flux loops at the intersections. It is equivalent to
say that the $e$-particle can condense on the rough boundary, while the
$m$-flux loop can condense on the smooth boundary. From a mathematical
perspective, these condensations form the Lagrangian subgroup on the boundary,
as detailed in [12]. We have a brief review of the Lagrangian subgroup in Sec.
II.3.

Figure 1: (a) The lattice of the 3D color code, and (b) its dual lattice. We
define the color of edges to be the color of the cells they connect. The lower
three polyhedrons are (c1) the truncated cuboctahedron, (c2) the truncated
octahedron and (c3) the cube, serving as the building blocks of the 3D color
code lattice discussed throughout this paper.

###  II.2 The 3D color code

The $3$-dimensional topological color code is defined on a lattice
$\mathcal{L}(\mathcal{V},\mathcal{E},\mathcal{F},\mathcal{C})$ with the
following constraints. First, each vertex on the lattice belongs to four
different edges. Second, all the cells are four-colorable, which means they
can be represented by four colors, and each pair of adjacent cells has
different colors.

The Hamiltonian of the $3$-dimensional color code can be written as

$\displaystyle H_{CC}=-\sum_{c}\prod_{v\subset c}X(v)-\sum_{f}\prod_{v\subset
f}Z(v).$ (4)

In general, one can choose arbitrary 4-colorable and 4-valent cellulations for
the $3$D color code. For the sake of convenience, we select a particular
cellulation for the three-dimensional topological color code, as depicted in
Fig. 1. It’s important to note that the results we obtain in this paper apply
generally and are not contingent on the specific cellulation chosen.

In the lattice shown in Fig. 1, we use yellow, green, purple, and red colors,
$\\{y,g,p,r\\}$, to label the color of cells and the edges that connect cells
with the same color, and we use two colors to label the plaquettes between two
cells with different colors, namely $\\{rp,rg,ry,pg,py,yg\\}$.

Following this convention, the $X$ and $Z$ stabilizers associated with purple
cells, for instance, are depicted as follows111Without specific statements, we
use magenta-colored circles to represent $Z$-operators and cyan-colored
circles to represent $X$-operators throughout the paper.:

<EMAIL_ADDRESS>(5)

Stabilizers for the other cells are defined in a similar manner. It can be
verified that the lattice’s two constraints ensure all Hamiltonian stabilizers
commute with each other.

Similar to the 3D toric code, strings of $Z$-operators generate point-like
topological charges (electric charges) at the endpoints. They are denoted by
$\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}},r_{\mathbf{z}}\\}$, with the
labels representing the colors of their corresponding cells. $Z$-type
excitations can be simultaneously created by applying a single $Z$-operator.

<EMAIL_ADDRESS>(6)

Membranes of $X$-operators create flux loops on their boundaries. We label
them as
$\\{rp_{\mathbf{x}},rg_{\mathbf{x}},ry_{\mathbf{x}},pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$.
Here we use two colors to label the plaquette between two cells of different
colors. These excitations (flux loops) can be created simultaneously by
applying a single $X$-operator.

<EMAIL_ADDRESS>(7)

To avoid confusion, we use a single color to label these excitations. To
create a single type of flux loop, one can apply $X$-operators on the
corresponding membrane and the flux loop is located on the boundary of that
membrane. For example, the $yg_{\mathbf{x}}$ flux loop is created by the
following membrane operator.

<EMAIL_ADDRESS>(8)

A more detailed discussion of the $X$-membrane operators and the flux loops
can be found in Sec. III.2.

There are also excitations created by $Y$-operators, which are the composites
of electric charges and magnetic flux loops.

In the reminder of this subsection, we provide a brief review of the fusion
rules of the 3D color code excitations. These excitations are abelian since
when two excitations with the same color label and the same Pauli label meet,
they annihilate themselves. We have

$\displaystyle(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}\times(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}=1,\quad(\mathbf{c}_{i})_{\mathbf{z}}\times(\mathbf{c}_{i})_{\mathbf{z}}=1,$
(9)

in which $\mathbf{c}_{i},\mathbf{c}_{j},\mathbf{c}_{k}\in\mathbf{C}$.

Then let us consider the excitations with the same Pauli label but with
different color labels. The fusion rules can be written as,

$\displaystyle\prod_{j,k}(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}=\prod_{i}(\mathbf{c}_{i})_{\mathbf{z}}=1$
(10)

in which we require $j\neq k$ and $j<k$. Eq. (10) indicates that charges with
different color labels are not entirely independent; only three among them are
independent.

###  II.3 Gapped boundaries of topological phases

In the previous subsections, we learned that in the presence of boundaries,
some excitations from the bulk of topological phases can condense on these
boundaries, while others remain gapped. From the perspective of operators,
this is equivalent to saying that the boundaries of certain excitation
creation operators can terminate on such boundaries, resulting in no
excitations being created at the intersections. The set of excitations that
can condense on the boundary forms a group, referred to as the Lagrangian
subgroup in finite abelian theories, as discussed in [22, 12].

The Lagrangian subgroup are defined as follows. Consider the set of
topological excitations in the bulk as $\mathsf{A}$. The condensation set
$\mathsf{M}$ on the boundary satisfies:

  * •

$\mathsf{M}$ is a subset of $\mathsf{A}$.

  * •

The excitations in $\mathsf{M}$ have trivial mutual and self statistics.

  * •

Any excitations that is not in $\mathsf{M}$ has non-trivial mutual statistics
with respect to at least one excitation in $\mathsf{M}$.

The set $\mathsf{M}$ that satisfies the above conditions are call the
Lagrangian subgroup of $\mathsf{A}$.

Using the boundaries of the 3D toric code as an example, we identify the
topological excitations in the bulk as $\mathbb{I}$ (vacuum), $e$ (electric
charge), $m$ (magnetic flux), and $em$ (composite of $e$ and $m$). There are
two different choices for gapped boundaries, specifically, $(\mathbb{I},e)$
and $(\mathbb{I},m)$. It can be verified that both sets of condensation meet
the previously mentioned conditions. Although $\mathbb{I}$ and $em$ also form
a group, $em$ cannot be part of the Lagrangian subgroup due to its non-trivial
self statistics.

In the abelian cases, the elements in the Lagrangian subgroup are always
irreducible representations of the groups (topological excitations).
Therefore, we can always use the generators to represent the condensation set.
In the cases of $(\mathbb{I},e)$ and $(\mathbb{I},m)$ boundaries, we can
simplify the notation and denote them as the $e$ and $m$ boundaries. It can be
generalized to the cases when we have several copies of 3D toric codes. For
example, consider there are two copies of 3D toric code with $e$-boundaries,
the Lagrangian subgroup is $(\mathbb{I},e_{1},e_{2},e_{1}e_{2})$. However, by
using the generating set, it simplifies to $(e_{1},e_{2})$. We use the
generating set of condensations to label boundaries throughout this paper
unless specified. In the reminder of the paper, we refer to the boundaries
that can be classified by the Lagrangian subgroup as the elementary
boundaries.

###  II.4 Unfolding the 3D Color Code without Boundaries

In this subsection, we provide an overview of the unfolding technique
introduced in Ref. [11], demonstrating that a $D$-dimensional color code is
equivalent to $D$ copies of toric codes, modulo local unitaries and
entangling/disentangling ancilla qubits. Our discussion primarily concentrates
on $3$-dimensional models, in alignment with the focus of this paper. It
should be noted that these results are applicable in general $D$ dimension.

Let $CC(\mathcal{L})$ be the stabilizer group of the topological color code
defined on a $3$-dimensional lattice $\mathcal{L}$, there exists an unfolding
(disentangling) unitary $U$ such that the color code can be transformed to $3$
copies of $3$-dimensional toric codes $TC(\mathcal{L}^{\prime}_{j})$ defined
on lattice $\mathcal{L}^{\prime}_{j}$, up to stabilizer group $\mathcal{S}$ of
the ancilla qubits, which is

$\displaystyle
U\left[CC(\mathcal{L})\otimes\mathcal{S}\right]U^{\dagger}=\otimes_{i=1}^{3}TC\left(\mathcal{L}^{\prime}_{j}\right).$
(11)

The mapping between lattice $\mathcal{L}$ and $\mathcal{L}^{\prime}_{j}$ can
be realized by the shrinking process (lattice deformation), and the unfolding
unitary transformation can be realized by applying local unitaries, shrinking
the lattice and disentangling qubits. We briefly summarize this process in the
following paragraphs.

Consider the lattice $\mathcal{L}$ associated with a $3$-dimensional
topological color code, as we defined in Sec. II. The unfolding process of
color $\mathbf{c}_{i},\mathbf{c}_{j},\mathbf{c}_{k}\in\mathbf{C}$, $i\neq
j\neq k$ on the lattice and the map between stabilizers can be described as
follows. First, apply local unitary operators to map all the Pauli operators
on vertices to Pauli operator on the $\mathbf{c}_{i}$-, $\mathbf{c}_{j}$-, and
$\mathbf{c}_{k}$-colored edges. Second, shrink all the $\mathbf{c}_{i}$-,
$\mathbf{c}_{j}$-, and $\mathbf{c}_{k}$-colored cells sequentially to points.
After this step, we obtain three copies of lattices with different colors.
Third, disentangle and discard the qubits on vertices. The outcome of the
above process yields three copies of 3D toric codes model defined on the
$\mathbf{c}_{i}$-, $\mathbf{c}_{j}$-, and $\mathbf{c}_{k}$-colored lattices,
respectively. In a broader sense, for a $D$-dimensional color code, selecting
and shrinking $D$ distinct colors can lead to $D$ copies of the
$D$-dimensional toric codes.

More concretely, consider the 3D color code defined on the lattice illustrated
in Fig. 1, we briefly summarize the result after the unfolding unitaries when
we select to shrink the green, yellow and purple cells.

  * •

By shrinking the green (yellow) cells, the lattice transforms into the
following configuration. The “…” in the figure indicates the lattice is
extended along the horizontal and vertical directions. The $X$-type
stabilizers are located at vertices, and each $X$-stabilizer consists of the
tensor product of $X$-operators on the nearest edges. The $Z$-type stabilizers
are given by the tensor product of $Z$-operators along the edges of
faces.222Acute readers may notice that based on the shrinking rule, there are
four qubits on each edge of the green (yellow) lattice. However, we can
further apply local unitaries to decouple these degrees of freedom and leave
only one qubit per edge, which are those of the 3D toric code. Details of this
process is discussed in Appx. A

  * •

By shrinking the purple cells, the lattice and stabilizers transform into the
following configuration.

An example of the transformations of stabilizers and lattice deformation on
the purple lattice is summarized in Fig. 2, and a detailed description of the
unfolding unitaries is discussed in Appx. A.

Figure 2: An example of the unfolding unitary transformation. The stabilizers
of the 3D color code are on the left-hand side of the arrows, while the
stabilizers of the 3D toric code are on the right-hand side. Here we choose
the unfolding process for the purple cells as an illustration. However, this
map works for every color and every color-code lattice. The map goes as
follows: (1) Map every $X$-operator to their nearest purple edges. (2) Map two
$Z$-operators at the vertices of a purple edge to a single $Z$-operator on the
same edge. (3) Shrink the purple cells to points. The remaining lattice model
is a 3D toric code defined on the purple lattice.

##  III Unfolding the 3D color code with boundaries

Figure 3: An illustration of the equivalence between the 3D color code with
different types of boundaries and three copies of toric codes with different
alignments. (a) A 3D color code with color boundaries (each pair of boundaries
can condense a distinct type of ($\mathbf{c}_{i})_{\mathbf{z}}$-particles.) is
equivalent to three copies of toric codes with distinct alignments of
boundaries. Therefore, each color boundary corresponds to a
$(e_{i},m_{j},m_{k})$-boundary of three copies of 3D toric codes. (b) A 3D
color code with two $Z$-boundaries and four $X$-boundaries is equivalent to
three copies of toric codes with the same alignment of boundaries. Therefore,
the $Z$-boundaries correspond to the $(e_{1},e_{2},e_{3})$-boundaries and the
$X$-boundaries correspond to the $(m_{1},m_{2},m_{3})$-boundaries. A miminal
model that can realize this boundary condition is explicitly constructed in
Appx. C

In Ref. [11], the authors show a correspondence between the excitations of two
copies of the 2D toric codes and those of the 2D color code. Analogously, we
establish a mapping between the excitations of three copies of the toric codes
and the 3D color code.333The map is equivalent under permutations of colors.

$\displaystyle y_{\mathbf{z}}\to e_{1},\quad pg_{\mathbf{x}}\to m_{1},$
$\displaystyle g_{\mathbf{z}}\to e_{2},\quad py_{\mathbf{x}}\to m_{2},$ (12)
$\displaystyle p_{\mathbf{z}}\to e_{3},\quad yg_{\mathbf{x}}\to m_{3}.$

For the convenience of our discussion, we define $\mathbf{c}_{1}=y$,
$\mathbf{c}_{2}=g$ and $\mathbf{c}_{3}=p$. In the remainder of this section,
we will demonstrate, following the original proof in Ref. [11], that the
$(e_{1},e_{2},e_{3})$ (all-rough) boundary of three copies of toric codes is
equivalent to the
$\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$-boundary ($Z$-boundary)
of the 3D color code, and that the $(m_{1},m_{2},m_{3})$ (all-smooth) boundary
is equivalent to the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary
($X$-boundary), up to local unitaries and entangling/disentangling ancilla
qubits. We illustrate the equivalence between the 3D color code with two
$Z$-boundaries and four $X$-boundaries, and three copies of the 3D toric codes
with the same boundary alignment in Fig. 3(b). A minimal model capable of
realizing this boundary condition is explicitly constructed in Appx. C.
Moreover, we explicitly construct the boundary Hamiltonians for the other
boundaries that can be classified by the Lagrangian subgroup.

###  III.1 The $Z$-boundary

######  Definition 1.

Consider a 3D color code with colors $\mathbf{C}=\\{y,g,p,r\\}$. The color of
the edges, denoted by $\mathbf{c}_{i}\in\mathbf{C}$, is defined as the color
of the cells they connect. Let us consider a surface that intersects only with
edges of a specific color. On one side of this surface, all the degrees of
freedom are removed. On the other side, the $X$-stabilizers and
$Z$-stabilizers that intersect with the surface are removed and truncated
respectively, while all the other stabilizers remain the same. This boundary
is referred to as the $Z$-boundary.

######  Theorem 1.

Consider the surface in Definition 1 intersects with color
$\mathbf{c}_{4}\in\mathbf{C}$. The condensations on the corresponding
$Z$-boundary are given by
$\\{(\mathbf{c}_{1})_{\mathbf{z}},(\mathbf{c}_{2})_{\mathbf{z}},(\mathbf{c}_{3})_{\mathbf{z}}\\}$,
where
$\mathbf{c}_{1},\mathbf{c}_{2},\mathbf{c}_{3},\mathbf{c}_{4}\in\mathbf{C}$ and
$\mathbf{c}_{1}\neq\mathbf{c}_{2}\neq\mathbf{c}_{3}\neq\mathbf{c}_{4}$. This
boundary is equivalent to the $(e_{1},e_{2},e_{3})$ boundary of three copies
of 3D toric codes up to local unitaries and entangling/disentangling ancilla
qubits.

###### Proof.

Without loss of generality, we choose $\mathbf{c}_{4}$ to be the red color,
and we use the lattice of 3D color code depicted in Fig. 1. To prove the
theorem, we first show that the condensations on this boundary are
$\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$. Then we illustrate that,
through the application of local unitaries, this boundary is equivalent to the
$(e_{1},e_{2},e_{3})$-boundary in three copies of 3D toric codes. This proof
is applicable in different colorizations and cell structures.

The lattice of the $Z$-boundary is shown below. Qubits are assigned on the
vertices of the lattice.

(13)

In addition to the $X$-stabilizers on the cells and $Z$-stabilizers on the
plaquettes, there are also truncated $Z$-stabilizers. We define these three
types of stabilizers to be $\mathcal{B}^{boundary}_{yg}(Z)$,
$\mathcal{B}^{boundary}_{pg}(Z)$, and $\mathcal{B}^{boundary}_{py}(Z)$,
respectively and display them below.

<EMAIL_ADDRESS>
There are no $X$-stabilizers on the truncated cells.

When a pair of $Z$-operators on the same edge with color
$\mathbf{c}_{i}\in\mathbf{C}$ is applied in the bulk, a pair of
$(\mathbf{c}_{i})_{\mathbf{z}}$ excitations on the connected
$\mathbf{c}_{i}$-colored cells are created. This is due to the fact that a
pair of $Z$-operators applying on a $\mathbf{c}_{i}$-colored edge commutes
with every stabilizer term except for the two $X$-stabilizers on the connected
$\mathbf{c}_{i}$-colored cells.

However, in a 3D color code with two $Z$-boundaries, there exist logical
strings of $Z$-operators that terminate on both boundaries and commute with
all the stabilizers. Unlike the 3D toric code, the 3D color code has three
inequivalent types of logical string operators between two $Z$-boundaries. In
the case of a 3D color code with $Z$-boundaries at the top and the bottom, as
illustrated in subsequent figures, the string operator for $y_{\mathbf{z}}$
excitations terminating on a $Z$-boundary is given by

<EMAIL_ADDRESS>
The string operator of $g_{\mathbf{z}}$ excitations that terminates on the
$Z$-boundary is given by The string operator of $p_{\mathbf{z}}$ excitations
that terminates on the $Z$-boundary is given by

It can be verified that these three logical string operators commute with all
stabilizers and cannot be decomposed into a product of local stabilizers.
Therefore, the condensations on the $Z$-boundary can be written as
$\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$.

Our next step is to show that this boundary, up to local unitary
transformations and entangling/disentangling ancilla qubits, is equivalent to
the $(e_{1},e_{2},e_{3})$-boundary of three copies of 3D toric codes. We apply
the unfolding unitaries introduced in Sec. II.4. To simplify the presentation
of the boundary, we have opted for a different alignments from the one shown
for the string operators. In this case, we consider the front and rear sides
of the green (and purple) lattice to be the $Z$-boundaries, as opposed to the
top and bottom sides as previously shown. The green sublattice is given by

<EMAIL_ADDRESS>
Here, the qubits are located on the edges of the lattice. The stabilizers on
this boundary are as follows.

<EMAIL_ADDRESS>
The excitations on this boundary are given by

<EMAIL_ADDRESS>
We see that magnetic loop excitations can be created on the boundary, while
electric charges can condense on this boundary. Therefore, this boundary is a
rough ($e$)-boundary for the green lattice, and similar arguments can be
applied to the yellow lattice due to the symmetry of the lattice.

Similarly, we apply the unfolding unitaries to the purple sublattice. The
$X$-stabilizers are intact. The $Z$-stabilizers on this boundary are as
follows.

<EMAIL_ADDRESS>
The excitations on this boundary are displayed below. We see that the electric
charges can condense on this boundary, while the magnetic condensations
cannot.

In conclusion, we show that the $Z$-boundary of the 3D color code is
equivalent to the rough boundaries of three copies of 3D toric codes up to
local unitary transformations and ancilla qubits. Explicitly, we show the
following map.

$\displaystyle\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}\longleftrightarrow(e_{1},e_{2},e_{3}).$
(14)

The arrow represents the equivalence up to local unitary transformation and
entangling/disentangling ancilla qubits. We illustrate this equivalence in
Fig. 3(b). ∎

###  III.2 The $X$-Boundary

######  Definition 2.

Consider a 3D color code with colors $\mathbf{C}=\\{y,g,p,r\\}$. The color of
the edges, denoted by $\mathbf{c}_{i}\in\mathbf{C}$, is defined as the color
of the cells they connect. Let us consider a surface that intersects only with
edges of a specific color. On one side of this surface, all the degrees of
freedom are removed. On the other side, the $Z$-stabilizers and
$X$-stabilizers that intersect with the surface are removed and truncated
respectively, while all the other stabilizers remain the same. This boundary
is referred to as the $X$-boundary.

######  Theorem 2.

Consider the surface in Definition 2 intersects with color
$\mathbf{c}_{4}\in\mathbf{C}$. The condensations on the corresponding
$X$-boundary are given by
$\\{(\mathbf{c}_{2}\mathbf{c}_{3})_{\mathbf{x}},(\mathbf{c}_{1}\mathbf{c}_{3})_{\mathbf{x}},(\mathbf{c}_{1}\mathbf{c}_{2})_{\mathbf{x}}\\}$,
where
$\mathbf{c}_{1},\mathbf{c}_{2},\mathbf{c}_{3},\mathbf{c}_{4}\in\mathbf{C}$ and
$\mathbf{c}_{1}\neq\mathbf{c}_{2}\neq\mathbf{c}_{3}\neq\mathbf{c}_{4}$. This
boundary is equivalent to the $(m_{1},m_{2},m_{3})$ boundary of three copies
of 3D toric codes up to local unitaries and entangling/disentangling ancilla
qubits.

###### Proof.

Without loss of generality, we choose $\mathbf{c}_{4}$ to be the red color,
and we use the lattice of 3D color code in Fig. 1. To prove the theorem, we
first show that the condensation on this boundary is
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$. Then we show that by
applying local unitaries, it is equivalent to the
$(m_{1},m_{2},m_{3})$-boundary of three copies of 3D toric codes. The proof
also works in other colorizations and other cell structures.

The lattice of this boundary is identical to the lattice shown in Eq. (13).
However, the stabilizers on it are different.

The $Z$-stabilizers are located on every plaquette. Since the cells of three
different colors are truncated, and the $X$-stabilizers are associated with
cells. Therefore, there are three types of truncated $X$-stabilizers on the
$X$-boundary. We denote them as $\mathcal{A}^{boundary}_{g}(X)$,
$\mathcal{A}^{boundary}_{y}(X)$, and $\mathcal{A}^{boundary}_{p}(X)$.

The truncated $X$-stabilizer on green cells, denoted as
$\mathcal{A}^{boundary}_{g}(X)$, is displayed as follows. The truncated
$X$-stabilizer on yellow cells, denoted as $\mathcal{A}^{boundary}_{y}(X)$,
has the same structure as those on green cells. Therefore, one can obtain
$\mathcal{A}^{boundary}_{y}(X)$ by translations of
$\mathcal{A}^{boundary}_{g}(X)$.

The truncated $X$-stabilizer on purple cells, denoted as
$\mathcal{A}^{boundary}_{p}(X)$, is displayed as follows.

<EMAIL_ADDRESS>
Since we chose to cut the red edges at the beginning, all the remaining red
cells are intact. Therefore, there are no truncated $X$-stabilizers of red
cells on this boundary.

Similar to the case with $Z$-boundaries, three independent types of membrane
operators can terminate on this boundary and commute with every stabilizer.
However, the membranes of the 3D color code are not flat, and their shapes
vary depending on the specific lattice. In our example, we display the
configurations of these (logical) membrane operators in Fig. 4. We define the
$X$-membrane operator on the $\mathbf{c}_{i}\mathbf{c}_{j}$-colored membrane
as $\overline{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ if it is also a logical
operator, and define it as $\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ if
it is not.

Figure 4: Three types of $X$-membrane operators. (a) The purple-green membrane
operator, $\widetilde{X}_{pg}$ (or $\overline{X}_{pg}$ if it is a logical
operator). (b) The purple-yellow membrane operator, $\widetilde{X}_{py}$ (or
$\overline{X}_{py}$). (c) The yellow-green membrane operator,
$\widetilde{X}_{yg}$ (or $\overline{X}_{yg}$).

The endpoints of the membrane that generates $pg_{\mathbf{x}}$ on this
boundary are given by

<EMAIL_ADDRESS>(15)

Here we show the intersection between the membrane operator and the
$X$-boundary. The endpoints of the membrane that creates $py_{\mathbf{x}}$ are
given by

<EMAIL_ADDRESS>(16)

The endpoints of the membrane that creates $yg_{\mathbf{x}}$ are given by

<EMAIL_ADDRESS>(17)

One can verify that all these logical membrane operators commute with the
stabilizers in the bulk and on the $X$-boundary. Therefore, the condensations
on the $X$-boundary are
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$.

Our next step is to demonstrate this $X$-boundary is equivalent to the
$(m_{1},m_{2},m_{3})$-boundary of three copies of 3D toric codes. By applying
the unfolding unitaries, the green lattice we get is depicted as follows.

<EMAIL_ADDRESS>
For the same reason, we choose the front and rear boundary of the above
lattice to correspond to the original $X$-boundary. The qubits are located on
the edges of the lattice. On this boundary, there are truncated
$X$-stabilizers on every vertex and intact $Z$ stabilizers on every plaquette.
The $X$ and $Z$-stabilizers on this boundary are given by

<EMAIL_ADDRESS>
The excitations on this boundary are shown below.

<EMAIL_ADDRESS>
We see that the magnetic loop becomes an arc and condenses on this boundary,
while pairs of electric charges can be created on this boundary. This suggests
that magnetic fluxes can condense on this boundary, while electric charges
cannot. Therefore, this boundary of the green lattice is a smooth boundary.

The yellow lattice has the same structure as the green one. Therefore, all the
arguments for the green lattice hold for the yellow lattice. After the
unfolding unitaries, the boundary of the yellow lattice is a smooth boundary
as well.

Finally, we apply the unfolding unitaries to the purple lattice. The lattice
is shown below.

<EMAIL_ADDRESS>
The truncated $X$-stabilizer and intact $Z$-stabilizer on this boundary are
depicted as follows.

<EMAIL_ADDRESS>
The excitations on this boundary are shown below We see that the magnetic flux
loop becomes an arc and condenses on this boundary, while pairs of electric
charges can be created on this boundary.

In conclusion, we show that the $X$-boundary of 3D color code is equivalent to
the smooth boundaries of three copies of 3D toric codes up to local unitary
transformations and ancilla qubits. Explicitly we show the following map.

$\displaystyle\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}\longleftrightarrow(m_{1},m_{2},m_{3}).$
(18)

The arrow represents the equivalence up to local unitary transformation and
entangling/disentangling ancilla qubits. We illustrate this equivalence in
Fig. 3(b). ∎

###  III.3 Other boundaries

For the completeness of our discussion, we briefly introduce the other
boundaries of the 3D color code that allow different condensation patterns
compared to the $X$\- or $Z$-boundaries.

####  III.3.1 The
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries

These boundaries are also referred to as the $\mathbf{c}_{i}$-color
boundaries, indicating their capacity to condense $\mathbf{c}_{i}$-colored
electric charges. Conventionally, the 3D color code is defined with three
distinct color boundaries and a folded boundary, in a tetrahedron geometry. By
applying unfolding unitaries, it is equivalent to three copies of the toric
codes with different alignments, as depicted in Fig. 3(a). We use the
$\\{y_{\mathbf{z}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary as an example
to show how to construct this type of boundaries. The boundary stabilizers,
$\mathcal{B}^{boundary}_{pg}(Z)$, $\mathcal{A}^{boundary}_{g}(X)$, and
$\mathcal{A}^{boundary}_{p}(X)$ are given by

<EMAIL_ADDRESS>(19)

This boundary is equivalent to the $(e_{1},m_{2},m_{3})$-boundary of three
copies of 3D toric codes after we apply the unfolding unitary transformation.

$\displaystyle\\{y_{\mathbf{z}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}\longleftrightarrow(e_{1},m_{2},m_{3}).$
(20)

The total number of distinct types of color boundaries is 3.

####  III.3.2 The
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries

This type of boundaries can condense two types of electric charges and one
type of flux loops. We use
$\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{x}}\\}$-boundary as an example
to illustrate how to construct this type of boundaries. The boundary
stabilizers, $\mathcal{B}^{boundary}_{pg}(Z)$,
$\mathcal{B}^{boundary}_{py}(Z)$, and $\mathcal{A}^{boundary}_{p}(X)$, are
given by

<EMAIL_ADDRESS>(21)

This boundary is equivalent to the $(e_{1},e_{2},m_{3})$-boundary after we
apply the unfolding unitary transformation.

$\displaystyle\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{x}}\\}\longleftrightarrow(e_{1},e_{2},m_{3})$
(22)

The total number of distinct types of
$\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{x}}\\}$-boundaries is 3.

####  III.3.3 The folded boundaries

Consider there are two copies of 3D toric codes, we can fold one copy of it,
and attach the folded boundary to either the $e$-boundary, or the $m$-boundary
of the second copy. We refer to this type of boundaries as the folded
boundaries, implying they are created by folding toric codes. Since one copy
of 3D toric code is folded along a codimension-1 (2D) submanifold in the bulk,
the condensation set on the folded interface is $(e_{i}e_{j},m_{i}m_{j})$.
Together with the boundary of the second copy, one can obtain two different
types of boundaries, namely, the $(e_{i}e_{j},m_{i}m_{j},e_{k})$-boundary and
the $(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary. According to our previous
discussion, this boundary correspond to the 3D color-code boundary with
condensations
$\\{(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{k})_{\mathbf{z}}\\}$
and
$\\{(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$,
respectively. For the simplicity of notations, we refer to the former boundary
as the fold$|e$-boundary and the latter as the fold$|m$-boundary in the
reminder of this paper, as these boundaries are composites of the fold of one
copy of 3D toric code and the $e$-boundary or $m$-boundary of another copy,
respectively.

Without loss of generality, we use explicity examples to show how to construct
these two types of boundaries. For the fold$|e$-boundary, we consider the
example when the condensation set is
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},p_{\mathbf{z}}\\}$.
Just like the cases for 3D toric code, as we illustrated in Appx. B.2.3, we
introduce ancilla qubits on the boundary to realize the condensation
property444Technically speaking, adding ancilla qubits is not necessary; one
can multiply $X$-stabilizers to eliminate the need for ancilla qubits.
However, ancilla qubits can serve as degrees of freedom that are purely
defined on the boundary. By applying local operators on the boundary,
typically single $X$ or $Z$ operators, one can observe how the condensations
emerge from the boundary and become gapped in the bulk.. We introduce one
ancilla qubit on each horizontal trancated yellow-green faces, as denoted by
the white circles in the following figure.

<EMAIL_ADDRESS>(23)

We define the $Z$-stabilizers on the boundary as

<EMAIL_ADDRESS>(24)

We then define the $X$-stabilizers as

<EMAIL_ADDRESS>(25)

Notice that both stabilizers are coupled with the ancilla qubits on the top
and the bottom. The first $Z$-stabilizer is given by the product of two
$Z$-stabilizers from each copy. In principle, any local product is a generator
in the stabilizer group. Therefore, the above figure is just one example that
satisfies the mutual commutation condition. The physical meanings of these
short $Z$-strings and their relation with the condensation of eletric charges
can be stated as follows: In the bulk, the short $Z$-strings can generate
pairs of electric charges at their endpoints. However, we add these terms on
the boundary and only keep the terms that commute with them in the
Hamiltonian. That is equivalent to say, the electric charges at the endpoints
are condensed on the boundary, or identified with the vacuum since the
corresponding operators commute with every other stabilizers on the boundary.
Therefore, any local products of the corresponding truncated $Z$-terms can be
regarded as a $Z$-stabilizer on the boundary, as long as they are coupled with
the ancilla qubits to make sure every terms are mutually commuting.

One can check the stabilizers above are mutually commuting, and commute with
the string or membrane operators that generate corresponding types of
excitations which can condense on it. The total number of distinct types of
fold$|e$-boundaries is 3.

We then consider the
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary
as an example for the fold$|m$-boundary. The stabilizers on this boundary are
similar to those of the previous type, except $\mathcal{B}^{boundary}_{yg}(Z)$
is substituted with $\mathcal{A}^{boundary}_{p}(X)$, which is

<EMAIL_ADDRESS>(26)

The total number of distinct types of fold$|m$-boundaries is 3.

####  III.3.4 The $pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$-boundary

The condensation set for the
$pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$-boundary is given by
$\\{y_{\mathbf{z}}g_{\mathbf{z}},g_{\mathbf{z}}p_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}\\}$.
Similar to the folded-boundary cases, we need to couple ancilla qubits to the
system to make the boundary stabilizers be mutually commuting. In general, we
need to have one ancilla qubit for every pair of
$\mathcal{A}^{boundary}_{y}(X)$, $\mathcal{A}^{boundary}_{g}(X)$ and
$\mathcal{A}^{boundary}_{p}(X)$ stabilizers, as suggested in the 3D toric code
model example in Appx. B.2.4. However, because of the lattice is asymmetry
between different colors, the numbers of $X$-stabilizers for each colors don’t
match. To solve this problem, we need to change the basis of generators for
the $\mathcal{A}^{boundary}_{p}(X)$ and $\mathcal{B}^{boundary}_{yg}(Z)$
stabilizers, which are given by

<EMAIL_ADDRESS>(27)

One can check they anti-commute with each other, while commute with all the
other stabilizers. Similar to the cases for the folded boundaries, we
introduce ancilla qubits and couple them with the stabilizers. We introduce
three ancilla qubits per set of $X$-stabilizers. Each set of $X$-stabilizers
contains one $X$-stabilizer from each color. For example, we add ancilla
qubits as follows:

<EMAIL_ADDRESS>(28)

Here we use different colors to distinguish ancilla qubits for different sets.
The $Z$-stabilizers can be defined as follow:

<EMAIL_ADDRESS>(29)

Other $Z$-stabilizers can be obtained by translations, and symmetric
rotations. However, the translation must be taken in the diagonal directions
to make sure the stabilizers are well defined. Similar to the cases of the
folded boundaries, the $Z$-stabilizers are given by the product of two
$Z$-stabilizers from different copies. As we argued in the previous
subsection, any local product is in the stabilizer group. Therefore, the above
figure is just one example that satisfies the mutual commutation condition.

We introduce two $X$-stabilizers on each truncated color cell. The
$X$-stabilizers on the truncated purple cells are given by

<EMAIL_ADDRESS>(30)

The $X$-stabilizers on the yellow cells are given by

<EMAIL_ADDRESS>(31)

The $X$-stabilizers on the truncated green cells are given by

<EMAIL_ADDRESS>(32)

One can check those terms are mutually commuting, and commute with the string
or membrane operators that generate corresponding types of excitations which
can condense on it. This boundary is equivalent to the
$(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$-boundary after we applying the
unfolding unitaries. Namely,

$\displaystyle\\{y_{\mathbf{z}}g_{\mathbf{z}},g_{\mathbf{z}}p_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}\\}\longleftrightarrow(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3}).$
(33)

The total number of distinct types of
$pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$-boundaries is 1. A 3D toric
code version of this boundary is constructed in Appx. B.2.4.

####  III.3.5 The $y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary

The condensation set for the
$y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary is given by
$\\{pg_{\mathbf{x}}py_{\mathbf{x}},py_{\mathbf{x}}yg_{\mathbf{x}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}$.
To construct the lattice Hamiltonian, we do the same trick as Eq. (27) to
match the number of stabilizers. However, in this case we couple one ancilla
qubit per $X$-stabilizer set, as suggested in the 3D toric code model example
in Appx. B.2.5. The ancilla qubits are placed as follows:

<EMAIL_ADDRESS>(34)

Different colors represent they are associated with different sets of
$X$-stabilizers. The $Z$-stabilizers are defined as follows:

$\displaystyle\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$},$
(35)

which is the product of three types of $Z$-stabilizers and the ancilla qubits.
Similar to the cases before, the $Z$-stabilizers are given by the product of
two $Z$-stabilizers from different copies. In principle, any local product is
in the stabilizer group. Therefore, the above figure is just one example that
satisfies the mutual commutation condition.

The $X$-stabilizers on the truncated purple cells are given by

<EMAIL_ADDRESS>(36)

The $X$-stabilizers on the yellow and green cells are given by

<EMAIL_ADDRESS>(37)

One can check those terms are mutually commuting, and commute with the string
or membrane operators that generate corresponding types of excitations which
can condense on it. This boundary is equivalent to the
$(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$-boundary after we applying the
unfolding unitaries. Namely,

$\displaystyle\\{pg_{\mathbf{x}}py_{\mathbf{x}},py_{\mathbf{x}}yg_{\mathbf{x}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}\longleftrightarrow(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$
(38)

The total number of distinct types of
$y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundaries is 1. A 3D toric code
version of this boundary is constructed in Appx. B.2.5.

####  III.3.6 Other boundaries with composite condensations

In general, boundaries with composite condensations of $X$ and $Z$ types can
also exist, provided they originate from different copies. Therefore, for the
completeness of our discussion, we list all other types of boundaries and
their counts for the readers’ reference. The boundaries and their
corresponding counts are as follows:

$\displaystyle\\{pg_{\mathbf{x}}g_{\mathbf{z}}p_{\mathbf{z}},py_{\mathbf{x}}y_{\mathbf{z}}p_{\mathbf{z}},yg_{\mathbf{x}}y_{\mathbf{z}}g_{\mathbf{z}}\\},\quad
1,$ (39)
$\displaystyle\\{pg_{\mathbf{x}}py_{\mathbf{x}}y_{\mathbf{z}}g_{\mathbf{z}},py_{\mathbf{x}}yg_{\mathbf{x}}y_{\mathbf{z}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\},\quad
1.$
$\displaystyle\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{k})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}}\\},\quad
3,$
$\displaystyle\\{(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\},\quad
3,$
$\displaystyle\\{(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}}(\mathbf{c}_{k})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}(\mathbf{c}_{i})_{\mathbf{z}}\\},\quad
3,$
$\displaystyle\\{(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{k})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}}\\},\quad
3.$

####  III.3.7 Summary of the elementary boundary types

In this section, we study the elementary boundaries of the 3D color code which
can be classified by the Lagrangian subgroup. We explicitly construct the
lattice model for the $X$-boundary (1), the $Z$-boundary (1), the color
boundaries (3), the
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries
(3), the folded boundaries (6), the
$pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$-boundary (1) and the
$y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary (1). The number
represents the distinct types of certain boundaries. Furthermore, we listed
all the other elementary boundaries that are classified by the Lagrangian
subgroup, which contains the composite of $Z$ and $X$ type excitations as
condensations. The number of distinct types of those boundaries is 14.
Therefore, the total number of distinct types of elementray boundaries of the
3D color code that are classified by the Lagrangian subgroup is 30.

##  IV Boundaries from attaching domain walls

It has been studied that for the code space of $D$-dimensional topological
stabilizer codes, the logical gate set generated by encoded gates
implementable by constant-depth circuits is contained in the $D^{\text{th}}$
level of the Clifford hierarchy [23, 24, 25]. When $D=3$, the logical gates
can be applied are in the third level of Clifford hierarchy, such as the
$\overline{T}$ gate [26] or the $\overline{\mathrm{CCZ}}$ gate [11, 27].

In Ref. [13], the authors showed that for a specific alignment of three copies
of the 3D toric codes with the $(m_{1},m_{2},m_{3})$-boundary, a transversal-
CCZ gate maps this boundary to an exotic boundary which is different from
either the usual $e$ or $m$ type boundaries. Neither $e$ nor $m$ type of
topological excitations can condense on this boundary. This suggests that this
type of boundary goes beyond the standard classification of gapped boundaries
based on the Lagrangian subgroup. Specifically, we have

$\displaystyle\widetilde{\mathrm{CCZ}}:(m_{1},m_{2},m_{3})\to(m_{1}s_{2,3}^{(2)},m_{2}s_{3,1}^{(2)},m_{3}s_{1,2}^{(2)}).$
(40)

This boundary is not equivalent to the original all-smooth boundary.

We briefly review the result of Ref. [13] in this section and establish the
equivalence of this boundary and the magic boundary we propose in this paper.

###  IV.1 Domain-wall sweeping

It has been understood that transversal gates, or more generally constant-
depth local circuits, are equivalent to sweeping gapped domain walls across
the system [28, 29, 30, 31, 32, 13]. Consider a transversal gate
$\widetilde{U}$. For $\widetilde{U}$ to also be a logical gate (denoted by
$\overline{U}$), it should satisfy the following condition,

$\displaystyle\widetilde{U}:\mathcal{H}_{C}\to\mathcal{H}_{C},$ (41)

in which $\mathcal{H}_{C}$ is the code space. The parent Hamiltonian of the
system is the following.

$\displaystyle H=H^{bulk}+H^{boundary},$ (42)

where $H^{bulk}$ and $H^{boundary}$ are the bulk and boundary Hamiltonians. To
satisfy the condition in Eq. (41), the transversal gate $\widetilde{U}$ should
satisfy the following condition

$\displaystyle P_{C}\widetilde{U}H\widetilde{U}^{\dagger}P_{C}=P_{C}HP_{C},$
(43)

where $P_{C}$ is the projector onto the code space $\mathcal{H}_{C}$. For a
translational invariant Hamiltonian, the above condition becomes

$\displaystyle
P_{C}\widetilde{U}H^{bulk}\widetilde{U}^{\dagger}P_{C}=P_{C}H^{bulk}P_{C}.$
(44)

In this case, we say $\widetilde{U}$ corresponds to a certain type of emergent
symmetries for the system, including 0-form symmetries which act on the entire
system and higher-form ($q$-form) symmetries which act on a codimension-$q$
submanifold555Here we only consider the spatial dimensions. Note that a
codimension-$q$ submanifold of a $d$-dimensional system corresponds to a
dimension-$(d-q)$ submanifold. An important property of a higher-form symmetry
operator is that it is equivalent up to local deformation as long as its
support, the codimension-$q$ submanifold, belongs to the same homotopy class.

Nevertheless, the existence of boundaries breaks the translational symmetry.
To satisfy the condition in Eq. (41), we further require

$\displaystyle
P_{C}\widetilde{U}H^{boundary}\widetilde{U}^{\dagger}P_{C}=P_{C}H^{boundary}P_{C}.$
(45)

We denote the code space of the boundary of the system as $\mathfrak{B}$. To
satisfy the condition in Eq. (41), the transversal gate $\widetilde{U}$ must
keep the boundary code space invariant. We have

$\displaystyle\widetilde{U}:\mathfrak{B}\to\mathfrak{B}.$ (46)

Figure 5: Illustration of domain wall sweeping. Consider a unitary operator
acting transversally on a region $\mathcal{R}$ (The blue-colored region in the
left figure). A gapped domain wall exists between the region $\mathcal{R}$ and
its complement. If the transversal $\widetilde{U}$ operation is a logical gate
of the code space, then as we sweep the domain wall across the system, the
gapped boundary can condense on the boundary $\mathfrak{B}$.  Figure 6: An
illustration of how topological excitations of 3D color code map upon
traversing the $S$-domain wall. We use two equivalent languages, the 3D color-
code language and the 3+1D TQFT/ toric-code language, to illustrate these
phenomena, and use black and blue colored fonts, respectively. (a) When
sweeping the $S$-domain wall across a non-contractible codimension-1
submanifold labeled by colors $\mathbf{c}_{i}$ and $\mathbf{c}_{j}$, as
depicted by the gray surface, electric charges remain unchanged, while
magnetic flux loops become attached to electric charges as the $S$-domain wall
is traversed. (b) When sweeping the $T$-domain wall across the entire system,
electric charges remain the same, while the magnetic flux loops get attached
to an $S$-domain wall as the $T$-domain wall is traversed. We denote this new
object as $(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{xs}}$.

The above statement can be alternatively interpreted by domain-wall sweeping.
Consider $\widetilde{U}_{\mathcal{R}}$ to be the transversal unitary operation
on region $\mathcal{R}$. As illustrated in Fig. 5, one can show that there
exists a domain wall along the boundary of the region $\mathcal{R}$. The
transversal gate $\widetilde{U}$ is a logical gate on the code space if the
corresponding domain wall can condense on the boundaries.

###  IV.2 An overview of domain walls and boundaries using the TQFT
description

In Ref. [14, 16, 31, 13, 17, 19], it has been shown that there exists two
types of emergent symmetries $\widetilde{U}$ and the corresponding domain
walls in three copies of 3D toric codes, which is equivalent to a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory in the
TQFT language [13, 17, 19].

The first type of domain wall is the codimension-1 (2D) CCZ-domain wall in the
context of 3D toric codes. Sweeping the CCZ-domain wall across the whole
system corresponds to applying a transversal-CCZ gate on all three copies of
the toric codes, which gives rise to a 0-form emergent symmetry acting on the
entire system. In the TQFT language, we denote such a domain wall by
$s_{1,2,3}^{(3)}$, where the susbscript means it couples toric-code copy 1, 2
and 3 while the superscript suggests the corresponding logical gate is in the
third-level of the Clifford hierarchy. As pointed out in Refs. [14, 15, 16,
17, 18, 19], $s_{1,2,3}^{(3)}$ is a codimension-1 gauged symmetry-protected
topological (SPT) defect generated by decorating a (3+1)D invertible phase
with a (2+1)D $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT
(also called a 2D cluster state) and then gauging the global
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry666Here,
(3+1)D and (2+1)D refer to the space-time dimension..

The second type is a codimension-2 (1D) CZ-domain wall in the toric-code
context. Sweeping the CZ-domain wall on a codimension-1 (2D) submanifold
corresponds to applying a transversal-CZ gate between two copies of the toric
codes, which gives rise to a 1-form emergent symmetry acting only on this
submanifold. In the TQFT language, we denote such a domain wall by
$s^{(2)}_{i,j}$, where the subscript means it couples toric-code copy $i$ and
$j$ while the subscripts suggest that the corresponding logical gate is in the
second-level of the Clifford hierarchy. As shown in Refs. [14, 15, 16, 17,
18], $s^{(2)}_{i,j}$ is a codimension-2 gauged SPT defect generated by
decorating a (3+1)D invertible phase with a (1+1)D
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT (also called a 1D cluster state) and
then gauging the global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry.

Note that both $s_{1,2,3}^{(3)}$ and $s^{(2)}_{i,j}$ are invertible defects
with them being their own inverse and satisfies the $\mathbb{Z}_{2}$ fusion
rules:

$s^{(3)}_{1,2,3}\times s^{(3)}_{1,2,3}=\mathbb{I},\quad s^{(2)}_{i,j}\times
s^{(2)}_{i,j}=\mathbb{I}.$ (47)

When the excitations in the model pass the codimension-2 domain wall
$s^{(2)}_{i,j}$, they are transformed as

$\displaystyle s^{(2)}_{i,j}:m_{i}\rightarrow m_{i}e_{j},\ m_{j}\rightarrow
e_{i}m_{j},\ e_{i}\rightarrow e_{i},\ e_{j}\rightarrow e_{j},$ (48)

where we have expressed transformation as a mapping of the domain wall
$s^{(2)}_{i,j}$ on the excitations. This is also illustrated in Fig. 6(a).

When the excitations pass the codimension-1 domain wall $s^{(3)}_{1,2,3}$,
they are transformed as

$\displaystyle s^{(3)}_{1,2,3}:$ $\displaystyle m_{1}\rightarrow
m_{1}s^{(2)}_{2,3},m_{2}\rightarrow m_{2}s^{(2)}_{3,1},m_{3}\rightarrow
m_{3}s^{(2)}_{1,2},$ $\displaystyle e_{1}\rightarrow e_{1},e_{2}\rightarrow
e_{2},e_{3}\rightarrow e_{3}.$ (49)

As we can see, both $s^{(2)}_{i,j}$ and $s^{(3)}_{1,2,3}$ induce no
transformation on the $e$-particles. This is due to the fact that both
$s^{(2)}_{i,j}$ and $s^{(3)}_{1,2,3}$ can be viewed as a condensation defect
of $e$ particles, as pointed out by Ref. [17]. On the other hand,
$s^{(2)}_{i,j}$ attaches an additional $e_{j}$-particles to the $m_{i}$-flux,
while $s^{(3)}_{1,2,3}$ attaches a domain wall $s^{(2)}_{j,k}$ to the
$m_{i}$-flux.

####  IV.2.1 The $(e_{i},m_{j},m_{k})$-boundary

Now we investigate the condensation properties of these domain walls on the
boundaries. In Ref. [13], it has been pointed out that both $s^{(2)}_{i,j}$
and $s^{(3)}_{1,2,3}$ can condense on the $(e_{i},m_{j},m_{k})$-boundary. The
condensation properties of $s^{(2)}_{i,j}$ can be shown as

$\displaystyle s^{(2)}_{i,j}:(e_{i},m_{j},m_{k})$
$\displaystyle\rightarrow(e_{i},e_{i}m_{j},e_{k})\equiv(e_{i},m_{j},m_{k})$
$\displaystyle s^{(2)}_{i,k}:(e_{i},m_{j},m_{k})$
$\displaystyle\rightarrow(e_{i},m_{j},e_{i}m_{k})\equiv(e_{i},m_{j},m_{k}),$
(50)

as illustrated in Fig. 9(b) in the later subsection. Here, the ‘$\equiv$’ on
the right side of the above equations essentially corresponds to a different
basis choice of the same condensations, which can also be considered as a
different choice of the generator set of the same Lagrangian subgroup. In
particular, we have used the fusion rule $m_{j}e_{i}\times e_{i}=m_{j}$. Since
$s^{(2)}_{i,j}$ and $s^{(2)}_{i,k}$ condense on the
$(e_{i},m_{j},m_{k})$-boundary, we can introduce the redundant notation for
this boundary
$(e_{i},m_{j},m_{k})\equiv(e_{i},m_{j},m_{k},s^{(2)}_{i,j},s^{(2)}_{i,k})$ by
adding the codimension-2 domain walls also into the condensations.

The condensation properties of $s^{(3)}_{1,2,3}$ can be shown as

$\displaystyle s^{(3)}_{1,2,3}:\
(e_{i},m_{j},m_{k})\equiv(e_{i},m_{j},m_{k},s^{(2)}_{i,j},s^{(2)}_{i,k})$
$\displaystyle\rightarrow$
$\displaystyle(e_{i},m_{j}s^{(2)}_{i,k},m_{k}s^{(2)}_{i,j},s^{(2)}_{i,j},s^{(2)}_{i,k})\equiv(e_{i},m_{j},m_{k},s^{(2)}_{i,j},s^{(2)}_{i,k})$
$\displaystyle\equiv$ $\displaystyle(e_{i},m_{j},m_{k}),$ (51)

where the ‘$\equiv$’ in the second line comes from the fusion rule is due to
the fact that two invertible domain walls fuse into vacuum
$s^{(2)}_{i,k}\times s^{(2)}_{i,k}=\mathbb{I}$ and hence
$m_{j}s^{(2)}_{i,k}$$\times$$s^{(2)}_{i,k}=m_{j}$. Therefore, the
$s^{(3)}_{1,2,3}$-domain wall also condenses on the
$(e_{i},m_{j},m_{k})$-boundary.

With the development of the TQFT description in Ref. [17], we can now also
understand the above condensation properties, which are deduced from simple
algebraic manipulations, by using the perspective of the $\mathbb{Z}_{2}^{3}$
gauge theory. One can obtain the $\mathbb{Z}_{2}$ gauge theory (toric code) by
gauging the $\mathbb{Z}_{2}$ global symmetry of an invertible paramgetic phase
of a transversal Ising model [17, 33, 34]. The $e$-boundary in the ungauged
model (Ising model) spontaneously breaks the $\mathbb{Z}_{2}$ global symmetry,
while the $m$-boundary preserves this symmetry. This asymmetric boundary
properties comes from the fact that $e$ is the $\mathbb{Z}_{2}$ symmetry
charge in the $\mathbb{Z}_{2}$ gauge theory, while $m$ is the symmetry flux.
This properties caries over to the $\mathbb{Z}_{2}^{3}$ gauge theory, which is
just three copies of $\mathbb{Z}_{2}$ gauge theories (toric codes). Since
$s^{(2)}_{i,j}$ is a gauged symmetry-protected topological (SPT) defect
according to Refs. [14, 15, 16, 17, 18, 19], the corresponding
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry will be spontaneously broken
when they reach a boundary containing a copy of $e_{i}$\- or $e_{j}$-boundary,
which means $s^{(2)}_{i,j}$ is trivialized and hence condenses on the
boundary. This is the reason that $s^{(2)}_{i,j}$ and $s^{(2)}_{i,k}$ condense
on the $(e_{i},m_{j},m_{k})$-boundary. Similarly, $s^{(3)}_{1,2,3}$ is also a
gauged SPT defect, the corresponding
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry will be
spontaneously broken when it reaches a boundary containing any $e_{1}$,
$e_{2}$ or $e_{3}$ boundary. This is the reason why $s^{(3)}_{1,2,3}$ can
condense on the $(e_{i},m_{j},m_{k})$-boundary.

In addition, we have the following mapping:

$\displaystyle s_{i,j}^{(2)}:(e_{i},m_{j},m_{k})\to(e_{i},m_{j}e_{i},m_{k}).$
(52)

Here, a codimension-2 (1D) boundary $(e_{i},m_{j}e_{i},m_{k})$ is produced at
the interface of two codiemsnion-1 (2D) boundaries $(e_{i},m_{j},m_{k})$. We
dub this type of boundaries as the nested boundaries, as illustrated in Fig.
9(a) in the later subsection.

From the TQFT perspective, the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry
of the $j^{\mathrm{th}}$ and the $k^{\mathrm{th}}$ copies is preserved.
Therefore, the $s_{j,k}^{(2)}$ domain wall cannot condense on this boundary.

####  IV.2.2 The $(e_{i},e_{j},m_{k})$\- and $(e_{1},e_{2},e_{3})$-boundaries

Besides the example of $(e_{i},m_{j},m_{k})$ discussed in Ref. [13], we also
consider the $(e_{i},e_{j},m_{k})$ and the $(e_{1},e_{2},e_{3})$ boundaries in
this paper. According to the above discussion of the spontaneous
$\mathbb{Z}_{2}$ symmetry breaking of the $e$-boundary, we can conclude that
$s^{(2)}_{i,j}$, $s^{(2)}_{i,k}$, $s^{(2)}_{j,k}$ and $s^{(3)}_{1,2,3}$ will
condense on the $(e_{i},e_{j},m_{k})$ and $(e_{1},e_{2},e_{3})$ boundaries.
One can check the condensation properties of the $(e_{i},e_{j},m_{k})$ with
the algebraic manipulations shown before:

$\displaystyle
s^{(2)}_{i,j}:(e_{i},e_{j},m_{k})\rightarrow(e_{i},e_{j},m_{k})$
$\displaystyle
s^{(2)}_{i,k}:(e_{i},e_{j},m_{k})\rightarrow(e_{i},e_{j},e_{i}m_{k})\equiv(e_{i},e_{j},m_{k})$
$\displaystyle
s^{(2)}_{j,k}:(e_{i},e_{j},m_{k})\rightarrow(e_{i},e_{j},e_{j}m_{k})\equiv(e_{i},e_{j},m_{k})$
$\displaystyle s^{(3)}_{1,2,3}:\
(e_{i},e_{j},m_{k})\equiv(e_{i},e_{j},m_{k},s^{(2)}_{i,j})$
$\displaystyle\rightarrow$
$\displaystyle(e_{i},e_{j},m_{k}s^{(2)}_{i,j},s^{(2)}_{i,j})\equiv(e_{i},e_{j},m_{k},s^{(2)}_{i,j})$
$\displaystyle\equiv$ $\displaystyle(e_{i},e_{j},m_{k}).$ (53)

We illustrate the condensation property of $s^{(2)}_{i,j}$ in Fig. 9(c) in the
later subsection.

For the $(e_{1},e_{2},e_{3})$-boundaries, the condensation follows obviously
from Eq. (48) and Eq. (IV.2) since both $s^{(2)}_{i,j}$ and $s^{(3)}_{1,2,3}$
act trivially on the $e$-particles.

####  IV.2.3 The folded boundaries

In this subsection we investigate the condensation properties of these domain
walls on the folded boundaries. The condensation properties of $s_{i,j}^{(2)}$
on the $(e_{i}e_{j},m_{i}m_{j},e_{k})$-boundary can be shown as

$\displaystyle s_{i,j}^{(2)}:(e_{i}e_{j},m_{i}m_{j},e_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}e_{i}e_{j},e_{k})$ (54)
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j},e_{k})$ $\displaystyle
s_{j,k}^{(2)}:(e_{i}e_{j},m_{i}m_{j},e_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}e_{k},e_{k})$
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j},e_{k})$ $\displaystyle
s_{i,k}^{(2)}:(e_{i}e_{j},m_{i}m_{j},e_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}e_{k},e_{k})$
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j},e_{k})$

Therefore, we conclude that all types of $s_{i,j}^{(2)}$ domain walls can
condense on the $(e_{i}e_{j},m_{i}m_{j},e_{k})$-boundary.

The condensation properties of $s_{i,j}^{(2)}$ on the
$(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary can be shown as

$\displaystyle s_{i,j}^{(2)}:(e_{i}e_{j},m_{i}m_{j},m_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}e_{i}e_{j},m_{k})$ (55)
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j},m_{k})$ $\displaystyle
s_{j,k}^{(2)}:(e_{i}e_{j},m_{i}m_{j},m_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}e_{k},m_{k}e_{j})$ $\displaystyle
s_{i,k}^{(2)}:(e_{i}e_{j},m_{i}m_{j},m_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}e_{k},m_{k}e_{i})$

Therefore, we conclude that $s_{i,j}^{(2)}$ can condense on the
$(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary, while $s_{j,k}^{(2)}$ and
$s_{i,k}^{(2)}$ cannot.

From the $\mathbb{Z}_{2}^{3}$ gauge theory perspective, the condensation of
the $s_{i,j}^{(2)}$ domain wall on the
$(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary and the
$(e_{i}e_{j},m_{i}m_{j},e_{k})$-boundary can be understood as follows: Within
the bulk of the $i^{\text{th}}$ and $j^{\text{th}}$ copies of the ungauged
model (3D Ising model), a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry
exists. However, this symmetry undergoes spontaneous breaking to the
$\mathbb{Z}_{2}^{diag}$ diagonal symmetry at the folded boundary. This occurs
because the two copies result from folding a single copy of the 3D Ising
model; hence, the real symmetry of these copies is the $\mathbb{Z}_{2}^{diag}$
symmetry. This is just the global symmetry of the unfolded 3D Ising model. The
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry associated with the
$s_{i,j}^{(2)}$ domain wall thus undergoes spontaneous breaking. Consequently,
the domain wall becomes trivialized and hence condenses on the boundary.

On the other hand, the reason why $s_{j,k}^{(2)}$ and $s_{i,k}^{(2)}$ domain
wall cannot condense on the $(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary can be
understood as follows: Let’s consider $s_{j,k}^{(2)}$ for example. As we
discussed before, the actual symmetry on the $j^{\text{th}}$ copy is the
$\mathbb{Z}_{2}^{diag}$ symmetry rather than the $\mathbb{Z}_{2}$ symmetry
that only acts on that copy. For the $k^{th}$ copy, the boundary is an
$m$-boundary, thereby preserving the $\mathbb{Z}_{2}$ symmetry. In the case of
the $j^{th}$ copy, the $\mathbb{Z}_{2}^{diag}$ symmetry remains unbroken since
the domain wall effectively remains within the bulk of the unfolded $i^{th}$
and $j^{th}$ copies of the 3D toric code. Since the
$\mathbb{Z}_{2}^{diag}\times\mathbb{Z}_{2}$ symmetry of the $j^{th}$ and
$k^{th}$ copies is preserved, the $s_{j,k}^{(2)}$ thus cannot condense on it;
instead, it attaches to it.

For the condensation of $s_{j,k}^{(2)}$ and $s_{i,k}^{(2)}$ domain walls on
the $(e_{i}e_{j},m_{i}m_{j},e_{k})$-boundary, the reason can be understood as
follows: The existence of the $e$-boundary of the $k^{\text{th}}$ copy
spontaneously breaks the $\mathbb{Z}_{2}^{diag}\times\mathbb{Z}_{2}$ symmetry
of the $s_{j,k}^{(2)}$ and $s_{i,k}^{(2)}$ domain walls. Therefore, they are
trivialized on this boundary and hence can condense on it.

Furthermore, we investigate the condensation properties of $s_{i,j,k}^{(3)}$
on both types of the folded boundaries, which are shown as

$\displaystyle s_{i,j,k}^{(3)}:$
$\displaystyle(e_{i}e_{j},m_{i}m_{j},m_{k})\equiv(e_{i}e_{j},m_{i}m_{j},m_{k},s_{i,j}^{(2)})$
(56)
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}s_{j,k}^{(2)}s_{i,k}^{(2)},m_{k}s_{i,j}^{(2)},s_{i,j}^{(2)})$
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j}s_{j,k}^{(2)}s_{i,k}^{(2)},m_{k})$
$\displaystyle s_{i,j,k}^{(3)}:$
$\displaystyle(e_{i}e_{j},m_{i}m_{j},e_{k})\equiv(e_{i}e_{j},m_{i}m_{j},e_{k},s_{j,k}^{(2)},s_{i,k}^{(2)})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}s_{j,k}^{(2)}s_{i,k}^{(2)},e_{k},s_{j,k}^{(2)},s_{i,k}^{(2)})$
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j},e_{k})$

For now, it seems like the $s^{(3)}_{i,j,k}$ cannot condense on the
$(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary since it leaves a
$m_{i}m_{j}s_{j,k}^{(2)}s_{i,k}^{(2)}$ condensation that cannot condense on
the initial boundary. However, we haven’t checked the condensation property of
the $s_{j,k}^{(2)}s_{i,k}^{(2)}$-domain wall yet. We have

$\displaystyle s_{j,k}^{(2)}s_{i,k}^{(2)}:$
$\displaystyle(e_{i}e_{j},m_{i}m_{j},m_{k})\to
s_{j,k}^{(2)}:(e_{i}e_{j},m_{i}m_{j}e_{k},m_{k}e_{i})$ (57) $\displaystyle\to$
$\displaystyle(e_{i}e_{j},m_{i}m_{j},m_{k}e_{i}e_{j})\equiv(e_{i}e_{j},m_{i}m_{j},m_{k}),$

which shows that the composite of domain walls, $s_{j,k}^{(2)}s_{i,k}^{(2)}$,
can condense on the boundary. We further have

$\displaystyle s_{i,j,k}^{(3)}:(e_{i}e_{j},m_{i}m_{j},m_{k})$
$\displaystyle\to(e_{i}e_{j},m_{i}m_{j}s_{j,k}^{(2)}s_{i,k}^{(2)},m_{k})$ (58)
$\displaystyle\equiv(e_{i}e_{j},m_{i}m_{j},m_{k})$

Therefore, the $s_{i,j,k}^{(3)}$ domain wall can condense on both types of the
folded boundaries.

From the TQFT perspective, the reason why the $s_{i,j,k}^{(3)}$ domain wall
can condense on the $(e_{i}e_{j},m_{i}m_{j},e_{k})$-boundary is that the
presence of the $e$-boundary spontaneously breaks the global symmetry.
Therefore, the domain wall can condense on it. Regarding the
$(e_{i}e_{j},m_{i}m_{j},m_{k})$-boundary, the symmetry of the system encounter
a symmetry breaking
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\to\mathbb{Z}_{2}^{diag}\times\mathbb{Z}_{2}$.
As a result, the $s_{i,j,k}^{(3)}$ domain wall, which is a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauged SPT defect,
gets trivialized on this boundary, and hence can condense on it.

####  IV.2.4 The $m_{1}m_{2}m_{3}$\- and $e_{1}e_{2}e_{3}$-boundaries

The $m_{1}m_{2}m_{3}$-boundary is given by the condensation set
$(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$, and the $e_{1}e_{2}e_{3}$-boundary
is given by the condensation set $(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$. We
investigate the condensation properties of the domain walls on these
boundaries. Let’s first consider the $s_{i,j}^{(2)}$ domain walls. We have

$\displaystyle s^{(2)}_{1,2}:$
$\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$ (59) $\displaystyle\to$
$\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3}e_{1}e_{2})$
$\displaystyle\equiv$ $\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$
$\displaystyle s^{(2)}_{1,2}:$
$\displaystyle(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$ $\displaystyle\to$
$\displaystyle(m_{1}m_{2}e_{1}e_{2},m_{2}m_{3}e_{1},e_{1}e_{2}e_{3}).$

Since the condensation has permutation symmetries, the results hold for other
$s_{i,j}^{(2)}$ domain walls. We thus conclude that the $s_{i,j}^{(2)}$ domain
wall can condense on the $m_{1}m_{2}m_{3}$-boundary, while cannot condense on
the $e_{1}e_{2}e_{3}$-boundary.

From the TQFT perspective, the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry in the bulk
breaks into the $\mathbb{Z}_{2}^{diag}$ symmetry on the
$(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$-boundary, which corresponds to the
charge conservation symmetry on the entire system. Since the $s_{i,j}^{(2)}$
domain walls are $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauged SPT defects,
while the boundary only have one $\mathbb{Z}_{2}^{diag}$ symmetry, it get
trivialized and hence can condense on the boundary.

In the case of the $e_{1}e_{2}e_{3}$-boundary, the bulk symmetry breaks into a
$\mathbb{Z}_{2}^{(110)}\times\mathbb{Z}_{2}^{(011)}$ symmetry.
$\mathbb{Z}_{2}^{(110)}$ means the product of $\mathbb{Z}_{2}$ symmetries on
the first and the second copies. Each $\mathbb{Z}_{2}$ symmetry here
corresponds to the charge conservation symmetry of two copies of 3D toric
code. Since the $s_{i,j}^{(2)}$ domain walls are
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauged SPT defects, the symmetry is
preserved on the boundary. Therefore, they cannot condense on it.

However, as we see in Eq. (57), even though one $s_{i,j}^{(2)}$ domain wall
cannot condense, the composite of them may condense. We have

$\displaystyle s^{(2)}_{2,3}s^{(2)}_{1,2}:$
$\displaystyle(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$ (60) $\displaystyle\to
s^{(2)}_{2,3}:$
$\displaystyle(m_{1}m_{2}e_{1}e_{2},m_{2}m_{3}e_{1},e_{1}e_{2}e_{3})$
$\displaystyle\to$
$\displaystyle(m_{1}m_{2}e_{1}e_{2}e_{3},m_{2}m_{3}e_{1}e_{2}e_{3},e_{1}e_{2}e_{3})$
$\displaystyle\equiv$ $\displaystyle(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3}),$

which shows that $s^{(2)}_{2,3}s^{(2)}_{1,2}$ can condense on this boundary.
The other two composite domain wall can also condense because of the
permutation symmetry.

From the TQFT perspective, the composite of domain walls,
$s^{(2)}_{2,3}s^{(2)}_{1,2}$, corresponds to two
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetries, while on the boundary the
symmetry is $\mathbb{Z}_{2}^{(110)}\times\mathbb{Z}_{2}^{(011)}$. Each
$\mathbb{Z}_{2}$ symmetry on the boundary corresponds to the symmetry breaking
of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry in the bulk. As a result, the
composite domain walls get trivialized, and hence can condense on the
boundary.

Finally let’s check whether $s_{i,j,k}^{(3)}$ condenses on those boundaries.
We have

$\displaystyle s_{i,j,k}^{(3)}:$
$\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$ (61)
$\displaystyle\equiv$
$\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3},s^{(2)}_{1,2},s^{(2)}_{2,3},s^{(2)}_{3,1})$
$\displaystyle\to$
$\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3}s^{(2)}_{1,2}s^{(2)}_{2,3}s^{(2)}_{2,3},s^{(2)}_{1,2},s^{(2)}_{2,3},s^{(2)}_{3,1})$
$\displaystyle\equiv$ $\displaystyle(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3}).$

Therefore, $s_{i,j,k}^{(3)}$ domain wall can condense on the
$m_{1}m_{2}m_{3}$-boundary.

For the $e_{1}e_{2}e_{3}$-boundary, we have

$\displaystyle s_{i,j,k}^{(3)}:$
$\displaystyle(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$ (62) $\displaystyle\to$
$\displaystyle(m_{1}m_{2}s^{(2)}_{2,3}s^{(2)}_{3,1},m_{2}m_{3}s^{(2)}_{3,1}s^{(2)}_{1,2},e_{1}e_{2}e_{3}).$

We know from Eq. (60) that, $s_{i,j}^{(2)}s_{j,k}^{(2)}$ can condense on this
boundary. As a result, we have

$\displaystyle(m_{1}m_{2}s^{(2)}_{2,3}s^{(2)}_{3,1},m_{2}m_{3}s^{(2)}_{3,1}s^{(2)}_{1,2},e_{1}e_{2}e_{3})$
(63) $\displaystyle\equiv$
$\displaystyle(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3}).$

$s_{i,j,k}^{(3)}$ domain wall can condense on the $e_{1}e_{2}e_{3}$-boundary.

In the perspective of TQFT, the symmetry on the boundary spontaneously breaks
into $\mathbb{Z}_{2}^{diag}$ symmetry for the $m_{1}m_{2}m_{3}$-boundary, and
$\mathbb{Z}_{2}^{(110)}\times\mathbb{Z}_{2}^{(011)}$ symmetry for the
$e_{1}e_{2}e_{3}$-boundary. In both case the original $\mathbb{Z}_{2}^{3}$
symmetry in the bulk is broken. Therefore, the $s_{i,j,k}^{(3)}$ domain wall
gets trivialized, and hence can condense on these boundaries.

####  IV.2.5 The $(m_{1},m_{2},m_{3})$-boundary

In Ref. [13], it has been shown that exotic boundaries can be generated by
attaching a domain wall which does not condense on the normal boundaries
discussed above. Namely, consider the condensations on the boundary of three
copies of 3D toric codes to be $(m_{1},m_{2},m_{3})$, sweeping the CCZ-domain
wall $s_{1,2,3}^{(3)}$ on the entire system and CZ-domain wall $s_{i,j}^{(2)}$
on a codimension-1 submanifold can generate two different types of exotic
boundaries respectively. By sweeping the $s_{1,2,3}^{(3)}$ domain wall, the
condensations of the all-smooth boundaries become

$\displaystyle
s_{1,2,3}^{(3)}:(m_{1},m_{2},m_{3})\to(m_{1}s_{2,3}^{(2)},m_{2}s_{3,1}^{(2)},m_{3}s_{1,2}^{(2)}).$
(64)

By sweeping the $s_{i,j}^{(2)}$ domain wall, we have the following map on the
condensations of the boundary:

$\displaystyle
s_{i,j}^{(2)}:(m_{i},m_{j},m_{k})\to(m_{i}e_{j},m_{j}e_{i},m_{k}),$ (65)

in which $i,j,k\in\\{1,2,3\\}$ and $i\neq j\neq k$.

As we see from Eq. (65), the boundary generated by attaching the
$s_{i,j}^{(2)}$ domain wall can only condense a charge-flux composite
$m_{i}e_{j}$ or $m_{j}e_{i}$ but not the pure flux $m_{i}$ or $m_{j}$. The
generated boundary labeled by $(m_{i}e_{j},m_{j}e_{i},m_{k})$ is different
from the original boundary $(m_{i},m_{j},m_{k})\equiv(m_{1},m_{2},m_{3})$
since there is no way to rewrite the boundary with a basis change like
previous cases. This also shows that the $s_{i,j}^{(2)}$ domain wall cannot
condense on the $(m_{1},m_{2},m_{3})$-boundary since the map does not keep the
boundary invariant according to Eq. (46).

Instead, the $s_{i,j}^{(2)}$ domain wall is attached to the
$(m_{1},m_{2},m_{3})$-boundary and produced an exotic nested boundary where a
codimension-2 (1D) boundary $(m_{i}e_{j},m_{j}e_{i},m_{k})$ is generated at
the interface of the two original codimension-1 (2D)
$(m_{1},m_{2},m_{3})$-boundaries, as illustrated in Fig. 8(a) in the later
subsection.

An even more exotic example is the one we show in Eq. (64), where the produced
boundary $(m_{1}s_{2,3}^{(2)},m_{2}s_{3,1}^{(2)},m_{3}s_{1,2}^{(2)})$ is also
different from the original $(m_{1},m_{2},m_{3})$-boundary since the CZ-domain
wall $s_{i,j}^{(2)}$ itself cannot condense on the
$(m_{1},m_{2},m_{3})$-boundary. In this case, neither the $e_{i}$ nor $m_{i}$
excitations can condense on this new boundary, which means the new gapped
boundary is beyond the classification based on the Lagrangian subgroup.
Instead, only the composite of the $m$-flux and the CZ-domain wall
$m_{i}s^{(2)}_{j,k}$ ($i\neq j\neq k$) can condense on this new magic
boundary, as illustrated in Fig. 7 in the later subsection.

The reason that neither $s_{i,j}^{(2)}$ nor $s_{1,2,3}^{(3)}$ condense on the
$(m_{1},m_{2},m_{3})$-boundary is because that the $m$-boundaries preserve the
$\mathbb{Z}_{2}$ symmetry in the unguaged model (Ising model) such that the
SPT defects cannot be trivialized on the boundary and hence condense.

####  IV.2.6 Other boundaries classified by the Lagrangian subgroup

There are also other boundaries that can be classified by the Lagrangian
subgroup. These boundaries contains condensations that are composites of
$e$-particles and $m$-flux loops. We listed them here for the readers’
reference. The domain-wall condensation properties are summarized in Fig. 10.

Namely, those boundary condensations are

$\displaystyle(m_{i}e_{j},m_{j}e_{i},m_{k}),$
$\displaystyle(m_{i}e_{j}e_{k},m_{j}e_{i},m_{k}e_{i}),$ (66)
$\displaystyle(m_{1}e_{2}e_{3},m_{2}e_{1}e_{3},m_{3}e_{1}e_{2}),$
$\displaystyle(e_{i},m_{j}e_{k},m_{k}e_{j}),$
$\displaystyle(m_{1}m_{2}e_{1}e_{2},m_{2}m_{3}e_{1},e_{1}e_{2}e_{3}),$
$\displaystyle(e_{i}e_{j},m_{i}m_{j}e_{k},m_{k}e_{j}).$

Those boundaries appeared as nested boundaries in our previous discussion, in
which case they are 1D lines nested on the 2D boundaries. However, the
boundaries can also has those condensations. In this case, the condensations
on the nested boundaries are just the boundary condensations we discussed
before, as illustrated in Fig. 10.

We know from Ref. [11] that, there is a correspondence between 3D color code
and three copies of 3D toric codes or equivalently the $\mathbb{Z}_{2}^{3}$
gauge theory. Then a natural question arises: what does the magic boundary and
the other boundaries look like in the corresponding color code? We answer this
question by explicitly constructing such kinds of boundaries in the 3D color-
code lattice model in the following subsections. Although we have already
derived the existence of these boundaries using the abstract TQFT language in
this subsection, an explicit lattice construction helps to verify the TQFT
prediction and understand the microscopic details of these boundaries such
that we can also use these models for fault-tolerant quantum computing. In
addition, we will also give a classification of the 3D color-code boundaries
in the following subsections according to our current understanding of the
defects and boundaries in the $\mathbb{Z}_{2}^{3}$ gauge theory.

###  IV.3 Transversal gates on the 3D color code

Following Ref. [11]’s prescription, we divide the lattice of 3D color code
into two sublattices and guarantee each vertex of one sublattice is connected
to the vertices of the other sublattice. When we apply a gate $\widetilde{U}$
transversally, we apply $U$ on the vertices of one sublattice and
$U^{\dagger}$ on the vertices of the other sublattice. This definition of
transversal gates can avoid the appearance of overall phase factors.

It is demonstrated in Ref. [11] that applying transversal-$C^{d-1}Z$ gates on
$d$ copies of toric codes in $d$-dimension is equivalent to applying
transversal-$R_{d}$ gate on its corresponding color code. In 3D, we are able
to apply $R_{2}=S$ and $R_{3}=T$ transversally.

Transversal-$S$ gate can be applied on the
$\mathbf{c}_{i}\mathbf{c}_{j}$-membrane. We denote this operator as
$\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}$. We have the following
equation.

$\displaystyle
P_{C}\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}\mathcal{A}_{\mathbf{c}_{k}}^{bulk}(X)\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}^{\dagger}P_{C}$
(67) $\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}_{\mathbf{c}_{k}}^{bulk}(X)\prod_{v\in\mathcal{M}}Z(v)P_{C}$
$\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}_{\mathbf{c}_{k}}^{bulk}(X)P_{C}P_{C}\prod_{v\in\mathcal{M}}Z(v)P_{C}$
$\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}_{\mathbf{c}_{k}}^{bulk}(X)P_{C},$

in which

$\displaystyle\mathcal{M}=\mathrm{supp}(\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}})\cap\mathrm{supp}(\mathcal{A}_{\mathbf{c}_{k}}^{bulk}),$
(68)

and the $\mathrm{supp}$ of an operator $\mathcal{O}$ means the set of qubits
where $\mathcal{O}$ acts non-trivially.

From the third line to the fourth line of Eq. (67) we use the fact that
$\prod_{v\in\mathcal{M}}Z(v)$ can be always expressed as the product of
$Z$-stabilizers, which we discuss in detail in the next subsection. Therefore,
this new $Z$-term remains a stabilizer, making it a logical identity within
the code space. In addition, since $S_{\mathbf{c}_{i}\mathbf{c}_{j}}$ are
composed of diagonal gates, it commutes with all the $Z$-stabilizers
$\mathcal{B}_{\mathbf{c}_{i}\mathbf{c}_{j}}^{bulk}(Z)$,
$\mathcal{B}_{\mathbf{c}_{j}\mathbf{c}_{k}}^{bulk}(Z)$ and
$\mathcal{B}_{\mathbf{c}_{i}\mathbf{c}_{k}}^{bulk}(Z)$. Therefore, we have
$P_{C}\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}H^{bulk}\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}^{\dagger}P_{C}=P_{C}H^{bulk}P_{C}$.
The transversal-$S_{\mathbf{c}_{i}\mathbf{c}_{j}}$ gate is thus an emergent
1-form symmetry of the code space, according to the definition using Eq. (44).

When $\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ is applied on a region
$\mathcal{R}$ with boundaries. Codimension-2 (1D) domain walls are created on
these boundaries and we denote them as the
$S_{\mathbf{c}_{i}\mathbf{c}_{j}}$-domain wall. When an electric excitation
passes the domain wall, it remains unchanged. However, when a magnetic flux
loop passes the domain wall, it gets attached to electric charges on the
intersections. We have the following map:

$\displaystyle\widetilde{S}_{\mathbf{c}_{i}\mathbf{c}_{j}}:(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}$
$\displaystyle\to(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},\
(\mathbf{c}_{i})_{\mathbf{z}}\to(\mathbf{c}_{i})_{\mathbf{z}},$ (69)
$\displaystyle(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}$
$\displaystyle\to(\mathbf{c}_{j})_{\mathbf{z}}(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}},\
(\mathbf{c}_{j})_{\mathbf{z}}\to(\mathbf{c}_{j})_{\mathbf{z}},$

which is illustrated in Fig. 6(a) and corresponds to the mapping Eq. (48) in
the TQFT language.

Transversal-$T$ gate can be applied on the entire system. After conjugation,
the X-stabilizers on the purple cell $\mathcal{A}_{p}^{bulk}(X)$ becomes

<EMAIL_ADDRESS>(70)

Similar conjugation can be applied to the red, green, and yellow cells.

One can notice that the new $XS$-operator can be viewed as a contractible
$S$-domain wall attached on the original $X$-stabilizer, which can itself be
considered as the smallest contractible $X$-membrane operator. The
contractible $S$-domain wall is a logical identity. To see this, we notice the
following identity,

<EMAIL_ADDRESS>(71)

The second operator on the right hand side is a stabilizer of the code, which
is itself a logical identity. That is to say, the contractible $S$-domain wall
and the $X$-stabilizer commute with each other within the code space, which
means the contractible $S$-domain wall is also a logical identity, namely

$\displaystyle P_{C}\mathcal{A}_{\mathbf{c}_{i}}^{bulk}(S)P_{C}=\mathbb{I}.$
(72)

We hence obtain

$\displaystyle
P_{C}\widetilde{T}\mathcal{A}_{\mathbf{c}_{i}}^{bulk}(X)\widetilde{T}^{\dagger}P_{C}$
(73) $\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}_{\mathbf{c}_{i}}^{bulk}(XS)P_{C}$ $\displaystyle=$
$\displaystyle
P_{C}\mathcal{A}_{\mathbf{c}_{i}}^{bulk}(S)P_{C}P_{C}\mathcal{A}_{\mathbf{c}_{i}}^{bulk}(X)P_{C}$
$\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}_{\mathbf{c}_{i}}^{bulk}(X)P_{C},$

which leads to
$P_{C}\widetilde{T}H^{bulk}\widetilde{T}^{\dagger}P_{C}$$=$$P_{C}H^{bulk}(X)P_{C}$
and satisfing the condition in Eq. (44). From the symmetry perspective, this
is equivalent to say that transversal-$T$ gate corresponds to an emergent
0-form symmetry of the code space in the bulk.

When the transversal-$T$ gate is applied on a region $\mathcal{R}$ with
boundaries. Codimension-1 (2D) domain walls are created on these boundaries
and we denote them as the $T$-domain wall. When an electric excitation passes
the domain wall, it remains unchanged. However, when a magnetic flux loop
passes the domain wall, it get attached to an $S$-domain wall. We have the
following map:

$\displaystyle\widetilde{T}:pg_{\mathbf{x}}\to pg_{\mathbf{xs}},\
py_{\mathbf{x}}\to py_{\mathbf{xs}},\ yg_{\mathbf{x}}\to yg_{\mathbf{xs}},$
(74)

which is illustrated in Fig. 6(b) and corresponds to the mapping Eq. (IV.2) in
the TQFT language.

In contrast, the $T$-symmetry does not hold when the system has the
$X$-boundaries. Similarly, the 1-form $S$ symmetry does not hold when its
support touches the $X$-boundary. This is due to the fact that the $T$ and $S$
domain walls do not condense on the $X$-boundary. By conjugating
transversal-$T$ gate on the corresponding color code with $X$-boundaries, one
can obtain a new type of boundaries of the 3D color code.

The $Z$-stabilizers on the boundary remain unchanged after applying the
transversal-$T$ gate, while the $X$-stabilizers on the boundary are turned
into non-Pauli stabilizers which are dubbed as as $XS$-stabilizers [35,
36].777More general cases are discussed in Ref. [35, 36]. Following the
notation in Ref. [36], a single operator can be represented by  $\displaystyle
XP_{N}(a|b|c)=\mathrm{exp}\left(\frac{a}{2N}2\pi i\right)X^{b}P^{c},\quad
P^{N}=\mathbb{I}.$ (75) Therefore, the $XS$ and $XS^{\dagger}$ operators can
be rewritten as  $\displaystyle XS\equiv XP_{4}(0|1|1),\quad
XS^{\dagger}\equiv XP_{4}(0|1|3)$ (76) The condensation set of the new
boundary is $\\{pg_{\mathbf{xs}},py_{\mathbf{xs}},yg_{\mathbf{xs}}\\}$,
representing the $\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$ flux
strings attached with $S$-domain walls, as illustrated in Fig. 7(c) and (d).

In the subsequent subsection, we demonstrate explicitly that this boundary
corresponds to the magic boundary previously outlined in the context of TQFT,
as illustrated in Sec.IV.2. Therefore, we also refer to this new boundary as
the magic boundary. The origin of this name is due to the fact that the
Hamiltonian of this boundary is beyond the Pauli stabilizers formalism, and
hence possess ‘magic’ [37]. The magic boundary is also an important ingredient
for performing fault-tolerant non-Clifford logical gates in fractal codes
which can replace the usual magic state distillation protocol as pointed out
in Ref. [13]. This magic boundary can also be viewed as a composite boundary,
comprising the $X$-stabilizers on the boundaries attached with $S$-domain
walls (equivalent to $s^{(3)}_{1,2,3}$), as depicted in Fig. 7(a).

###  IV.4 The magic boundary

Figure 7: An illustration of the $T$ ($\mathrm{CCZ}$)-domain wall sweeping and
its effect on the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary, topological
excitations and logical operations. We use two equivalent languages, the 3D
color-code language and the 3+1D TQFT/ toric-code language, to illustrate
these phenomena, and use black and blue colored fonts, respectively. (a) The
magic boundary can be obtained by sweeping the $T$-domain wall across the
entire system and attaching it to the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$ boundary. (b) When
traversing the $T$-domain wall, the electric charges
$(\mathbf{c}_{i})_{\mathbf{z}}$ remain unchanged. However, magnetic flux loops
become attached to $S$-domain walls and cannot condense on the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary. This implies
that neither electric charges nor magnetic flux loops can condense on the
magic boundary. (c) Upon traversing the $T$-domain wall, the $S$-domain walls
attached to the magnetic flux loops get re-absorbed into the $T$-domain wall.
Consequently, the remaining $(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}$
excitation can condense on the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary.
Equivalently, this means that magnetic flux loops with $S$-domain walls
attached can condense on the magic boundary. (d) Upon sweeping the $T$-domain
wall, the logical $X$-operator becomes attached to an $S$-membrane, forming
the $\overline{XS}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ operator. This membrane
operator generates a magnetic flux loop with an $S$-domain wall attached to
it. As the domain wall sweeps across the entire system, it can condense on the
magic boundary.

The magic boundary is exotic since the condensations are beyond the Lagrangian
subgroup. One can see it by observing neither $X$-type nor $Z$-type
excitations can condense on it.

Consider the $X$-membranes we introduce in Fig. 4. These three membrane
operators create $pg_{\mathbf{x}},yg_{\mathbf{x}}$, and $yg_{\mathbf{x}}$ on
their boundaries respectively. After conjugating the transversal-$T$ gate on
the 3D color code, these $X$-membrane operators do not commute with the
Hamiltonian terms on cells ($XS$-stabilizers). To better study the interplay
between the membrane operators and the $XS$-stabilizers, we define the
following group commutator,

$\displaystyle K(T,V):=V^{-1}T^{-1}VT,$ (77)

where $T$ and $V$ are both operators. In the case $T$ and $V$ commute with
each other, we have $K=\mathbb{I}$. In the case $T$ and $V$ anti-commute with
each other, we have $K=-\mathbb{I}$. In general cases, $K$ can be a non-
trivial matrix. As suggested in Refs. [17, 18], Eq. (77) is an example of
higher group structure when $T$, $V$ and $K$ are symmetry operators acting on
the Hilbert space.

We have the following algebraic relation between $XS$, $XS^{\dagger}$ and
$X$-operators,

$\displaystyle K(X,XS)=iZ,\quad K(X,XS^{\dagger})=-iZ.$ (78)

It suggests that $\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ and
$\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS)$ do not generally commute. However,
when these operators act on the code space of the 3D color code, the resulting
$Z$-terms are equivalent to product of $Z$-stabilizers for the
$XS$-stabilizers in the bulk. Consequently, they commute with each other
within the code space. This relationship can be illustrated by the following
equation:

$\displaystyle
P_{C}\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS)P_{C}$
(79) $\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS)\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}K\left(\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS),\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}\right)P_{C}$
$\displaystyle=$ $\displaystyle
P_{C}\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS)\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}P_{C},$

in which $P_{C}$ is the projector onto the code space $\mathcal{H}_{C}$,
$\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ is the $X$-membrane operator
with color $\mathbf{c}_{i}\mathbf{c}_{j}$ and
$\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS)$ is the $XS$-stabilizers on a
$\mathbf{c}_{k}$-colored cell in the bulk.

In the remainder of this subsection, we demonstrate that this equation is true
by showing
$K\left(\mathcal{A}^{bulk}_{\mathbf{c}_{k}}(XS),\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}\right)$
is equivalent to product of $Z$-stabilizers, which acts trivially on the code
space.888In the following discussions, when we analyze the commutator between
two operators, we consistently consider the scenario that they have a non-
trivial intersection of supports. When the intersection of supports is trivial
(non-overlapping), the commutator equals to $\mathbb{I}$. Eq. (79) still
holds. Although the illustration is based on the 3D color code lattice
depicted in Fig. 1, it holds for other cellulations and colorizations as well.

We first discuss the $X$-membrane operator on the purple-yellow membrane as
shown in Fig. 4(b), namely $\widetilde{X}_{py}$. The intersection between the
support of this membrane operator and a yellow cell in the bulk is shown
below.

<EMAIL_ADDRESS>(80)

Here, we utilize gray circles to denote the intersecting vertices between
$\widetilde{X}_{py}$ and $\mathcal{A}^{bulk}_{y}(XS)$. As shown by Eq. (78),
upon exchanging the order of $\widetilde{X}_{py}$ and
$\mathcal{A}^{bulk}_{y}(XS)$, an additional $Z$-operator, accompanied by a
phase factor of $i$ or $-i$, emerges at each gray circle. These phase factors
cancel themselves in our convention.

The commutator of $\widetilde{X}_{py}$ and $\mathcal{A}^{bulk}_{y}(XS)$ is
given by the following $Z$-membrane,

<EMAIL_ADDRESS>(81)

This operator is equivalent to the product of four $Z$-stabilizers on the
hexagons and one on the bottom octagon. Consequently, it acts trivially on the
code space. As a result, $\widetilde{X}_{py}$ commutes with the
$\mathcal{A}^{bulk}_{y}(XS)$ within the code space.

The commutator of $\widetilde{X}_{py}$ and $\mathcal{A}^{bulk}_{g}(XS)$ in the
bulk is the following.

<EMAIL_ADDRESS>(82)

Similarly, this operator is equivalent to the product of four $Z$-stabilizers
on the bottom hexagons. As a result, $\widetilde{X}_{py}$ commutes with
$\mathcal{A}^{bulk}_{g}(XS)$ in the bulk within the code space.

The commutator of $\widetilde{X}_{py}$ and $\mathcal{A}^{bulk}_{r}(XS)$ in the
bulk is the following.

<EMAIL_ADDRESS>(83)

This operator is equivalent to the product of two $Z$-stabilizers on hexagons.
Therefore, $\widetilde{X}_{py}$ commutes with $\mathcal{A}^{bulk}_{r}(XS)$ in
the bulk within the code space.

Finally, the commutator of $\widetilde{X}_{py}$ and
$\mathcal{A}^{bulk}_{p}(XS)$ in the bulk is the following.

<EMAIL_ADDRESS>(84)

This operator is equivalent to a $Z$-stabilizer on the square.

In conclusion, we demonstrate that $\widetilde{X}_{py}$ commutes with all the
stabilizers in the bulk within the code space.

A similar argument can be applied to $\widetilde{X}_{pg}$ as shown in Fig. 4
(a) and $\mathcal{A}^{bulk}_{\mathbf{c}_{i}}(XS)$ due to the symmetry of the
lattice.

Then we discuss the commutator between $\widetilde{X}_{yg}$, as shown in Fig.
4 (c), and $\mathcal{A}^{bulk}_{\mathbf{c}_{i}}(XS)$. The commutator of
$\widetilde{X}_{yg}$ and $\mathcal{A}^{bulk}_{y}(XS)$ in the bulk is the
following.

<EMAIL_ADDRESS>(85)

This operator is equivalent to the product of four $Z$-stabilizers on hexagons
and four on squares.

The commutator between $\widetilde{X}_{yg}$ and $\mathcal{A}^{bulk}_{g}(XS)$
is the same as shown in Eq. (85) due to the symmetry of the lattice and the
membrane operator. Therefore, they also commute with each other within the
code space.

The commutator of $\widetilde{X}_{yg}$ and $\mathcal{A}^{bulk}_{r}(XS)$ is the
following.

<EMAIL_ADDRESS>(86)

And the commutator of $\widetilde{X}_{yg}$ and $\mathcal{A}^{bulk}_{p}(XS)$ is
given by

<EMAIL_ADDRESS>(87)

Either term is equivalent to a $Z$-stabilizer on a square.

Up to now, we have demonstrated that $\widetilde{X}_{py}$,
$\widetilde{X}_{pg}$, and $\widetilde{X}_{yg}$ commute with the
transversal-$T$ gate conjugated Hamiltonian in the bulk within the code space.

However, in the presence of boundaries, translation symmetry is broken, and
the stabilizers on the boundaries may not be identical to those in the bulk.
Consequently, after conjugating transversal-$T$ gate on the entire system,
$\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ cannot terminate on the magic
boundary, namely, the
$\\{pg_{\mathbf{xs}},py_{\mathbf{xs}},yg_{\mathbf{xs}}\\}$-boundary.

We follow the same procedure to check the commutation relations on the magic
boundary. Define $\mathcal{A}^{boundary}_{\mathbf{c}_{i}}(XS)$ to be the
$XS$-stabilizers on the truncated $\mathbf{c}_{i}$-cells on the boundary,
which we introduce in Sec. III.2. We first consider the commutator between
$\widetilde{X}_{py}$, as shown in Fig. 4(b), and
$\mathcal{A}^{boundary}_{y}(XS)$, which is given by

<EMAIL_ADDRESS>(88)

From Def. 2, we know that this operator is not equivalent to the product of
$Z$-stabilizers on this boundary. Therefore, $\widetilde{X}_{py}$ and
$\mathcal{A}^{boundary}_{y}(XS)$ do not commute with each other within the
code space.

Then, we consider the same membrane operator and
$\mathcal{A}^{boundary}_{g}(XS)$. The commutator is the following.

<EMAIL_ADDRESS>(89)

We see the resultant operator is a product of two $Z$-stabilizers on hexagons.
Therefore, $\widetilde{X}_{py}$ and $\mathcal{A}^{boundary}_{g}(XS)$ commute
with each other within the code space.

Finally, we consider the same membrane operator and
$\mathcal{A}^{boundary}_{p}(XS)$. The commutator is the following.

<EMAIL_ADDRESS>(90)

Following the same argument, one can demonstrate that the resulting
$Z$-operator is not equivalent to the product of stabilizers within the code
space. Consequently, $\widetilde{X}_{py}$ does not commute with
$\mathcal{A}^{boundary}_{p}(XS)$. As $\widetilde{X}_{py}$ generates a
$py_{\mathbf{x}}$ flux loop on its boundary, the above argument is equivalent
to say that $py_{\mathbf{x}}$ cannot condense on the magic boundary.

In general, one can show that $\widetilde{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$
does not commute with $\mathcal{A}^{boundary}_{\mathbf{c}_{i}}(XS)$ and
$\mathcal{A}^{boundary}_{\mathbf{c}_{j}}(XS)$ on the magic boundary, while
commutes with $\mathcal{A}^{boundary}_{\mathbf{c}_{k}}(XS)$, in which
$\mathbf{c}_{i}\in\mathbf{C}$ is the color of the 3D color code,
$i,j,k\in\\{1,2,3\\}$, and $i\neq j\neq k$.

Furthermore, since the $p_{\mathbf{z}},y_{\mathbf{z}}$, and $g_{\mathbf{z}}$
excitations do not condense on the $X$-boundary, as we argued in Sec. III.2,
it is straightforward to check they do not condense on the magic boundary as
well.

Figure 8: An illustration of the $S$ $(\mathrm{CZ})$-domain wall sweeping
across a codimension-1 submanifold in the presence of the $X$-boundary. (a)
After sweeping the $S$-domain wall across a codimension-1 submanifold, it
attaches to the $X$-boundary and cannot condense on it. We refer to this type
of boundaries as the nested boundaries. (b) When traversing the $S$-domain
wall, the magnetic flux loops become attached to electric charges. As a
result, they cannot condense on the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary. (c) When
traversing the $S$-domain wall, the electric charges attached to the magnetic
flux loops get re-absorbed into the domain wall. As a result, they can
condense on the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary. It is
equivalent to say that the magnetic flux loops with electric charges attached
to them can condense on the
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary with an
$S$-domain wall attached.

In summary, we show that
$p_{\mathbf{z}},y_{\mathbf{z}},g_{\mathbf{z}},pg_{\mathbf{x}},$
$py_{\mathbf{x}}$ and $yg_{\mathbf{x}}$ cannot condense on the magic boundary,
as illustrated in Fig. 7(b), and this boundary is beyond the classification of
gapped boundaries based on the Lagrangian subgroup.

###  IV.5 The other elementary boundaries

In this subsection, we study the other elementary boundaries of 3D color code,
and their domain-wall condensation properties. We summarize the properties of
all the boundaries we study in Table 1 and 2.

####  IV.5.1 The
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries

When the boundary can condense at least one $Z$-type excitation, for example,
$(\mathbf{c}_{i})_{\mathbf{z}}$, then after conjugation of transversal
$T$-gate, $\overline{X}_{\mathbf{c}_{i}\mathbf{c}_{j}}$ and
$\overline{X}_{\mathbf{c}_{i}\mathbf{c}_{k}}$ can still terminate on it, for
$i\neq j\neq k$.

Without loss of generality, let us consider the boundary that condenses
$\\{y_{\mathbf{z}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$, as we introduce in
Sec. III.3.1.

After conjugation of transversal-$T$ gate, the $X$-stabilizers become
$XS$-stabilizers and the $Z$-stabilizers remain invariant. The condensations
thus become $\\{y_{\mathbf{z}},py_{\mathbf{xs}},yg_{\mathbf{xs}}\\}$.

We follow the same procedure in Sec. IV.4. The commutator between
$\overline{X}_{py}$ and $\mathcal{A}^{boundary}_{p}(XS)$ is the following

<EMAIL_ADDRESS>(91)

It is precisely the $Z$-stabilizers on the boundary. Therefore,
$\overline{X}_{py}$ and $\mathcal{A}^{boundary}_{p}(XS)$ commute with each
other on the boundary within the code space.

The commutator between $\overline{X}_{py}$ and
$\mathcal{A}^{boundary}_{g}(XS)$ is the following.

<EMAIL_ADDRESS>(92)

This operator is the product of two $Z$-stabilizers thus $\overline{X}_{py}$
and $\mathcal{A}^{boundary}_{g}(XS)$ commute with each other on the boundary
within the code space.

Finally, we examine the commutator between $\overline{X}_{py}$ and
$\mathcal{B}^{boundary}_{pg}(Z)$, and find that they also commute with each
other. A similar argument can be applied to the case of $\overline{X}_{yg}$.

Therefore, we conclude that $yg_{\mathbf{x}}$, $py_{\mathbf{x}}$ and
$y_{\mathbf{z}}$ can condense on the
$\\{y_{\mathbf{z}},py_{\mathbf{xs}},yg_{\mathbf{xs}}\\}$-boundary. This is
equivalent to say

$\displaystyle\\{y_{\mathbf{z}},py_{\mathbf{xs}},yg_{\mathbf{xs}}\\}\equiv\\{y_{\mathbf{z}},py_{\mathbf{x}},yg_{\mathbf{x}}\\},$
(93)

which is equivalent to Eq. (IV.2.1) in the TQFT description in Sec. IV.2.
Although the 3D color code has color permutation symmetry, by purely permuting
the color labels, the boundaries we get are physically equivalent. Therefore,
we conclude that there are 3 distinct types of boundaries which can condense
one $Z$-type excitations on them. For the 3D color code with these type of
boundaries, the transversal-$T$ gate corresponds to an emergent 0-form
symmetry of the entire code space.

In general, the relation can be written in the following form:

$\displaystyle\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{xs}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{xs}}\\}\equiv\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\},$
(94)

in which $\mathbf{c}_{i},\mathbf{c}_{j},\mathbf{c}_{k}\in\mathbf{C}$ and
$i\neq j\neq k$. This general relation also corresponds to Eq. (IV.2.1).

According to the case we discuss in Eq. (52) in the TQFT description, there
are two types of $S$-domain wall that can condense on this boundary,
$S_{\mathbf{c}_{i}\mathbf{c}_{j}}$\- and
$S_{\mathbf{c}_{i}\mathbf{c}_{k}}$-domain wall, while
$S_{\mathbf{c}_{j}\mathbf{c}_{k}}$-domain wall cannot condense on this
boundary. We leave the discussion of this case to the next subsection.

####  IV.5.2 The
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundary

Without loss of generality, we consider the
$\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{x}}\\}$-boundary to illustrate
the properties of this type of boundary. The boundary stabilizers can be found
in Sec. III.3.2.

After conjugation of the transversal-$T$ gate, the condensations on the
boundary become $\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{xs}}\\}$. The
commutator between $\overline{X}_{py}$ and $\mathcal{A}^{boundary}_{g}(XS)$
equals to identity. Therefore, $\mathcal{A}^{boundary}_{g}(XS)$ and
$\overline{X}_{py}$ commute with each other within the code space and
$yg_{\mathbf{x}}$ can condense on the boundary, which is equivalent to say

$\displaystyle\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{xs}}\\}\equiv\\{y_{\mathbf{z}},g_{\mathbf{z}},yg_{\mathbf{x}}\\}.$
(95)

Same as the TQFT description in Eq. (53), all three types of $S$-domain walls
can condense on this boundary. Therefore, the total number of distinct types
of the
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$-boundaries
is 3.

####  IV.5.3 The $\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$-boundary

We also discuss the
$\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$-boundary ($Z$-boundary or
the $(e_{1},e_{2},e_{3})$-boundary in the TQFT description). The boundary
condensations remain invariant under the conjugation of transversal-$T$ gate
or under the conjugation of transversal-$S$ gate on codimension-1 submanifold.
Therefore, the total number of distinct types of the $Z$-boundaries is 1.

####  IV.5.4 The folded boundaries

In this subsection, we investigate the condensation properties of the folded
boundaries introduced in Sec. III.3.3.

Consider the
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary
as an example for the fold$|m$-boundaries. After conjugation of
transversal-$T$ gate, the boundary condensation set becomes
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{xs}}py_{\mathbf{xs}},yg_{\mathbf{x}}\\}$.
One can check the commutator between $\overline{X_{pg}X_{py}}$ and the
boundary $XS$-stabilizer
$\mathcal{A}_{y}^{boundary}(XS)\mathcal{A}_{g}^{boundary}(XS)$ is equivalent
to the product of $Z$-stabilizers on the boundary. Therefore, we have the
following relation:

$\displaystyle\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{xs}}py_{\mathbf{xs}},yg_{\mathbf{x}}\\}\equiv\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},yg_{\mathbf{x}}\\}$
(96)

In the TQFT description, this corresponds to Eq. (55), Eq. (56) and Eq. (58).
The total number of distinct types of fold$|m$-boundary is 3.

For the fold$|e$-boundaries, let’s consider the
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},p_{\mathbf{z}}\\}$-boundary
as an example. After conjugation of the transversal-$T$ gate, the condensation
set becomes
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{xs}}py_{\mathbf{xs}},p_{\mathbf{z}}\\}$.
Similarly, one can check the commutator between $\overline{X_{pg}X_{py}}$ and
the boundary $XS$-stabilizer
$\mathcal{A}_{y}^{boundary}(XS)\mathcal{A}_{g}^{boundary}(XS)$ equals identity
within the code space. Therefore, $pg_{\mathbf{xs}}py_{\mathbf{xs}}$ can
condense on the bounday, which is equivalent to say

$\displaystyle\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{xs}}py_{\mathbf{xs}},p_{\mathbf{z}}\\}\equiv\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},p_{\mathbf{z}}\\}$
(97)

This relation corresponds to Eq. (56). Furthermore, one can check all the
three types of $S$-domain walls can condense on this boundary, which
corresponds to Eq. (54) in the perspective of TQFT. The total number of
distinct types of the fold$|e$-boundary is 3.

####  IV.5.5 The $pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}$\- and the
$y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundaries

Consider the lattice model we introduce in Sec. III.3.4 and III.3.5. After the
conjugation of transversal-$T$ gate, the boundary condensation set becomes
$\\{y_{\mathbf{z}}g_{\mathbf{z}},g_{\mathbf{z}}p_{\mathbf{z}},pg_{\mathbf{xs}}py_{\mathbf{xs}}yg_{\mathbf{xs}}\\}$
and
$\\{pg_{\mathbf{xs}}py_{\mathbf{xs}},py_{\mathbf{xs}}yg_{\mathbf{xs}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}$,
respectively.

For the former case, one can check that the commutator between the
$\overline{X_{pg}X_{py}X_{yg}}$ operator and the $XS$-stabilizers on the
boundary can be expressed as product of $Z$-stabilizers (product of short
$Z$-strings corresponding to two different copies). Therefore, we have the
following relation:

$\displaystyle\\{y_{\mathbf{z}}g_{\mathbf{z}},g_{\mathbf{z}}p_{\mathbf{z}},pg_{\mathbf{xs}}py_{\mathbf{xs}}yg_{\mathbf{xs}}\\}$
(98)
$\displaystyle\equiv\\{y_{\mathbf{z}}g_{\mathbf{z}},g_{\mathbf{z}}p_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}\\}.$

For the latter case, one can check that the commutator between
$\overline{X_{pg}X_{py}}$ and the boundary $XS$-stabilizers can be expressed
as product of $Z$-stabilizers (product of short $Z$-strings corresponding to
three different copies). Therefore, we have the following relation:

$\displaystyle\\{pg_{\mathbf{xs}}py_{\mathbf{xs}},py_{\mathbf{xs}}yg_{\mathbf{xs}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}$
(99)
$\displaystyle\equiv\\{pg_{\mathbf{x}}py_{\mathbf{x}},py_{\mathbf{x}}yg_{\mathbf{x}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}.$

In the perspective of TQFT, these relations correspond to Eq. (61), (62) and
(63).

###  IV.6 The nested boundaries

Figure 9: An illustration of the $S$ $(\mathrm{CZ}$)-domain wall sweeping
across a codimension-1 submanifold in the presence of an boundary with at
least one $Z$-type condensation. (a) When the boundary has one $Z$-type
condensations, for example $y_{\mathbf{z}}$, the $S$-domain wall on the
$pg$-membrane cannot condense on it. (b) However, the other two $S$-domain
walls on the $py$\- and $yg$-membranes can condense on this boundary. (c) When
the boundary has two or more $Z$-type condensations, all of the three types of
$S$-domain walls can condense on it.

As suggested in Sec. IV.2, there are four types of boundaries that some of the
$S$-domain walls cannot condense on, i.e., $(e_{i},m_{j},m_{k})$,
$(m_{1},m_{2},m_{3})$, $(e_{i}e_{j},m_{i}m_{j},m_{k})$ and
$(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$ boundaries in the TQFT descriptions.
The corresponding color-code boundaries are the $\mathbf{c}_{i}$-color
boundaries, the $X$-boundary, the fold$|m$-boundaries, and the
$y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary, respectively. We study
the properties of these boundaries in this subsection, and summarize them in
Fig. 10.

####  IV.6.1 The $\mathbf{c}_{i}$-color boundary

Consider the
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}\\}$-boundary,
the $S_{\mathbf{c}_{i}\mathbf{c}_{j}}$\- or
$S_{\mathbf{c}_{i}\mathbf{c}_{k}}$-domain wall can condense on the boundary.
However, the $S_{\mathbf{c}_{j}\mathbf{c}_{k}}$-domain wall can not condense
on it. It attaches to the boundary and generates a codimension-2 (1D) domain-
wall defect. The condensation set on that defect becomes
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{k})_{\mathbf{z}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{j})_{\mathbf{z}}(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$,
while other part of the boundary is unchanged. The corresponding condensation
in the TQFT description is $(e_{i},m_{j}e_{k},m_{k}e_{j})$, which has been
introduced in Sec. IV.2. The total number of distinct types of the
$\mathbf{c}_{i}$-color boundaries with non-condensing $S$-domain wall is 3. We
denote this type of boundary as the nested boundary, as illustrated in Fig. 8,
Fig. 9(a), and Fig. 10(d).

####  IV.6.2 The $X$-boundary

Similarly, in the case the boundary condensation is
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$ (X-boundary),
$S$-domain walls on all the three-colored membranes cannot condense on this
boundary, and introduce a codimension-2 defect on it, as depicted in Fig. 8.
For example, consider the $S_{yg}$-domain wall attaches to the $X$-boundary.
The condensation set of this codimension-2 domain-wall defect becomes
$\\{g_{\mathbf{z}}pg_{\mathbf{x}},y_{\mathbf{z}}py_{\mathbf{x}},yg_{\mathbf{x}}\\}$,
while the other part of the boundary is unchanged. In the TQFT description,
this nested defect is denoted by $(e_{2}m_{1},e_{1}m_{2},m_{3})$. In the
general cases, one can attach composites of $S$-domain walls on the
$X$-boundary. The total number of distinct $X$-boundaries with nested defects
is 7, as illustrated in Fig. 10(a).

####  IV.6.3 The fold$|m$-boundary

Consider the
$\\{y_{\mathbf{z}}g_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}},yg_{\mathbf{x}}\\}$-boundary
as an example. the $S_{yg}$ domain wall can condense on it. However, the
$S_{pg}$ and $S_{yp}$ domain wall cannot. At the intersection of this boundary
and the $S_{pg}$-domain wall, the condensation set becomes
$\\{y_{\mathbf{z}}g_{\mathbf{z}},p_{\mathbf{z}}pg_{\mathbf{x}}py_{\mathbf{x}},g_{\mathbf{z}}yg_{\mathbf{x}}\\}$,
while the other part of the boundary remain unchanged. In the TQFT
description, this boundary is denoted as
$(e_{1}e_{2},e_{3}m_{1}m_{2},e_{2}m_{3})$. The total number of distinct types
of fold$|m$-boundaries with nested boundaries is 3, as illustrated in Fig.
10(c).

Figure 10: This figure shows the web of domain-wall attachment. Each pair of
condensations can be viewed as a pair of the boundary condensation and the
nested condensation. For example, consider the pair $(m_{1},m_{2},m_{3})$ and
$(m_{1}e_{2},m_{2}e_{1}e_{3},m_{3}e_{2})$, If $(m_{1},m_{2},m_{3})$ is the
condensation on the boundary, them by attaching the composite domain wall
$s_{1,2}^{(2)}s_{2,3}^{(2)}$, one could get a nested boundary with
$(m_{1}e_{2},m_{2}e_{1}e_{3},m_{3}e_{2})$ condensation on the 1D intersection
and $(m_{1},m_{2},m_{3})$ condensation on other part. The inverse process is
also possible, as indicated by the arrows in the figure. One can start with
the $(m_{1}e_{2},m_{2}e_{1}e_{3},m_{3}e_{2})$-boundary, and attach
$s_{1,2}^{(2)}s_{2,3}^{(2)}$ domain wall on it. The condensation set at the
intersection becomes $(m_{1},m_{2},m_{3})$. The total number of distinct types
of the nested boundaries is 70, in which 56 are associated with (a), 2 are
associated with (b), 6 are associated with (c), and 6 are associated with (d).

####  IV.6.4 The $y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary

The condensation set on the
$y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$-boundary is
$\\{pg_{\mathbf{x}}py_{\mathbf{x}},py_{\mathbf{x}}yg_{\mathbf{x}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}$.
The lattice construction is given in Sec. III.3.5. One can show that by
sweeping the $S$-domain wall across the codimension-1 submanifold, at the
intersection with the boundary, the $S$-domain wall cannot terminate. In the
TQFT description, this relation corresponds to Eq. (59). The total number of
distinct types of nested boundaries associated with this boundary is 1, as
illustrated in Fig. 10(b).

####  IV.6.5 The other boundaries

We summarize the the cases for the other boundaries that some $S$-domain walls
cannot condense in Fig. 10.

###  IV.7 Summary of boundary types

In summary, we identify 101 distinct types of boundaries for the 3D color
code, with 1 of them being the magic boundary and 70 of them being the nested
boundaries. While a trivial color permutation ($y\to g$, $g\to p$, $p\to r$,
and $r\to y$) can generate additional boundary types, the new boundaries
created by this permutation are physically equivalent. Therefore, we exclude
these from our count. We summarize the properties of these boundaries in Table
1, Table 2, and Fig. 10.

Condensations on boundaries | TQFT/toric code | Distinct | Types | Sections
---|---|---|---|---
| counterpart | types |  |
$\\{y_{\mathbf{z}},g_{\mathbf{z}},p_{\mathbf{z}}\\}$ | $(e_{1},e_{2},e_{3})$ | 1 | Elementary | III.1
$\\{pg_{\mathbf{x}},py_{\mathbf{x}},yg_{\mathbf{x}}\\}$ | $(m_{1},m_{2},m_{3})$ | 1 | Elementary | III.2
$\\{(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$ | $(m_{i}e_{j},m_{j}e_{i},m_{k})$ | 3 | Elementary | III.3.6
$\\{(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}}(\mathbf{c}_{k})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}(\mathbf{c}_{i})_{\mathbf{z}}\\}$ | $(m_{i}e_{j}e_{k},m_{j}e_{i},m_{k}e_{i})$ | 3 | Elementary | III.3.6
$\\{pg_{\mathbf{x}}g_{\mathbf{z}}p_{\mathbf{z}},py_{\mathbf{x}}y_{\mathbf{z}}p_{\mathbf{z}},yg_{\mathbf{x}}y_{\mathbf{z}}g_{\mathbf{z}}\\}$ | $(m_{1}e_{2}e_{3},m_{2}e_{1}e_{3},m_{3}e_{1}e_{2})$ | 1 | Elementary | III.3.6
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$ | $(e_{i},m_{j},m_{k})$ | 3 | Elementary | III.3.1
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{k})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}}\\}$ | $(e_{i},m_{j}e_{k},m_{k}e_{j})$ | 3 | Elementary | III.3.6
$\\{(\mathbf{c}_{i})_{\mathbf{z}},(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$ | $(e_{i},e_{j},m_{k})$ | 3 | Elementary | III.3.2
$\\{(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{k})_{\mathbf{z}}\\}$ | $(e_{i}e_{j},m_{i}m_{j},e_{k})$ | 3 | Elementary | III.3.3
$\\{(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}\\}$ | $(e_{i}e_{j},m_{i}m_{j},m_{k})$ | 3 | Elementary | III.3.3
$\\{(\mathbf{c}_{i})_{\mathbf{z}}(\mathbf{c}_{j})_{\mathbf{z}},(\mathbf{c}_{j}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{i}\mathbf{c}_{k})_{\mathbf{x}}(\mathbf{c}_{k})_{\mathbf{z}},(\mathbf{c}_{i}\mathbf{c}_{j})_{\mathbf{x}}(\mathbf{c}_{j})_{\mathbf{z}}\\}$ | $(e_{i}e_{j},m_{i}m_{j}e_{k},m_{k}e_{j})$ | 3 | Elementary | III.3.6
$\\{y_{\mathbf{z}}g_{\mathbf{z}},g_{\mathbf{z}}p_{\mathbf{z}},pg_{\mathbf{x}}py_{\mathbf{x}}yg_{\mathbf{x}}\\}$ | $(e_{1}e_{2},e_{2}e_{3},m_{1}m_{2}m_{3})$ | 1 | Elementary | III.3.4
$\\{pg_{\mathbf{x}}py_{\mathbf{x}},py_{\mathbf{x}}yg_{\mathbf{x}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}\\}$ | $(m_{1}m_{2},m_{2}m_{3},e_{1}e_{2}e_{3})$ | 1 | Elementary | III.3.5
$\\{pg_{\mathbf{x}}py_{\mathbf{x}}y_{\mathbf{z}}g_{\mathbf{z}},py_{\mathbf{x}}yg_{\mathbf{x}}y_{\mathbf{z}},y_{\mathbf{z}}g_{\mathbf{z}}p_{\mathbf{z}}$} | $(m_{1}m_{2}e_{1}e_{2},m_{2}m_{3}e_{1},e_{1}e_{2}e_{3})$ | 1 | Elementary | III.3.6