The boundaries and twist defects of the color code and their
applications to topological quantum computation
Markus S. Kesselring
1
, Fernando Pastawski
1
, Jens Eisert
1
, and Benjamin J. Brown
2,3
1
Dahlem Center for Complex Quantum Systems, Freie Universit
¨
at Berlin, 14195 Berlin, Germany
2
Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
3
Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
October 25, 2018
The color code is both an interesting ex-
ample of an exactly solvable topologically or-
dered phase of matter and also among the most
promising candidate models to realize fault-
tolerant quantum computation with minimal
resource overhead. The contributions of this
work are threefold. First of all, we build upon
the abstract theory of boundaries and domain
walls of topological phases of matter to com-
prehensively catalog the objects as they are re-
alizable in color codes. Together with our clas-
sification we also provide lattice representa-
tions of these objects which include three new
types of boundaries as well as a generating set
for all 72 color code twist defects. Our work
thus provides an explicit toy model that will
help to better understand the abstract theory
of domain walls. Secondly, building upon the
established framework, we discover a number
of interesting new applications of the cataloged
objects for devising quantum information pro-
tocols. These include improved methods for
performing quantum computations by code de-
formation, a new four-qubit error-detecting
code, as well as families of new quantum error-
correcting codes we call stellated color codes,
which encode logical qubits at the same dis-
tance as the next best color code, but us-
ing approximately half the number of physi-
cal qubits. To the best of our knowledge, our
new topological codes have the highest encod-
ing rate of local stabilizer codes with bounded-
weight stabilizers in two dimensions. Finally,
we show how the boundaries and twist defects
of the color code are represented by multiple
copies of other phases. Indeed, in addition to
the well studied comparison between the color
code and two copies of the surface code, we
also compare the color code to two copies of
the three-fermion model. In particular, we
find that this analogy offers a very clear lens
through which we can view the symmetries of
the color code which gives rise to its multitude
of domain walls.
Markus S. Kesselring: markus.kesselring@fu-berlin.de
Contents
1 Introduction 2
2 Anyon models 3
3 The two-dimensional color code 4
3.1 The stabilizer formalism . . . . . . . . 4
3.2 The color code lattice and stabilizers . 4
3.3 Color code anyons . . . . . . . . . . . 5
3.4 Fusion . . . . . . . . . . . . . . . . . . 5
3.5 Self-exchange and braiding . . . . . . . 6
4 Boundaries 7
4.1 Lagrangian subgroups . . . . . . . . . 7
4.2 The boundaries of the color code . . . 8
4.3 Lattice representations of the color
code boundaries . . . . . . . . . . . . . 9
5 Small color codes and fault-tolerant
quantum computation 10
5.1 Code deformation . . . . . . . . . . . . 10
5.2 A four-qubit color code . . . . . . . . 10
6 Bulk domain walls 11
6.1 Symmetries in abelian anyon models . 11
6.2 Symmetries in the color code . . . . . 12
6.3 Lattice representation of domain walls 12
7 Twists 14
7.1 Twist defects . . . . . . . . . . . . . . 14
7.2 Fusion between twists . . . . . . . . . 15
7.3 Anyon localization and the quantum
dimensions of twists . . . . . . . . . . 16
7.4 Interplay between twists and domain
walls . . . . . . . . . . . . . . . . . . . 17
7.5 Confinement of twists at boundaries . 18
7.6 Detaching corners and twist condensa-
tion . . . . . . . . . . . . . . . . . . . 19
8 Stellated color codes 21
9 Unfolding boundaries and twists 22
10 Conclusions and future work 24
Acknowledgments 24
A Fermions in the color code 25
B The three-fermion model and the color
code 25
C Pachner moves and color code twists 26
D Stellated surface codes 29
References 29
1
arXiv:1806.02820v3 [quant-ph] 24 Oct 2018
1 Introduction
Topological phases of matter are of significant inter-
est both from a condensed matter perspective [1] as
well as for their potential application in fault-tolerant
quantum computation [29]. Indeed, among the lead-
ing architectures that have been proposed for scal-
able quantum-information processing are topological
quantum error-correcting codes. The physics of these
phases is enriched further when their low-energy ex-
citations are connected by symmetries, in which case
they give rise to non-trivial domain walls and point-
like topological defects [1016] that have been utilized
in a number of instances [12, 1727] to improve pro-
tocols for topological quantum computation and to
reduce overhead requirements. Advancing the study
of topological defects and their corresponding symme-
tries gives us new tools to improve architectures for
topological quantum computation and, moreover, ex-
tends our understanding of the underlying topological
phases that support them.
The color code [28] is a particularly interesting
model due to both its symmetries and its ability to
realize fault-tolerant quantum computational opera-
tions. The importance of the use of different bound-
aries to implement the Clifford gates fault-tolerantly
using transversal rotations has been recognized since
its inception [28]. Further work on the computational
power of color codes and their higher-dimensional
counterparts have also been studied directly with var-
ious boundary configurations in Refs. [2932] and
in the guise of the unfolded surface code model in
Ref. [33, 34]. Remarkably, these models are known
to saturate bounds on the computational power of lo-
cal models using local gates [3538]. It has also been
shown that the boundaries of the color code can be
exploited to perform Clifford gates using code defor-
mation [39, 40] either by the braiding of punctures [41]
or by use of lattice surgery [42] in Refs. [27, 43]. Fur-
ther to this, certain color code twist defects [17, 44],
and their application to quantum computation [17, 27]
have also been studied.
At a more fundamental level, a study in Ref. [45]
examines the symmetries of the color code model to
identify continuous domain walls; the fundamental
objects that are used to generate twist defects. Be-
yond this, the symmetries and twist defects with the
color code model have been expressed using a tensor
network formalism in Ref. [46]. This is complemented
by Ref. [47] where a general study of twist defects and
domain walls using the framework of tensor networks
is given.
A recent surge in activity [4854] has shown that we
may be on the verge of experimental demonstrations
of the fault-tolerant components that will make up a
scalable quantum computer. Moreover, with its afore-
mentioned versatility for performing logical gates, to-
gether with its high threshold error rates [55] and
its relatively economical resource demands [56], the
color code is regarded among the leading candidate
architectures for the realization of topological quan-
tum computation. As such, we have recently seen ex-
perimental proposals to realize color code models with
a number of physical implementations and platforms.
In particular Majorana devices [5762] may be suit-
able as they allow for the measurement of relatively
high-weight stabilizers.
The color code model evidently has a very rich
structure [45, 46] that we may be able to exploit fur-
ther to discover better and more resource efficient pro-
tocols to perform quantum-computational operations
in a fault-tolerant manner. It is the primary goal
of the present manuscript to systematically explore
the different objects that can be realized with mod-
ifications to the color code lattice through a study
of its underlying topological excitations. We discover
three new types of boundary, together with lattice re-
alizations of all 72 different twist defects that can be
realized with the color code model. We catalog all
of these objects and quantify their properties for use
with topological quantum computation.
We go on to find a number of new applications of
the boundaries and twist defects of the color code for
quantum information processing. For example, we
argue that the new boundaries we discover, which we
refer to as Pauli boundaries, may be of practical ben-
efit when performing topological quantum computa-
tion. Specifically, we find that, like with the surface
code model [40], we can move punctures with Pauli
boundaries in the color code model using single-qubit
measurements. In contrast, to the best of our knowl-
edge, we know of no way that we can move punctures
with the known color code boundaries without the use
of two-qubit Bell-type measurements [41]. We addi-
tionally find a new four-qubit error detecting code as
a small instance of a new variant of the color code
that has Pauli boundaries, one that could well serve
as a blueprint for an experimental realization of quan-
tum error correcting protocols, reminiscent of the ones
presented in Refs. [32, 50].
Beyond these examples, we find that our study of
twist defects reveals new local codes with high encod-
ing rates. It is known that two-dimensional commut-
ing projector codes obey the bound kd
2
cn [63],
where k is the number of encoded qubits, d is the
code distance, n is the number of physical qubits in
the code and c is a constant to maximize. By consid-
ering color codes where we place a twist defect in the
center of the lattice, we find a code where c converges
towards 4 as the rotational symmetry of the code
increases. In particular we present two codes with
bounded stabilizer weights w, for w 8 we find a code
with c = 4, and for w 6 a code with c = 8/3 2.7.
In contrast, the twisted surface code [22] has a value
c = 4/3 and the standard color code has a value c = 2
(c = 4/3) when w 8 (w 6). The standard sur-
Accepted in Quantum 2018-10-09, click title to verify 2
face code has a value c = 1 for a lattice that is ap-
propriately oriented [20, 64], or when punctures are
arranged in a suitable pattern [65]. With our simple
example we see that there may be many other benefits
to be gleaned by exploring all of the twists available
with the color code. We present our new family of
stellated color codes, as well as our new four-qubit
error-detecting code, in self-contained sections that
can be understood with only prerequisite knowledge
of the stabilizer formalism for the reader interested
only in the quantum information applications of our
work.
Finally, we study how the new objects we introduce
to the color code lattice can be represented on multi-
ple copies of other topological phases through unfold-
ing [26, 33, 46, 6668]. In particular, in addition to
representing the new objects we discover on two copies
of the surface code, we also explore a lesser known [69]
connection between the color code and two copies of
the three-fermion model [70]. We find this connection
to give a very natural presentation of the underlying
structure of the color code model. Developing the
connections between the color code model and copies
of other, simpler phases may help to discover new and
improved schemes for topological quantum computa-
tion. Examples of which have been demonstrated, for
instance in Refs. [26, 71]. Similar mappings have also
provided new ways of writing decoding algorithms for
the color code [66].
The remainder of the manuscript is structured as
follows. In Sec. 2 we review the general theory of
anyon models and in Sec. 3 we review the stabilizer
formalism, the color code and the low-energy excita-
tions of this model. In Sec. 4 we examine the theory
of boundaries in abelian topological phases and dis-
cover new boundaries with the color code. Using these
new boundaries we discuss improved methods of de-
forming color code punctures and a new four-qubit
error-detecting code in Sec. 5. In Sec. 6 we present
the symmetries of the color code model and introduce
domain walls to the lattice that map excitations non-
trivially while preserving the basic data of the parti-
cle model. Then, in Sec. 7 we extend the discussion
of domain walls by terminating lattice dislocations to
introduce the twists of the color code. Through the
study of twists we find new two-dimensional codes
with high encoding rates, discussed in Sec. 8. Lastly,
in Sec. 9 we study the connection between the color
code and two copies of the surface code model to show
how color code defects are represented in this simpler
model. We finally conclude by proposing new direc-
tions of research that make use of the toolbox we have
elucidated in Sec. 10.
We also give supplemental information on the color
code in the Appendices. In App. A we discuss the six
fermionic excitations of the color code, and in App. B
we discuss the connection between the color code and
two copies of the three-fermion model. In App. C
we discuss the dual color-code lattice, and consider
systematically introducing certain twists to the model
with Pachner moves. Finally, in App. D we present
a new family of surface codes, namely, the stellated
surface codes, which are related to the new color codes
we present.
2 Anyon models
We require notation to describe the anyonic low-
energy excitations of the model of interest without
explicitly referring to the underlying system. Here
we review topological quantum field theory which de-
scribes anyonic particles while remaining agnostic of
microscopic details. For a more detailed and complete
discussion of the theory of anyons see Refs. [46, 15]
or the Appendix E of Ref. [10].
The low-energy excitations of a topological phase of
matter are described by an anyon model. An anyon
model is a set of excitations together with additional
data that describes interactions between these exci-
tations. We write a set of labels a, b, c, ··· C to
denote the distinct charges of the system. Included
in all anyon models is the vacuum charge, 1 C. The
vacuum is the label that indicates ‘no charge’.
Fusion describes the combination of two particles
a, b C. The fusion product of two particles a and b
is denoted with the binary operation a×b. The fusion
rules of an anyon model are described by the fusion
multiplicities N
c
a,b
, such that
a × b =
X
c
N
c
a,b
c, (1)
where c C. Each anyon a has a unique antiparticle
labeled
a which can fuse with it to the vacuum charge,
N
1
a,a
= 1.
Anyons have a Hilbert space associated with them
known as a fusion space. It is possible for the di-
mension of the fusion space to increase as anyons are
added to the system. The factor with which the fusion
space increases for an anyon a is called its quantum
dimension, denoted d
a
1, which is determined by
the equation
d
2
a
=
X
c∈C
d
c
N
a
a,c
, (2)
where d
1
= 1. Anyon a is abelian if d
a
= 1 and is
non-abelian if d
a
> 1. An anyon model is abelian if it
contains only abelian anyons. In fact, for all abelian
anyon models, fusion outcomes are deterministic such
that for each a, b there exists a unique label c such
that N
c
a,b
= 1, and N
d
a,b
= 0 for all d 6= c.
Lastly, we address the exchange and braid statistics
of anyons. Exchanging two anyons of charge a results
in a phase θ
a
, called the topological spin. Diagram-
Accepted in Quantum 2018-10-09, click title to verify 3
matically we express this with the equation
θ
a
a a
=
a a
, (3)
where the lines indicate the trajectories of particles.
Note that θ
a
= θ
a
. We call a bosonic if it exchanges
trivially with itself, i.e. θ
a
= +1, and fermionic if
θ
a
= 1.
Braiding two abelian anyons a and b clockwise re-
sults in a phase M
a,b
, called the monodromy. We can
similarly express this operation with a diagrammatic
equation
M
a,b
a b
a b
=
a b
a b
. (4)
This equation can be transformed by adding a braid
to the two sides of Eq. (4) to obtain
M
a,b
a b
=
a b
, (5)
an expression which will provide useful to calculate
the monodromy of anyons in the considered model. If
M
a,b
= +1, we say a and b braid trivially.
Lastly, we can derive the data involving anyons if
we know enough about the particles into which we
can decompose it. If an anyon c is the only fusion
product of a and b, as is the case in abelian anyon
models, then its self-exchange statistic is
θ
c
= θ
a
θ
b
M
a,b
, (6)
and the monodromy with another anyon d C is given
by
M
c,d
= M
a,d
M
b,d
. (7)
3 The two-dimensional color code
We next review the two-dimensional color code [28],
and summarize how the microscopic description gives
rise to its corresponding anyon model. Refs. [2931,
56] provide alternative reviews of the model.
3.1 The stabilizer formalism
A useful tool to describe a large class of quan-
tum error-correcting codes and many-body quantum
states is the stabilizer formalism [72]. Stabilizer codes
are specified by their stabilizer group, S. This is
an abelian subgroup of the Pauli group acting on n
qubits. We denote the generators of the Pauli group
X
j
and Z
j
, where indices 1 j n denote individual
spin-half particles, or qubits, of the system.
The stabilizer group defines a code subspace
spanned by basis vectors |ψi. The code subspace, or
Figure 1: (a) the 6.6.6 and (b) the 4.8.8 color code lattice.
Physical qubits live on the vertices. Each plaquette p hosts
two stabilizers, one measuring the Z parity of the qubits on
the boundary of p, and the other measures the X parity.
Examples of stabilizers are shown in panel (a).
code-space for short, is the common +1 eigenspace of
all the stabilizers of the stabilizer group. Specifically,
we have
s |ψi = |ψi, (8)
for all s S. The additional requirement that
6∈ S guarantees that the stabilized subspace is
non-empty. The dimension of the code-space is 2
k
,
where k = nm, is the difference between the number
of physical qubits n and the number of independent
generators of the stabilizer group m. When interpret-
ing the stabilized subspace as a logical code-space, k
is the number of logical qubits it supports.
Pauli operators which commute with the stabilizer
group but are not included in it are logical opera-
tors. More formally, the set of logical operators is
such that L
:
= N(S)\S, where N(S) is the normal-
izer of S. Operators in L preserve the code-space
yet act non-trivially on it. The weight w(P ) of a
Pauli operator is the number of qubits upon which
P acts non-trivially. It allows defining the code dis-
tance, d
:
= min
P ∈L
w(P ), which is commonly used as
a first order approximation to quantify the degree of
protection provided by the code-space.
3.2 The color code lattice and stabilizers
The color code is defined on a trivalent three-colorable
lattice, regular examples of which are shown in Fig. 1.
Specifically, each vertex of the lattice has exactly
three incident edges and each plaquette, indexed p,
can be assigned one of three colors, red, r; green, g;
or blue, b, such that no two plaquettes of the same
color touch.
With the lattice defined we consider a system where
a qubit is placed on each vertex of the lattice. The
stabilizer group of the color code is then generated by
operators associated to the plaquettes of the lattice.
Specifically we have two stabilizer generators,
s
x
p
=
Y
jp
X
j
and s
z
p
=
Y
jp
Z
j
, (9)
for each plaquette p, where the product is taken over
all the qubits that lie on the boundary of p. We show
examples of stabilizers on the lattice in Fig. 1.
Accepted in Quantum 2018-10-09, click title to verify 4
3.3 Color code anyons
We have claimed that the color code model gives
rise to anyonic quasiparticle excitations. However,
we have thus far only introduced the color code as a
code-space of some larger physical Hilbert space using
the stabilizer formalism. In fact, we have also implic-
itly defined a Hamiltonian system whose low-energy
excitations are anyons [2, 28]. The code-space of the
model is also the ground state subspace of the exactly
solvable Hamiltonian H, whose interaction terms are
stabilizer generators, which for the color code is
H =
X
p
s
z
p
X
p
s
x
p
. (10)
Applying Pauli matrices, P , to states in the code-
space yields excited eigenstates of H. Plaquettes
where at least one stabilizer generator is violated by,
i.e., anti-commutes with, P are then interpreted as
quasi-particle excitations, namely, anyons. The type
of anyonic quasi-particle is determined by which sta-
bilizer(s) are violated on a given plaquette, and the
color of the plaquette as we will describe in more de-
tail below. For a detailed discussion on color code
excitations, see Ref. [28] and for an in depth discus-
sion on the connection between the stabilizer code and
commuting Pauli Hamiltonians see Ref. [8]. Given
the connection between the Hamiltonian presented in
Eq. (10), we will continue the discussion in terms of
stabilizer states as we defined them previously.
We can characterize the sixteen anyonic charges of
the color code by the operators that create them. The
operators that create excitations are Pauli operators
with string-like support. Importantly, string opera-
tors commute with all the plaquette operators except
for those at their endpoints. As such, anyonic excita-
tions are created at the endpoints of string operators
where plaquette operators are violated.
To study the sixteen color code anyons, we first fo-
cus on a subset of nine charges. This subset is an over-
complete generating set, in that it can generate any
charge from the color code anyon model through fu-
sion. Even though four generators would also suffice,
we find this generating set particularly convenient to
characterize the the model. The nine charges in this
set can be denoted by two labels which are a Pauli la-
bel, P = x, y, z, and a color label, C = r, g, b. Given
that such excitation is specified by one of three colors
and one of three Pauli labels we arrive at nine anyons
which are labeled as CP . Shortly, it will become con-
venient to arrange the generating charges in a 3×3
grid as in Tab. 1 where color labels are arranged in
columns, and Pauli labels in rows.
The excitations we create on the lattice are deter-
mined by the string-operator used to create the ex-
citation. The color label is determined by the color
of the violated stabilizer at the endpoint of a string
and the Pauli label corresponds to the Pauli opera-
tor which was applied to create the excitation. For
rx gx bx
ry gy by
rz gz bz
Table 1: The nine generating anyons in the color code can be
labeled using a color label and a Pauli label. The three colors,
r, g, and b are give in the columns, the Pauli labels x, y, and
z are associated with rows. This arrangement of anyons will
prove useful in what follows, as it succinctly conveys braid
statistics and fusion rules of the color code anyon model.
instance, in Fig. 2 (a) we show a string operator that
creates two gx excitations at its endpoints. This is
because the string is composed of Pauli-X operators,
and the operator shown in the figure violates the two
green s
Z
p
plaquettes at its endpoints. One can readily
check that the other stabilizers on the lattice commute
with this operator.
3.4 Fusion
We now offer a brief intuition on how the fusion rules
arise from the Hamiltonian of the color code, Eq. (10).
A more detailed discussion is given in Ref. [28]. Fol-
lowing the example shown in Fig. 2 (a) we have al-
ready seen that two anyons of the same type are cre-
ated at the endpoints of the string. In fact, we can
extend the string by applying Pauli-X rotations on
the pairs of qubits on the green edges adjacent to the
plaquettes that support the excitations. Green edges
are those that connect two green plaquettes.
In general, it holds that pairs of excitations of a
Figure 2: The fusion rules for color code anyons. (a) Each
color code anyon is its own antiparticle. Here, as an example,
an operator is shown which annihilates (or creates) a pair of
gx anyons. Corresponding operators can be found for pairs
of all color code anyons. (b) Three anyons sharing the Pauli
label but each with a different color label annihilate. This can
also be read as a fusion (or decay) process: From bottom to
top for example, it corresponds to the fusion of a red and a
green anyon to form a blue anyon. Notice, the Pauli label x
was chosen arbitrarily; the same holds for y or z. (c) Three
anyons sharing the same color label but each with one of
the three Pauli labels can also be annihilated. Again, and
alternative reading is the fusion of an x anyon and a y anyon
to form a z anyon. The red color label can be replaced by
either green or blue.
Accepted in Quantum 2018-10-09, click title to verify 5
given color are created by applying strings of the ap-
propriate Pauli rotations to edges of its corresponding
color [28]. Given that we can separate pairs of exci-
tations of the same type CP by extending a string of
Pauli operators P type via edges of color C, we arrive
at the fusion rule
CP × CP = 1. (11)
Physically, this fusion rule represents
2
charge con-
servation as it shows that anyons of a given type
must be created in pairs. This rule can also be in-
terpreted to say that color code anyons are their own
anti-particles.
We can similarly study string operators acting on
the color code lattice to determine fusion rules be-
tween anyons with different color and Pauli labels. In
Fig. 2 (b) a Pauli-X operator is applied to a single site.
This operator violates stabilizers on its three adjacent
plaquettes, each of which has a different color due to
the three-colorability of the lattice. The excitations
created by this operator can be separated arbitrarily
far from the other excitations with the application of
another appropriately chosen string. We thus arrive
at the fusion rule
C
1
P × C
2
P = C
3
P, (12)
for any Pauli label P and pairwise distinct color la-
bels C
1
, C
2
, C
3
. We thus observe a conservation law
between the color labels of the color code excitations:
Two charges of a common Pauli-type and distinct col-
ors fuse to give the charge of the same Pauli type and
the third color.
Finally, we can also observe a fusion rule between
different Pauli labels. In Fig. 2 (c) we show a string
operator composed of Pauli-X, Pauli-Y and Pauli-Z
operators with three red endpoints where stabilizers
are violated. At two of the endpoints we have a sin-
gle violated s
x
p
(s
z
p
) stabilizer which indicates an rz
(rx) excitation. At the third endpoint both stabilizer
generators are violated, indicating an ry excitation at
this location. Generalizing, we obtain the fusion rule
that connects Pauli-labels of the excitations
CP
1
× CP
2
= CP
3
, (13)
for arbitrary color C and pairwise distinct Pauli labels
P
1
, P
2
, P
3
. The reader may recognize a duality be-
tween the color label and the Pauli label in Eq. (12)
and Eq. (13). Indeed, this duality is a key feature
of the color code anyon model as will become clear
throughout the manuscript.
With the fusion rules we have developed so far we
begin to see the utility of Tab. 1. Indeed, the table
echos Eq. (12) as the fusion product of two distinct
excitations from the same row give the third excita-
tion of the given row. Similarly, the fusion product
of two excitations of different Pauli label from a com-
mon column fuse to give the third excitation from the
given column. This reflects the fusion rule shown in
Eq. (13). More trivially, we have that the fusion prod-
uct of two excitations from the same element of the
table fuse to give the vacuum excitation as in Eq. (11).
With the rows and columns of Tab. 1 now sepa-
rately understood, it remains to consider the fusion
product of pairs of charges that share neither a row
nor a column. As we will show in the following subsec-
tion, the nine excitations in Tab. 1 are in fact bosonic
excitations. Aside from the vacuum particle, the exci-
tations of the color code that are not shown in Tab. 1
are fermionic excitations. All of the remaining exci-
tations can be expressed as the fusion product of two
of elements. In fact, the color code gives rise to six
distinct fermions. For more details on the fermionic
excitations of the color code we refer the reader to
App. A. Moreover, in App. B we show that we can
describe the excitations of the color code using two
non-interacting copies of a model of three fermions.
3.5 Self-exchange and braiding
We referred to color code anyons as either bosonic or
fermionic. To justify these names we now derive their
self-exchange statistics. Consider Fig. 3, where we
compare two sets of string operators which move rz
anyons along closed loops. On the left, two separate
loops are shown. This is the left-hand side of Eq. (6),
only that the bottom-left(right) and top-left(right)
ends of the trajectories are here connected. On the
right-hand side of Fig. 3, a single closed loop of an rz
operator which crosses itself in the middle is depicted.
This is the right-hand side of Eq. (6), again with the
bottom-left(right) end of the trajectories connected
Figure 3: Bosons in the color code have trivial spin θ
a
= +1.
As an example compare string operators which move an rz
anyon along a closed loop. On the left, two circular paths
looping back to their initial positions are shown. Each of
these operators is the product of the z type stabilizers on
the blue and green plaquettes they enclose and hence itself
again a stabilizer. On the right however, the middle section
is changed, such that path crosses itself once in the middle
to form a single larger loop. However, this redrawing of the
path did not change how the operators act, and the left-hand
side is exactly equal to the right-hand side. Further more,
according to Eq. (3), the difference between these operators
gives us the spin of the moved anyon, here θ
rz
= +1. The
same holds for all nine anyons shown in Tab. 1.
Accepted in Quantum 2018-10-09, click title to verify 6
Figure 4: To obtain the braid statistics of the color code
anyons, we study the commutators of string operators. As
an illustrative example, an rx anyon is moved from the left to
the right out of frame. Then, the line along which said anyon
was moved is crossed top to bottom by three other anyons.
Comparing the two different orders in which these operators
are applied reveals the phase obtained by braiding the anyons,
see Eq. (5). In (a) we see that strings moving anyons sharing
the color label (red in our example) exchange trivially. Here,
not a single qubit is acted upon by both strings. In general
the number of qubits supporting two crossing strings of the
same color is even. (b) Crossing strings moving anyons with
the same Pauli label also exchange trivially, as strings moving
them act in the same Pauli basis. (c) shows the third case,
neither the Pauli label nor the color label of the two anyons
associated with the crossing string operators match. This
leads to an odd number of qubits on which the strings act in
anti-commuting bases. In our example one qubit is acted on
first by an X and then by Z. This leads to a phase of 1 if
the anyons are braided.
to the top left(right). Hence, the difference between
the left and right half of the figure corresponds to the
self-exchange statistic of rz. But clearly, the effect of
the shown operators is exactly the same and hence
θ
rz
= +1, rz is a boson. This is true for all nine
non-trivial bosons, as can be seen by changing the
Pauli basis in which the considered operators act, or
by moving the string operators such that they move
anyons of different colors. The self-exchange of the
color code fermions can be derived using Eq. (6) and
Eq. (14) given below together with any decomposi-
tion of fermions into two bosons shown in Eq. (23)
and Tab. 8 in App. A.
Finally, we look at the phase obtained by braiding
pairs of color code excitations. Braiding a bosonic
anyon a with color label C
a
and Pauli label P
a
with
a second boson b with labels C
b
and P
b
results in the
phase
M
a,b
= (1)
δ(C
a
6=C
b
)δ(P
a
6=P
b
)
(14)
where δ(statement) is an indicator function that re-
turns 1 if statement is true and 0 otherwise.
Like for the fusion rules, we find Tab. 1 proves a
useful tool to visualize the braid statistics between
bosonic excitations. This is to say, anyons a and b
which share a row or a column braid trivially, i.e.
M
a,b
= +1. If a and b share neither a row nor a
column however, we have that M
a,b
= 1. To prove
the above, we compare the order in which the cross-
ing operators moving anyons a and b are applied. As
in Eq. (5), the difference reveals the phase M
a,b
. As
can be seen in Fig. 4, pairs of string operators asso-
ciated to anyons with the same color label or Pauli
label exchange trivially, and thus we find M
a,b
= +1.
When both, the color label and the Pauli label is dif-
ferent however, the operators anticommute and we
get M
a,b
= 1. The braiding rules involving fermions
can be derived from the above, using Eq. (7) and any
decomposition of fermions into two bosons shown in
Eq. (23) and Tab. 8 in App. A.
4 Boundaries
Unless we are dealing with a closed 2D manifold, the
spatial extent of a topological phase must end at some
boundary, such as a gapped domain wall with the triv-
ial phase. It is important to gain a better grasp on
the physics at the boundary of a topological phase.
Not only is this interesting from a condensed mat-
ter perspective [13, 14, 7376], but can be exploited
in quantum computation [23, 3941, 77, 78]. In this
section, after reviewing concepts of the boundaries of
abelian topological codes, we introduce lattice real-
izations of three new color code boundaries and show
that this completes the boundaries available to the
model. We finally discuss the prospective applications
of these boundaries for quantum computational tasks
as well as small scale experimental implementations
of quantum error-correcting codes.
4.1 Lagrangian subgroups
Gapped boundaries are characterized by Lagrangian
subgroups. These mathematical objects are subsets
of anyons in the anyon model C which can condense
at the boundary of interest. Anyon condensation is
the process wherein an anyon meets and is absorbed
by the boundary. The reverse process is also possible
where a single excitation is created or ‘emitted’ by a
boundary. Anyons which do not condense are said to
be confined by the boundary. Boundaries that can
condense excitations are important for quantum in-
formation applications as these are topologically pro-
tected degrees of freedom whose state is determined
by the charges they have absorbed.
The subset of anyons that can condense at a bound-
ary is constrained to obey certain conditions that
we will now briefly review. Indeed, it is shown in
Refs. [73, 74] that the subsets of excitations in C that
are absorbed at gapped boundaries correspond to La-
grangian subgroups M. A Lagrangian subgroup ful-
fills three conditions: Firstly, as one might expect,
a Lagrangian subgroup is a group, and thus closed
under fusion. Further, if a particle can condense at
a given boundary, so can its antiparticle. Likewise,
the vacuum particle can be condensed at any gapped
Accepted in Quantum 2018-10-09, click title to verify 7
Figure 5: The conditions on Lagrangian subgroups picto-
rially summarized. The gray line represents the boundary
between the topological phase at the bottom and the trivial
phase above. (I) If two anyons a and b can condense, so
can the outcome of their fusion c = a × b. And the emis-
sion of an anyon a can be read as the condensation of its
antiparticle a. (II.a) Anyons which can condense must have
bosonic self-exchange statistics. The shown paths are stabi-
lizers, hence the left and the right situation are equal, which
implies trivial self-exchange for condensible anyons. (II.b)
Two anyons which can condense must braid trivially. Again,
the depicted paths are stabilizers and must hence commute,
which implies the trivial braiding of the anyons. (III) An
anyon d which can not condense must exchange non-trivially
with at least one anyon which can condense. This means the
set of condensible anyons is maximal. Sets of anyons which
fulfill these conditions are called Lagrangian subgroups of the
anyon model.
boundary. Secondly, all of the anyons within the sub-
group are bosons and braid trivially with all other
anyons in the subgroup. The third condition is that
M is maximal in the sense that no additional anyon
can be added to the subgroup such that the other
conditions are fulfilled. In the following, we revisit
the argument given in Sec. IV of Ref. [73].
First, we start by showing that M is closed under
fusion, a M and b M = a × b M. a × b
can condense by decaying into a and b, which can
then condense separately. These is pictorially shown
on the left of Fig. 5 (I). Before showing that M does
actually form a group, we first turn out attention to
the other claims.
The second requirement of the boundary can be
understood by considering string operators s
a
. These
operators act such that they pull an anyon a M
from the boundary, transport it along a trivial tra-
jectory through the bulk and condense it back at the
same boundary such that s
a
leaves the charge at the
boundary unchanged. Given that the string opera-
tors follow a trivial trajectory and do not change the
net charge of the boundary, these operators do not
affect the logical state of the system. Furthermore,
given that such operators do not create excitations,
we can regard these operators as stabilizers such that
s
a
|ψi = |ψi for all ground state vectors |ψi.
Using such operators, we can argue that all anyons
which can condense must be bosons as follows: Con-
sider the four operators shown in Fig. 5 (II.a). Com-
paring s
a
, s
0
a
to ˜s
a
, ˜s
0
a
, we see the endpoints have
been exchanged. Hence, the left and the right in
Fig. 5 (II.a) differ, according to Eq. (3), by a phase θ
a
given by the spin of charge a. But since these four op-
erators are all stabilizers, they act trivially and hence
θ
a
= +1. This holds for all a M, and thus all
condensible anyons are bosons.
To show the trivial braiding between condensible
bosons, consider operators s
a
and s
b
, which move dif-
ferent anyons a, b M. Their trajectories intersect
in a single point, as depicted in Fig. 5 (II.b). Note
how the order in which they are applied is changed on
the right compared to the left. Again, since s
a
and
s
b
are stabilizers, the left and the right situation are
exactly equivalent. And according to Eq. (5), com-
paring the two cases reveals the phase M
a,b
obtained
when braiding a and b. We conclude that anyons in a
Lagrangian subgroup braid trivially with each other
such that M
a,b
= 1 a, b M.
The third claim about the maximality of these
subgroups can be intuitively understood by requir-
ing braiding non-degeneracy. This means every any-
onic excitation can be measured by at least one other
anyon through braiding, the only exception being
charges in M. See the left half of Fig. 5 (III).
This braiding non-degeneracy also holds close to a
boundary, if for example anyon d is condensed in
or confined at the boundary. It can be stated as,
d 6∈ M a M s.t. M
a,d
6= +1. In this case, the
only anyons which can perform a full braid around d
are anyons which are able to condense, a M. This
can be seen on the right half of Fig. 5 (III). Thus, an
anyon close to the boundary can either be seen by at
least one anyon a M, or is itself in M.
Now, we can address that M indeed forms a group.
We already discussed that it is closed under fusion.
From the maximality we can conclude that the triv-
ial charge 1 is also in M, as it braids trivially with
all anyons. To see that for each charge a M its
antiparticle a M holds, we can for example use
Eq. (6). Choosing b = a and c = 1 = a × a and using
θ
a
= θ
a
, we find M
a,a
= 1.
4.2 The boundaries of the color code
To find all boundaries of the color code, we explore
the Lagrangian subgroups of the excitations of the
model. Given that Lagrangian subgroups consist only
of bosonic excitations, we concentrate only on the
bosonic excitations of the color code. We find Tab. 1
offers a particularly convenient representation of the
bosons of the color code model for identifying La-
grangian subgroups. Precisely, subsets consisting of
the trivial anyon and three bosons from a row or col-
umn constitute the Lagrangian subgroups of the color
code.
The fact that these sets of anyons are closed under
fusion follows readily from the discussion in Sec. 3.4.
Accepted in Quantum 2018-10-09, click title to verify 8