Fermat's Library | The boundaries and twist defects of the color code and their applications to topological quantum computation annotated/explained version.

The boundaries and twist defects of the color code and their

applications to topological quantum computation

Markus S. Kesselring

, Fernando Pastawski

, Jens Eisert

, and Benjamin J. Brown

2,3

Dahlem Center for Complex Quantum Systems, Freie Universit

at Berlin, 14195 Berlin, Germany

Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia

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

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 [2–9]. 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 [10–16] that have been utilized

in a number of instances [12, 17–27] 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. [29–32] 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 [35–38]. 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 [48–54] 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 [57–62] 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

≤ 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, 66–68]. 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. [4–6, 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

a,b

, such that

a × b =

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,

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

≥ 1, which is determined by

the equation

c∈C

a,c

, (2)

where d

= 1. Anyon a is abelian if d

= 1 and is

non-abelian if d

> 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

a,b

= 1, and N

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

, called the topological spin. Diagram-

Accepted in Quantum 2018-10-09, click title to verify 3

matically we express this with the equation

a a

, (3)

where the lines indicate the trajectories of particles.

Note that θ

= θ

. We call a bosonic if it exchanges

trivially with itself, i.e. θ

= +1, and fermionic if

= −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

a,b

a b

. (4)

This equation can be transformed by adding a braid

to the two sides of Eq. (4) to obtain

a,b

a b

, (5)

an expression which will provide useful to calculate

the monodromy of anyons in the considered model. If

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

a,b

, (6)

and the monodromy with another anyon d ∈ C is given

c,d

= M

a,d

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. [29–31,

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

and Z

, 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

where k = n−m, 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,

j∈∂p

and s

j∈∂p

, (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 = −

−

. (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

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

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

P × C

P = C

P, (12)

for any Pauli label P and pairwise distinct color la-

bels C

, C

. 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

) 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

= CP

, (13)

for arbitrary color C and pairwise distinct Pauli labels

, P

. 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

= +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

= +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

= +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

and Pauli label P

with

a second boson b with labels C

and P

results in the

phase

a,b

= (−1)

δ(C

6=C

)δ(P

6=P

)

(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.

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, 73–76], but can be exploited

in quantum computation [23, 39–41, 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

. 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

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

|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

, s

to ˜s

, ˜s

, 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

given by the spin of charge a. But since these four op-

erators are all stabilizers, they act trivially and hence

= +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

and s

, 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

and

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

, 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

The braiding requirement is met as well, as discussed

in Sec. 3.5. There we argued that bosons in the ta-

ble braid trivially with each other if and only if they

share a row or a column. It is readily checked that the

subgroups of the rows and columns of Tab. 1 are max-

imal. We consider trying to add additional anyons to

the Lagrangian subgroups of a row or column. The

addition of a fermion would obviously violate the self-

exchange criterion of the subgroup. And trying to

add another boson only leaves bosons from a diﬀer-

ent column or row. But they braid non-trivially with

two of the elements of the Lagrangian subgroup. We

conclude that, together with the trivial charge 1, the

three rows and three columns of Tab. 1 correspond

exactly to the six possible Lagrangian subgroups of

the color code anyon model.

Alternatively, if the anyon model in question can be

decomposed into two layers of a simpler anyon model,

boundaries can be found that way. Namely, bound-

aries correspond to folds between the two layers of

the simpler model with its domain walls, see Sec. 6

and Ref. [74]. The famous equivalence of the color

code and two copies of the toric code is explored in

Sec. 9. A second equivalence between the color code

and two copies of the three-fermion model is discussed

in App. B.

We next go on to ﬁnd lattice representations of

these boundaries where the respective label subsets

condense.

4.3 Lattice representations of the color code

boundaries

We have identiﬁed six Lagrangian subgroups of the

color code anyon model. However, to the best of our

knowledge, only three of these subgroups have been

explored previously. In what follows we review the

boundaries that have already been studied and we

ﬁnd and discuss the new boundaries that correspond

to the other three Lagrangian subgroups of the color

code model.

The Lagrangian subgroups that have been consid-

ered explicitly form the columns of Tab. 1. Each of

the columns share a common color label, as such, we

ﬁnd that we can obtain boundaries that absorb bosons

with a common color label. We call these the red,

green and blue color boundaries. These color bound-

aries were presented in the original work introducing

the color code [28]. We show an example of a blue

color boundary in Fig. 6 (a). We remark that the

stabilizer generators at the boundary of the model re-

spect the same rules presented in Sec. 3 where each

plaquette p supports two stabilizers s

j∈∂p

and s

j∈∂p

. The ﬁgure also shows string op-

erators that create single blue excitations from the

boundary.

In addition to the three Lagrangian subgroups that

correspond to the columns of Tab. 1, we also ﬁnd three

Figure 6: The color code boundaries and corners between

them. (a) shows a blue boundary as an example of a color

boundary, as well as the condensation of the three blue

bosons. (b) An example of a Pauli boundary. Here the z

boundary is shown and the condensation of the three bosons

with Pauli label z. (c) Corners between two color bound-

aries appear if a qubit is only part of a single plaquette of

the third color. (d) A corner between two Pauli boundaries

is characterized by a qubit which is only part of two striped

boundary plaquettes. (e) Corners between Pauli and color

boundaries are also possible, here shown a z Pauli bound-

ary meeting a blue color boundary. (f) shows the stabilizers.

As usual, hexagons carry associated X- and Z-type weight-

six stabilizer. The color boundaries shown have quadrilateral

plaquettes which host weight-four X-type and Z-type sta-

bilizers. The striped triangular plaquettes featured in Pauli

boundaries are only stabilized by a single stabilizer. We indi-

cate the basis in which the triangular plaquettes are measured

by the direction of the stripes.

Lagrangian subgroups that correspond to the rows of

the table. We can therefore expect the existence of

boundaries that absorb all of the bosonic excitations

with a common Pauli label, independent of their color

label. These boundaries will be referred to as the

x, y and z Pauli boundaries. A similar conjecture

has been made in Ref. [33] by consideration of the

unfolded color code, i.e., two copies of the toric code.

We discuss this in more detail in Sec. 9.

In Fig. 6 (b) we show a lattice realization of the

z Pauli boundary, which condenses bosons with the

Pauli label P = z. At the boundary we show weight-

three plaquettes. In principle we could choose any

number of stabilizers acting at these plaquettes p,

j∈∂p

, s

j∈∂p

or s

j∈∂p

But since we require all stabilizers to commute, we see

that we can only choose one stabilizer per plaquette.

This is because s

, s

and s

mutually anti commute

as they each share three qubits. Similarly, two of these

boundary stabilizers on adjacent plaquettes must be

of a common Pauli type in order to commute with one

another. Adjacent boundary stabilizers have common

support only on one single qubit. In the ﬁgure we ar-

bitrarily choose s

stabilizers, and we show examples

of diﬀerently colored excitations with a z Pauli label

that are emitted from the boundary. To the best of

our knowledge, the Pauli boundary has not been ex-

plicitly represented on the color code lattice before.

The following table summarizes the discussion

Accepted in Quantum 2018-10-09, click title to verify 9

rx gx bx

ry gy by

rz gz bz

Pauli

boundaries

color

boundaries

Table 2: To visualize the color code boundaries, we can

return to the arrangement of the bosonic anyons presented

in Tab. 1. The nine non-trivial bosonic anyons in the color

code are arranged such that anyons in the same column or

in the same row exchange trivially. Each row and column

corresponds to a boundary. Anyons which can be condensed

at a common color (Pauli) boundary share their color (Pauli)

label.

given above. It explicitly shows that the columns

of the table represent the Lagrangian subgroups that

give rise to color boundaries and the rows repre-

sent the Lagrangian subgroups that give rise to Pauli

boundaries.

Finally, we present the interfaces between the dif-

ferent boundary types. General interfaces between

boundaries have attracted some interest in the con-

densed matter community [79]. For quantum in-

formation applications, points where boundary type

changes are referred to as corners [23].

We show diﬀerent corners in Fig. 6 (c), (d) and (e).

The ﬁrst interface separates two color boundaries, see

Fig. 6 (c). This interface has already been utilized in

Ref. [28] to ﬁnd a two-dimensional color code capa-

ble of transversally performing all Cliﬀord operations.

See also Refs. [29, 30] for a more detailed discussion

on similar boundary interfaces. In Fig. 6 (d) we show

an interface between two diﬀerent Pauli boundaries.

We show the corner between an x Pauli boundary

and a z Pauli boundary which generalizes immediately

to all Pauli-Pauli corners. At the top-right corner of

the ﬁgure we show two boundary plaquettes that are

commonly supported on two qubits where two Pauli

boundaries can meet such that the boundary stabiliz-

ers commute. Such corners can be used to construct

codes with Pauli boundaries, as discussed in Sec. 5.2.

Finally, in Fig. 6 (e), we show a corner where a Pauli

boundary meets a color boundary. Once again, it is

readily checked that the stabilizers represented in the

ﬁgure commute. This example can also be generalized

easily to obtain all nine color/Pauli corners.

5 Small color codes and fault-tolerant

quantum computation

In addition to giving us a better understanding of

the fundamental physics that underlies the color code

phase, a second beneﬁt of systematically laying out all

of the diﬀerent features of the color code is that it al-

lows us to discover new modes of fault-tolerant quan-

tum computation with the model. In what follows we

give two such examples of applications of Pauli bound-

aries. Speciﬁcally, we show that the new boundaries

we have introduced give rise to better methods of per-

forming code deformations, and we also ﬁnd a new ex-

ample of a four-qubit quantum error correcting code

which can be interpreted as the smallest instance of a

color code with Pauli boundaries.

5.1 Code deformation

We ﬁrst consider code deformations, and argue that

braiding punctures with Pauli-boundaries to perform

fault-tolerant entangling gates will be advantageous

over equivalent operations using punctures with col-

ored boundaries as have been considered previously

in the literature [41].

One of the leading paradigms for performing logi-

cal gates is through the braiding of punctures in some

topological quantum error-correcting code. This, tra-

ditionally, put the color code at a disadvantage com-

pared to the toric code. In the toric code, single

qubit measurements are suﬃcient to create and move

punctures. In contrast, the creation and movement of

punctures featuring the color boundaries of the color

code require two qubit Bell measurements [41]. We

ﬁnd that we can move punctures with Pauli bound-

aries using only single-qubit measurements. This is

readily checked because we can measure a patch of

individual qubits of the color code lattice using single-

qubit Pauli measurements in a common Pauli basis to

introduce or increase the size of a puncture. In doing

so, we remain in a known eigenstate of the stabiliz-

ers at the boundary of the new puncture that has

been produced by the measurements. We leave a de-

tailed derivation of this argument as an exercise to the

reader. This property of Pauli boundaries is advan-

tageous for practical implementations for color code

quantum computation as single-qubit measurements

are invariably easier to perform than two-qubit par-

ity measurements and are therefore less likely to in-

troduce noise during deformations. This feature may

also play an important role for measurement based

fault-tolerant quantum computation [39] using color

codes.

5.2 A four-qubit color code

Our extended toolbox has also enabled us to ﬁnd a

new four-qubit color code which we expect will be of

interest for near-term experimental realization with

small-scale quantum devices. They complement what

has been seen in small-scale trapped ion quantum

computers [50, 54] and relates to insights into the

smallest interesting color code [32].

To describe the features of our new code, we ﬁrst

present the family of triangular color codes with Pauli

boundaries. These codes feature an x Pauli bound-

ary, a y Pauli boundary and a z Pauli boundary. The

Accepted in Quantum 2018-10-09, click title to verify 10

Figure 7: Examples of small triangular color codes with Pauli

boundaries. The stabilizers acting on bulk plaquettes are the

usual. On the boundary plaquettes, indicated by the stripes,

only one stabilizer is measured. These boundary stabilizers

are shown in the top left. (a) shows the smallest member

of this family, a J4, 1, 2K code. It consists of four physical

qubits, three weight-three stabilizers and hosts one logical

qubit. Its logical operators are, for instance, X

= X

= Y

and Z

= Z

. (b) and (c) show the next

smallest codes with Pauli boundaries, consisting of nine and

sixteen qubits respectively. In (c) examples of two logical

operators, X

and Z

, are shown.

three smallest members from this family of codes are

shown in Fig. 7. Bulk plaquettes feature the two reg-

ular stabilizers given in Eq. (9), the striped boundary

stabilizers host a single weight-three stabilizer each,

the basis in which it acts is indicated by the direction

of its stripe pattern. These codes encode a single logi-

cal qubit each. Examples of their logical operators are

shown in Fig. 7 (c). Remarkably, the smallest instance

of this code, shown in Fig. 7 (a), is a new example of

a J4, 1, 2K code with only weight-three stabilizers.

6 Bulk domain walls

In this section we introduce transparent domain walls,

or invertible domain walls [14], to the model. These

domain walls change the charge labels of anyons which

cross them in a non-trivial way, such that fusion,

braiding and self-exchange of the anyons is conserved.

Speciﬁcally, we discuss closed domain walls here; open

domain walls and in particular their terminal points

are considered in Sec. 7. As we are interested in ap-

plying the theory to the color code, we restrict the dis-

cussion to abelian anyon models and refer the readers

to Refs. [15, 80] for the general case.

We ﬁrst provide a comprehensive study of trans-

parent domain walls for abelian anyon models in Sub-

sec. 6.1. Then we apply this theory in our concrete ex-

ample of the color code, whose symmetries we study in

Subsec. 6.2. Lattice representations of domain walls

are given in Subsec. 6.3.

6.1 Symmetries in abelian anyon models

Transparent domain walls change the labels of anyons

crossing them according to a map. This map is an

automorphism, ϕ : C → C which maps the anyon

model onto itself. Acting on an anyon a ∈ C it returns

Figure 8: The consistency conditions on a domain wall can

be derived by studying contractible loops crossing the domain

wall. From left to right the ﬁgures correspond to Eqs. (15-

17).

again an anyon ϕ(a) = a

∈ C. We require these

automorphisms to leave the basic data of the anyon

model, i.e its fusion, self-exchange and braid statistics,

invariant. See Ref. [15]. Explicitly, this means



a,b



= N

a,b

, (15)

ϕ (θ

) = θ

= θ

, (16)

ϕ (M

a,b

) = M

= M

a,b

. (17)

These conditions can be obtained by requiring that

anyons moved along contractible loops crossing the

domain wall constitute stabilizers and hence leave the

state invariant. Considering a branching loop, like

on the left of Fig. 8, we can follow Eq. (15). If an

anyon c on the left of a domain wall decays into a

and b which are then moved over the domain wall,

then on the right a

and b

must fuse to c

. And us-

ing that stabilizers commute with each other, we can

deduce Eq. (16) from the paths shown in the middle

of Fig. 8. We observe that any phase θ

picked up by

the self-exchange on the left must be canceled by the

phase θ

∗

obtained by the reversed exchange on the

right. Lastly, to get Eq. (17), consider the two un-

linked loops on the right of Fig. 8. Again, any phase

a,b

picked up by the braid performed on the left

must be canceled out by the phase M

∗

which is

obtain by the reverse braid on the right.

The automorphisms fulﬁlling these conditions for

an anyon model C are called the symmetries of C.

And each transparent domain wall can be described

Figure 9: Two closed domain wall, drawn in purple. The

arrows are to indicate the directionality of the domain wall.

The domain wall drawn as a solid line is described by the

automorphism ϕ, the dashed one by ϕ

. Where the domain

walls are fused, a new domain wall is obtained which is de-

scribed by ϕ

◦ϕ. Note that domain walls have an orientation

and changing it corresponds to taking the inverse ϕ in the

group of automorphisms.

Accepted in Quantum 2018-10-09, click title to verify 11

and labeled by one such automorphism ϕ. The set of

symmetries of an anyon model forms a group [15]. The

group identity is the trivial map, corresponding to the

absence of a domain wall, and the group multiplica-

tion is the composition of maps. Thus, composition

maps also fulﬁll the conditions and are therefore also

symmetries of the anyon model. Physically, compos-

ing maps corresponds to multiple domain walls being

combined or fused. See Fig. 9 for a schematic of how

domain walls change anyons and how the fusion of

domain walls corresponds to the composition map.

6.2 Symmetries in the color code

To classify the transparent domain walls in the color

code, it proves once again useful to go back to the 3×3

grid of bosonic color code anyons, shown in Tab. 1. As

mentioned earlier, fusion of two diﬀerent anyons in a

common row or column results in the third anyon in

said row or column. As such we ﬁnd that the fusion

rules are preserved by all transformations on the grid

that respect that all elements of a given row or column

remain together in any other row or column. Likewise,

we know that excitations braid trivially if and only if

they share a column or row of the grid. Like fusion

then, the mutual braid statistics of this generating

set of excitations shown in the table are respected

provided all excitations that share a common row or

column remain in rows and columns with the same

excitations.

The above discussion implies that the symmetries of

the generating set of bosonic excitations, and thus the

excitations of the color code in general, are obtained

by ﬁnding all the transformations that respect the

rows and columns of the grid. Indeed, these are row

permutations, column permutations, and the trans-

position of the grid. Given there are three rows and

three columns, there are six diﬀerent permutations of

the rows, and of the columns, and ﬁnally one trans-

pose. We thus arrive at 6 ·6 ·2 = 72 symmetries of the

color code. This agrees with the exhaustive numerical

search given in Ref. [45].

Further insight on the symmetry group can be

gained using the following decomposition: (S

× S

. One S

comes from permutations of the columns,

i.e. colors, the second from the rows, i.e. the Pauli

labels. Since permuting rows and columns commutes,

the corresponding group is given by the direct prod-

uct, S

× S

. Finally, the

matrix transposition

exchanging color and Pauli labels acts via the semi

direct product (n). As a compact notation we can use

the wreath product (o) to write the symmetry group

as S

. This decomposition follows directly from

interpreting the color code anyon model as two copies

of the three-fermion model, see App. B for more de-

tails.

To label symmetries which exchange two colors, we

use a capital letter of the third, unchanged color. For

example, the symmetry which exchanges the green

and the blue color label while leaving the red label,

we denote as R. Equivalently, G and B respectively

leave green and blue color labels invariant while ex-

changing the other two. Similarly, the exchanges of

the Pauli labels we denote as X, Y or Z, for symme-

tries which leave the x, y or z Pauli label invariant.

Finally, we use D for the duality symmetry between

color and Pauli labels, corresponding to the transpo-

sition of the 3×3 anyon grid, shown in Tab. 3. All of

the symmetries mentioned so far are of order 2, i.e.

applying them twice leads to the trivial symmetry.

Combining two of the above order 2 color symme-

tries creates one of the two cyclic color permutations.

Thus we can label them RB and BR, for example.

Conjugation of one color exchange with a second re-

sults in the third, e.g. RGR = B. The analogous

statement holds for the permutations of Pauli labels;

XZ and ZX are the cyclic Pauli label permutations

and we can check that XY X = Z. Conjugating

a color exchange with D creates a symmetry which

solely exchanges Pauli labels, for example DBD = Z,

or conversely DZD = B.

r g b

Table 3: The eﬀect of the three generating symmetries on

the anyons. B is the blue invariant color permutation which

exchanges the red and green color label. Y the y-invariant

Pauli permutation which exchanges x and z Pauli labels. And

D the duality symmetry which exchanges color and Pauli

labels as shown.

To generate all 72 symmetries, three are already

suﬃcient. A minimal choice might be, for instance,

X, Y and D, two diﬀerent row permutations and the

duality. However, here, we choose B, Y and D, a row

exchange, a column exchange and the transposition,

as shown in Tab. 3. This is to demonstrate all three

diﬀerent types of domain walls in what follows.

6.3 Lattice representation of domain walls

Now, we give lattice representations of color code do-

main walls which correspond to the generating sym-

metries above. In particular, these are the B, Y and

D symmetries. They can generate all 72 color code

domain walls, as discussed in Sec. 6.2. And they

also constitute examples for three diﬀerent classes of

twists, one only acting on the color label, one on

the Pauli label and duality transformation which ex-

changes the two. To obtain these closed domain walls,

we transform the stabilizers along a closed path. This

provides an apparent physicality to the location of the

Accepted in Quantum 2018-10-09, click title to verify 12

Figure 10: A 6.6.6 color code lattice with a blue-invariant

domain wall B, marked by the purple line. Along the domain

wall a fourth color, yellow, is introduced. Two commuting

stabilizers are measured on each plaquette, one x-type and

one z-type. Anyons crossing the domain wall keep their Pauli

label, but red and green color labels are exchanged. Note

how we omitted the arrow indicating the orientation of the

domain walls, as it corresponds to an order two symmetry

B = B.

domain walls, which although artiﬁcial provides a way

to later emphasize their terminal points whose local-

ity is physical. These terminal points are called twist

defects and are the focus of Sec. 7. We emphasize

that neither of the transformations can be achieved

by means of a local unitary with support only where

the domain wall will come to lie.

An example of a domain wall changing the color

label is shown in Fig. 10. It exchanges solely the red

and the green color label of anyons crossing it. It cor-

responds to the symmetry B, shown in Tab. 3. Along

the domain wall, each plaquette hosts two stabiliz-

ers, an x- and a z-type. Notice how in this example

a fourth color is used to color the faces. The brak-

ing of three-colorability is in fact a general feature of

color permuting domain walls. In App. C we show

how color permuting domain walls can be introduced

by applying a series of Pachner moves on the dual

lattice. Using the Pachner move technique also al-

lows us to generate domain walls which permute the

three color labels in a cyclic fashion, see Fig. 36 (a)

and Fig. 37 (a). The two other color exchanging

domain walls, R and G can be drawn equivalently.

In Refs. [17, 44, 81], similar domain walls have been

demonstrated for equivalent models.

Next, we turn our attention to domain walls per-

muting Pauli labels. They can be constructed with-

out changing the underlying lattice. Instead, just the

basis in which stabilizers act along the domain wall

are measured is altered. Such a domain wall is rep-

resented by a violet line in Fig. 11. To generate a y-

invariant domain wall Y , the two stabilizers for each

plaquette lying on top of the domain are changed.

The ﬁrst one acts in the x-basis on qubits on one side

of the domain wall, and in the z-basis on qubits on the

other side. The second one acts in the z-basis on one

side and the x-basis on the other side. This is equiv-

Figure 11: A 6.6.6 color code with a y-invariant Pauli domain

wall Y represented by a purple line. Along the line, every

plaquette hosts two stabilizers, they are shown in the bottom

left. They can be obtained by conjugating the basis in which

the stabilizers act by a Hadamard on one side of the domain

wall.

alent to conjugating the basis in which the domain

wall stabilizers act with a Hadamard on one side of

the domain wall, as discussed in Ref. [45]. These also

correspond to locality preserving gates in the topo-

logical quantum ﬁeld theory, which were studied in

Ref. [82]. Domain walls which permute the Pauli label

of anyons in a cyclic manner can also be constructed

in a similar fashion by conjugating them partially by

a unitary which cyclically permutes x, y and z.

Lastly, we give a lattice representation of the do-

main wall which transposes the 3×3 arrangement of

anyons. It corresponds to the symmetry D and we re-

fer to it as the domino domain wall, since it features

stabilizers that resemble domino bricks, see Fig. 12.

Such a domain wall can be constructed by pulling a

zigzag line of edges apart which duplicates all of the

qubits lying along the line. In the created space we

color squares of the lattice which deﬁne rectangular

stabilizers that each act on six qubits which are sup-

ported on two adjacent squares that make a rectangle.

Each of these new stabilizers has two colors, one on

Figure 12: An example of an domino domain wall D. The

dislocation line hosts rectangular stabilizers supporting six

qubits. Like domino bricks, they are colored in a diﬀerent

color on either half. Their colors dictate the basis in which

they act, shown in the bottom left.

Accepted in Quantum 2018-10-09, click title to verify 13

either side, resembling domino bricks. These colors

are determined by the colors of the neighboring pla-

quettes. To each of these domino-brick shaped cells

we assign one weight-six stabilizer, where the colors

of the brick determine which operator acts on rectan-

gle. Stabilizers colored green and blue are measured

in the x-basis, red and blue in the y-basis, and red and

green in the z-basis. This results in the symmetry D

which transposes the 3×3 grid. This domain wall it

leaves the bosons rx, gy and bz that cross it invariant.

See Fig. 12 for a lattice representation of the domain

wall and the eﬀect it has on anyons. Changing what

pair of colors is associated with which Pauli basis, six

diﬀerent domain walls can be obtained. To the best

of our knowledge, this is the ﬁrst time a lattice rep-

resentation of this domain wall has appeared in the

literature.

7 Twists

In the previous section, we considered continuous

transparent domain walls that change the charge la-

bels of an abelian anyon model while preserving the

fusion rules and braid statistics between the charges.

In what follows we build on our discussion by ter-

minating the domain walls. Again, we restrict our

discussion to abelian anyon models. At the terminal

points of these domain walls, we realize twist defects

or simply twists [11, 12, 14, 15]. These remarkable

point-like objects have many properties in common

with non-abelian anyons. This is especially intriguing

considering that we started with an abelian theory. As

such they are of particular interest for applications in

quantum information processing.

We begin by introducing twist defects in Sub-

sec. 7.1. In Subsec. 7.2 the fusion of twists is dis-

cussed. We then study the twist defects of the color

code and discuss how they interact with its anyons

in Subsec. 7.3, with domain walls in Subsec. 7.4 and,

lastly, with boundaries and corners in Subsec. 7.5 and

Subsec. 7.6.

7.1 Twist defects

We start by labeling the twist defects. To this end,

we consider an anyon a that is moved once around

a twist before returning to its initial position. The

anyon will cross the domain wall ϕ which the twist

terminates once under this monodromy, and as such

is mapped onto the excitation ϕ(a). We label a twist

ϕ if it acts on C with the mapping ϕ under a single

clockwise monodromy.

With this deﬁnition we observe particle anti-

particle like creation of pairs of twists when adding an

open domain wall to the bulk of a phase, see Fig. 13.

Indeed, we observe a twist ϕ at one terminal point of

the domain wall and ϕ at the other.

Figure 13: Twists terminating a domain wall correspond to

the automorphism as the domain wall. Consider a domain

wall which maps an anyon a according to ϕ if the anyon gets

transported in one direction across. If the anyon goes the

other way, the inverse ϕ is applied. The two twists lying on

the terminal point of the domain wall are drawn as purple

squares. Moving the anyon a counter-clockwise around the

twists applies the automorphism ϕ for one twist and ϕ for

the other twist terminating the domain wall.

We now go on to discuss the explicit example of

twist defects in the color code. Speciﬁcally, we con-

sider a generating set of twist defects, B, Y and D,

which are the terminal points of their corresponding

domain walls. This generating set will be suﬃcient to

generate all of the twists of the color code using twist

composition rules we discuss later in Sec. 7.2.

We ﬁrst look at a lattice realization of the blue-

invariant twist B lying at the terminal point of the

blue-invariant domain wall, see Fig. 14. On the lat-

tice, the terminal point of the domain wall lies at a

plaquette with an odd number of vertices. This means

it can no longer support both an x-type and a z-type

stabilizer, as they would not commute. Instead, a

single basis has to be chosen in which the parity is

measured.

A Y twist lies at the terminal point of the domain

wall that permutes the Pauli-X and Pauli-Z label of

the color code bosons. Unlike a B twist we do not

need to change the structure of the lattice to introduce

a Y twist. However, at the terminal plaquette of the

domain wall we can impose only a single stabilizer of

Pauli-Y terms, as is depicted in Fig. 15.

Figure 14: Shown is a lattice representation of the B twist.

It features a weight-seven stabilizer which is just measured

in one basis, the y-basis for example. If anyonic charges are

moved once around it, their color label changes according to

the automorphism B.

Accepted in Quantum 2018-10-09, click title to verify 14

Figure 15: A y-invariant domain wall can be terminated by

a plaquette which hosts one single stabilizer acting in the

y-basis. This constitutes a Y twist which exchanges x and

z Pauli labels of color code anyons under monodromy.

We ﬁnally look at the twist D that terminates the

domino-brick domain wall. As shown in Fig. 16, at

the terminal point of the domain wall we have one

weight-ﬁve and two weight-seven stabilizers.

While in some cases it is possible to propose twist

defect stabilizers via an educated guess, the stabilizers

can be rather complex; this is particularly true in the

case of D twists. In fact, it is possible to essentially

eliminate any guess work and derive the stabilizers for

the twist defect by considering a sequence of Pauli op-

erations which by construction preserves the ground

space. First, we create a pair of anyons, a and ¯a.

Excitation a is then transported around the defect `

times, following its label change every time it crosses

the domain wall ϕ such that ϕ

(a) = a. Upon com-

pleting this repeated monodromy we can annihilate

the two excitations to obtain a new stabilizer opera-

tor from the product of the Pauli operators used to

realize this process. In order to ﬁnd low-weight stabi-

lizers, the resulting string operator can be contracted

by multiplying it with conventional stabilizers that lie

on the lattice.

Figure 16: The domino twist D has three changed stabiliz-

ers. One of the Domino brick stabilizers is cut so that it only

supports ﬁve qubits. Additionally we have two weight-seven

stabilizers, overlapping on six oft their qubits.

Figure 17: Similar to how the fusion of domain walls cor-

responds to composing maps, the same holds for twists. In

this example, two twists described by the maps ϕ and ϕ

are

combined to create a twists described by ϕ

◦ ϕ.

7.2 Fusion between twists

The fusion of domain walls can be described by com-

position maps, see Sec. 6.1. Twists correspond to

terminal points of domain walls, and hence the fu-

sion of twists is also described by the composition

of maps, see Fig. 17. Given that the symmetry of

an anyon model forms a group, the fusion of pairs of

twists result in other twists. This means places where

a domain wall changes type or where multiple domain

walls meet are also twists.

For an example, in the color code we can consider

the fusion of two color exchanging twists, R and B, to

form a cyclic color permuting twist, RB, see Fig. 18.

Note that the lattice representation of the RB twist is

not presented in the main text, we refer the reader to

App. C, in particular to Fig. 36 (a). This fusion can

be visualized nicely on the dual lattice using Pachner

moves, see Fig. 38 in App. C.

In Fig. 17 we have deformed the domain walls up-

wards at the point where the twists meet. We thus

obtain a twist ϕ

◦ ϕ. Instead we could have chosen

to bend the domain walls downwards, such that the

twist maps a according to ϕ ◦ ϕ

(a). This ambiguity

in the label is a general feature of twist fusion. To

rectify this, we need to consider conjugacy classes of

the symmetry group. The label used to describe a

twists does in general depend on an arbitrarily cho-

Figure 18: An R twist on the left and a B twist on the right

are fused to form an RB twist. The RB twist features two

plaquettes with odd weight, where only one stabilizer each is

measured.

Accepted in Quantum 2018-10-09, click title to verify 15

Figure 19: This ﬁgure depicts the fusion of two twists de-

scribed by the symmetries ϕ and ϕ

. On the left, we illustrate

how the reference point used to label the fused twist matters.

If the reference point is chosen to lie above the twist, as in

the top left, we obtain a map ϕ

◦ ϕ. Whereas a reference

point below the twist results in the description ϕ ◦ ϕ

, as

shown in the bottom left. However, one description can be

changed to the other by wrapping a domain wall around the

twists. This is seen on the right, where the lower descrip-

tion is obtained by enclosing the three twist with a closed ϕ

domain wall.

sen starting point. Consider Fig. 19, where a twist is

labeled by two diﬀerent maps on the left half of the

ﬁgure. Shown on the right, we see that the two maps

are equivalent up to conjugation. Here, conjugating

the twist is equivalent to changing gauge or engulf-

ing the entire twist compound inside a closed domain

wall. If we do this in the example of Fig. 19, as shown

on the right, we can convince our selves that the two

maps are equivalent up to conjugation with ϕ.

In a general example, where multiple domain walls

meet, even more than two diﬀerent but valid descrip-

tions might be found. And in general, without ﬁxing a

global gauge and precise conventions, twists are only

identiﬁed up to conjugacy.

In the color code, twists belong to one of nine equiv-

alence classes. Tab. 4 lists the conjugacy classes A to

I of all 72 twists. Within an equivalence class twists

X Y Z XZ ZX

A B B B D D

R B C C C E E

G B C C C E E

B B C C C E E

RB D E E E F F

BR D E E E F F

D ◦ () X Y Z XZ ZX

G H H H I I

R H G I I H H

G H I G I H H

B H I I G H H

RB I H H H I G

BR I H H H G I

Table 4: The equivalence classes of the 72 color code twists.

Each entry corresponds to a twist obtained by fusion of the

twist labeling the row and column. In the second table, every

twist is additionally fused with the domino twist D. We ﬁnd

nine conjugacy classes, labeled A to I here.

Figure 20: Shown is a twist described by the map ϕ. We

consider an anyon a which can decay into ϕ(b) and b. If

the ϕ(b) anyon is now moved counter-clockwise around the

twist, it crosses the domain wall described by ϕ. Hence it

turns into a b anyon, which can fuse with b to form the trivial

particle 1. This process is called localization of anyon a at

twist ϕ.

share many properties, such as their order or their

quantum dimension (see Tab. 5).

7.3 Anyon localization and the quantum di-

mensions of twists

Similar to how certain anyons can condense at bound-

aries, twists can also absorb anyons. This process is

called localization of an anyon at a twist. We can ex-

press the localization of an anyon a at twist ϕ with

the fusion rule

ϕ × a = ϕ. (18)

To determine the anyons that can localize at a twist,

we study its automorphism ϕ together with the fusion

multiplicities of the anyon model N

a,b

. In particular,

if there exists some excitation b ∈ C such that

ϕ(b),b

= 1, (19)

then we have that ϕ × a = ϕ. To show why this is

true, we study Fig. 20. The ﬁgure shows a twist ϕ and

an anyon a which decays into a pair of particles ϕ(b)

and b. ϕ(b) is then moved counter-clockwise around

the twist such that the resulting excitation ϕ ◦ϕ(b) =

b annihilates with b and restores the system to the

ground state.

To illustrate anyon localization in the color code,

consider the three examples shown in Fig. 21. The B

twist can localize all blue bosonic anyons by decay-

ing them into red and green anyons. If one is moved

around the twist it changes color and can annihilate

with the other anyon. Cyclic color permuting twists

can annihilate all sixteen color code anyons. This is

because all anyons are composed of the nine bosonic

anyons, which in turn can decay into two anyons of

diﬀerent color. If one is moved around the twist one or

two times, it can be brought to match color with the

other one, so that they annihilate. Lastly, the D twist

can condense one of the fermionic particles. Fermions

in the color code are composed of two bosonic parti-

cles with diﬀering color and Pauli labels, see App. A.

Accepted in Quantum 2018-10-09, click title to verify 16

Figure 21: Left: A B twist, which exchanges the red and

the green color labels of anyons is shown. A blue anyon with

any Pauli label can localize at the twist. We decay it into a

red and a green anyon, which when moved around the twist

match color and annihilate. Middle: A cyclic color twist

can localize anyons of any color. This is because an anyon

can obtain all three color labels if moved around the twist

suﬃciently many times. Right: The D twist can condense a

single fermionic anyon. In this case, the f

= rx × bz anyon

gets localized. The dashed line indicates the diﬀerent Pauli

label of the red anyon.

The domino twist D as we deﬁned it can localize the

fermion f

= rx × bz. To see this, we take one of its

constituent bosons, say rx, move it around the twist

once to transform it into bz which annihilates with

the other constituent part of f

The process of anyon localization at twists, as de-

scribed by Eq. (18), is reminiscent of the fusion in non-

abelian anyon models. Further, as with non-abelian

anyons, we can encode logical qubits in the fusion

space of twists. To increase the number of logical

qubits, more twists can be added to the system. The

quantum dimension of a twist defect is the factor by

which the fusion space dimension grows, as a twist

of that type is added. We calculate the quantum di-

mension of twist ϕ, denoted d

, with a formula that

is analogous to that of non-abelian anyons given in

Eq. (2)

a∈C

ϕ,a

, (20)

where N

ϕ,a

tells us if the anyon a can localize at the

twist ϕ. In addition to calculating the quantum di-

mension of a twist defect directly from its fusion rules,

we also point out work where, like with non-abelian

anyons [83, 84], the quantum dimension of twist de-

fects can be evaluated by studying the entanglement

entropy of the ground state of topological phases that

include twist defects [85–87].

Since all anyons in the color code are abelian, their

dimension is 1 and hence the quantum dimension of a

twist is just the square root of the number of anyons

it can localize. As mentioned above, the B twist for

example can, apart from the trivial charge, localize

all three blue bosonic anyons. Thus, its quantum di-

mension is d

= 2. Cyclic color twists can localize all

sixteen anyons and have quantum dimension d

= 4.

The domino twist D can localize only a single fermion

and the vacuum and has dimension d

√

2. The

X Y Z XZ ZX

1 2 2 2 4 4

R 2 2 2 2 4 4

G 2 2 2 2 4 4

B 2 2 2 2 4 4

RB 4 4 4 4 2 2

BR 4 4 4 4 2 2

D ◦ () X Y Z XZ ZX

√

Table 5: The quantum dimension of the 72 color code

twists. Each entry corresponds the twist obtained by fus-

ing the twists labeling the row and column. In the second

table, every twist is additionally fused with the domino twist

quantum dimension of all 72 color code twists is given

in Tab. 5.

Anyons which do not localize at a twist might get

identiﬁed with other non-trivial anyons. Say anyon

a localizes at twist ϕ and hence a × ϕ = ϕ, and an

anyon c can decay to a and b, i.e. c = b ×a. Then, we

cannot distinguish the two twist-anyon compounds,

c × ϕ = b × a × ϕ = b × ϕ. Microscopically, twist-

anyon compounds are anyons located close to the

twist defect. Mathematically, the twist-anyon com-

pounds form a G-crossed braided tensor category, for

more details see Ref. [15]. In this manuscript we are

solely concerned with the ground-state space, or code-

space when viewed as a quantum code. This means

we will focus on twists which are locally equivalent to

the twist with the trivial particle 1 localized.

7.4 Interplay between twists and domain walls

We now study crossing domain walls and the twists

which terminate them. This is relevant when twists

are moved over domain walls using, for instance, code

deformations [12, 17, 21, 44, 88]. To ﬁgure out how

the twist changes after crossing a domain wall we con-

sider closed contractible loops engulﬁng the crossing

point of the domain walls. If an anyon was moved

along such a contractible path until it reaches its orig-

inal position, it should return to its initial charge la-

bel. Consider a twist which acts as the map ϕ on

the anyon labels, as depicted in Fig. 22. After it

crosses a domain wall ϕ

it transforms into a twist

acting as ϕ

= ϕ

◦ ϕ ◦ ϕ

. An anyon a which moves

on the aforementioned path then gets mapped by

◦ ϕ

◦ ϕ ◦ ϕ

= . Remember, the equality stems

from the fact that the loop followed by the anyon is

contractible. This means the symmetry describing the

Accepted in Quantum 2018-10-09, click title to verify 17

Figure 22: Consider a twist described by ϕ, here drawn at the

bottom, terminating a solid domain wall. It crosses a dotted

domain wall ϕ

. This transforms the twist to ϕ

. An anyon

a starting in the top-left quadrant and traveling counter-

clockwise around the crossing point gets ﬁnally mapped to

◦ ϕ

◦ ϕ ◦ ϕ

(a). Since its path is contractible, the ﬁnal

label has to be equal to its original label a. Following this

equivalence for a generating set of anyons identiﬁes the twist

= ϕ

◦ ϕ ◦ ϕ

as the conjugation of ϕ with ϕ

transformed twist is obtained by conjugating its origi-

nal symmetry with the one of the crossed domain wall.

Hence, twists crossing domain walls will stay within

the same equivalence class, see Sec. 7.2.

Again, let us illustrate this with a color code exam-

ple. Consider a color exchanging twist, say G which

crosses the domino domain wall described by the dual-

ity symmetry D. Using the procedure described above

for a generating set anyons, rx, rz, bx and bz shows

that the twist transforms to a Pauli exchanging twist

Y , see Fig. 23. This is to show the duality transfor-

mation D does aﬀect twists in very much the same

way as it does anyons. The Pauli label of twists and

the color labels are exchanged.

As a second example we consider a color exchang-

ing twist G crossing a color exchanging domain wall

R which turns it into a B twist. This follows from the

Figure 23: A G twist (bottom) transforms to a Y twist (top)

if moved over a domino domain wall D (middle). This can be

seen by starting with a generating set of four anyons, rx, rz,

bx and bz, in the top-left quadrant. Moving them counter-

clockwise around the crossing point to the top right, they

transform to rz, rx, bz and bx, respectively. Moving them

back to their initial position over the top half of the vertical

domain wall should return them to their initial charge. The

only domain wall with this eﬀect is the one terminated by a

Y twist.

Figure 24: Lattice representation of a twist crossing a do-

main wall. A G twist (bottom) is moved over an R domain

wall (middle) and transforms to a B twist (top). Apart from

the plaquettes hosting the twists marked by purple squares,

all plaquettes host two independent stabilizers.

fact that GRG = B. Again, we see that the domain

wall has an equivalent action on the twist label as it

would on anyon labels, see Fig. 24 for a lattice repre-

sentation. This example can again be understood nat-

urally using the Pachner move construction for twists

and domain walls, see Fig. 41 in App. C.

7.5 Conﬁnement of twists at boundaries

In this subsection, we explore how twists and domain

walls interact with boundaries. In particular, a do-

main wall covering a boundary can change the bound-

ary type. This implies that a twist defect conﬁned at

the boundary corresponds to a corner. A corner is

the point where two diﬀerent boundaries meet, as de-

scribed in Sec. 4. If the boundary type is not changed

by the automorphism applied by the domain wall then

the corresponding twist can condense at said bound-

ary.

Let us begin by investigating how boundaries

change when domain walls are present. Consider a do-

main wall described by the automorphism ϕ close to

a boundary which is characterized by its Lagrangian

subgroup M, see in Fig. 25 (a). If an anyon a is con-

densible at M, i.e. a ∈ M, then the anyon ϕ(a) can

condense at the boundary by ﬁrst moving over the

domain wall. This transforms it to ϕ ◦ ϕ(a) = a ∈

M. Clearly, an anyon ϕ(a) can still condense when

the domain wall covers the boundary, as depicted in

Fig. 25 (b).

In fact, the anyons ϕ(a) form a Lagrangian sub-

group M

= ϕ(M) which may condense at the re-

sulting boundary. This follows from the properties of

Lagrangian subgroups, discussed in Sec. 4.1, and the

conditions put on the symmetries of the anyon model,

see Eqs. (15-17). To illustrate this, the covering do-

main wall was removed in Fig. 25 (c) and the new

boundary M

drawn as a dashed line. In this pic-

ture the twists correspond to the corners between the

diﬀerent boundaries. In an alternative reading of the

above we have that the corners correspond to conﬁned

Accepted in Quantum 2018-10-09, click title to verify 18

Figure 25: Corners, points where the boundary type changes,

can be identiﬁed with conﬁned twist defects. In (a), a pair of

twists ϕ and ϕ sit close to a boundary M where anyons a ∈

M can condense. The boundary behind the domain wall is

able to condense anyons ϕ(a), if moved over the domain wall

ﬁrst. (b) shows same situation as (a), but the twists and their

connecting domain wall are now covering the boundary. This

corresponds to (c), where boundary covered by twists and

the domain wall now correspond to corners and a changed

boundary denoted by the dashed gray line. If the original

boundary is described by the Lagrangian subgroup M, then

the dashed domain wall is described by M

= ϕ(M). Note,

this encapsulates trivial corners, i.e. the case where M =

. We call twists ϕ

for which ϕ

(M) = M condensible

at the boundary M.

twists. In contrast, if an automorphism ϕ

leaves the

boundary type unchanged, i.e. ϕ

(M) = M, we say

the twist ϕ

can condense at the boundary M. Twist

condensation is discussed in more detail in Sec. 7.6.

This leads to an ambiguity, corners can not, in gen-

eral, be uniquely identiﬁed with a single twist, as they

might fuse with condensible twists. If ϕ(M) = M

and ϕ

(M) = M, then ϕ ◦ϕ

(M) = M

. This means

a twist ϕ ◦ ϕ

, the fusion outcome of a condensible

twist ϕ

with a corner twist ϕ is again a valid inter-

pretation of the corner.

Let us now turn to the corners in the color code for

concrete examples of the aforementioned ambiguity in

the association of corners with twists. Consider the

common triangular color code with the three color

boundaries, red, green and blue. This code can be

constructed in the several ways from a disk with a

single red boundary and adding open domain walls to

segments of the uniform red boundary. We show two

such ways in Fig. 26. We ﬁrst consider applying a do-

main wall that implements the G symmetry to cover

one red colored boundary segment which transforms

it to the blue color boundary. Similarly, the B domain

wall creates a green boundary when covering the sec-

ond part of the red boundary. Hence, the corner that

lie between the red-blue(red-green) boundary can be

interpreted as a G (B) twist. The blue-green cor-

ner then is the fusion of the two twists which we call

GB = BR. This is depicted on the right in Fig. 26.

Alternatively we can use the cyclic domain walls RB

and BR to transform a segment of the red boundary

onto a blue or green boundary, respectively. Hence,

as shown on the left in Fig. 26, we can label each

Figure 26: The identiﬁcation of color code corners with

twists is ambiguous. We start with a color code with just a

single red boundary, as shown in the ﬁrst row. Consecutively,

we transform one boundary after the other by covering them

with domain walls, until the common triangular color code in

the bottom center is obtained. This construction shows the

non-uniqueness of the interpretation of corners as twists. On

the left, we end up with three BR twist, one in each corner,

where on the right only one corner is identiﬁed with a cyclic

twist.

of the three corners as a cyclic color permuting twist

BR. The two approaches to realize the color code

with three diﬀerently colored boundaries diﬀer by an

R twist fused with the two corner twists that lie at

the red boundary. However, given the red boundary

is invariant if we cover it with a red domain wall, we

see that the physics of the two models we have pro-

duced is invariant under the fusion of the G and B

twist of Fig. 26 with red twists. We thus see that the

two pictures we have produced are consistent.

Corners between Pauli boundaries can be identiﬁed

equivalently with Pauli permuting twists. Corners be-

tween Pauli and color boundaries host twists shown

in the second half of Tab. 4, who involve the D twist

when generated from the standard set of symmetries.

This is shown in more detail in Fig. 28, only that

there, the corners are detached and pulled into the

bulk as twists. This process is discussed in Sec. 7.6.

7.6 Detaching corners and twist condensation

Not only can we identify corners with certain twists,

we can also detach corner twists and move them into

the bulk. Consider Fig. 27 (a), where we interpret

the M

boundary as a domain wall ϕ covering the

M boundary, and the corners as twists. These twists

can be moved, for example through code deformation,

such that they come to lie in the bulk, see Fig. 27 (b).

This leads to Fig. 27 (c), the twist is detached from

the corner and pulled into the bulk.

Notice how this framework also encapsulates the

Accepted in Quantum 2018-10-09, click title to verify 19

Figure 27: (a) Shows the same situation as depicted in

Fig. 25 (b), where the M

boundary is seen as a domain

wall covering the M boundary, and the corners as twists. In

(b), the domain wall is extended such that the ϕ twist now

lies in the bulk. This corresponds to (c), where the domain

wall covering the boundary is omitted to illustrate that it

constitutes a diﬀerent boundary. Now, the situation looks

like a single twist whose domain wall terminates between the

M and M

boundaries. This can be read as a detached

corner which is pulled into the bulk as a ϕ twist. (d) shows

the case where M = ϕ

(M), the trivial corner. Twists which

map the Lagrangian subgroup of a boundary back to itself are

able to condense at and being emitted from said boundary.

trivial corner, i.e. the case M = M

. This means

a twist ϕ

which maps the Lagrangian subgroup de-

scribing a boundary back onto itself, i.e. ϕ

(M) = M,

can condense at or be emitted from said boundary, see

Fig. 27 (d).

To illustrate the above with practical examples, let

us turn our focus again to the color code model. Start-

ing with twists condensation, we can observe that all

Pauli twists can condense at color boundaries, since

they map the Lagrangian subgroups of color bound-

aries back onto themselves. Additionally, color ex-

changing twists which leave one color invariant can

condense at and be emitted from the boundary of said

color. As an example, consider Fig. 28 (a), where an

R twist is emitted from a red boundary. Pauli bound-

aries can emit Pauli exchanging twists which leave the

relevant label invariant, as well as all purely color per-

muting twists and twists which are the fusion product

of these examples. See Fig. 28 (b) for an example of

an X twist emitted from an x Pauli boundary. Let

us now consider the diﬀerent color code corners, dis-

cussed in Sec. 4.3 and depicted in Fig. 6. Corners

between two color boundaries can be interpreted as

the twist which exchanges them. For instance, a cor-

ner between a blue and a red color boundary can be

detached and pulled into the bulk as a G twist, see

Fig. 28 (c). Alternatively, the cyclic color twist RB

can be pulled into the bulk as a detached blue-red cor-

ner, see Fig. 28 (d). The same holds for Pauli/Pauli

corners, consider Fig. 28 (e) showing a Y twist as a

detached x-z corner. The last type of corner where

color and Pauli boundaries meet, corresponds to D

type twists. Fig. 28 (f) shows a detached color/Pauli

corner as a D twist in the bulk.

Twist condensation and detaching corner twists

Figure 28: Lattice representations of twists condensing at

boundaries and detached corner twists. (a) an R twist emit-

ted from a red color boundary. (b) an X twist emitted from

x Pauli boundary. (c) a G twist as a detached blue-red

color/color corner. (d) a BR twist as a detached blue-

red color/color corner. (e) a Y twist as a detached x-z

Pauli/Pauli corner. Notice, on the boundary plaquettes the

changed orientation of the stripe pattern indicates a change

in basis. (f) a D twist as a detached red-x color/Pauli corner.

can be used in fault-tolerant quantum computation.

Topological degrees of freedom associated to twist de-

fects, in which twist encoded logical qubits can be

stored, become local degrees of freedom when the

twist is condensed at a boundary. Measuring these

local degrees of freedom then corresponds to measure-

ments on logical qubits. In the reverse process, where

twists are emitted from corners or boundaries, the

logical degrees of freedom increase. This is similar to

considering non-abelian anyons and a boundary where

such anyon excitations condense. This is similar to

how merging and splitting codes using lattice surgery

changes the number of logical qubits [42]. This is

no coincidence, as discussed in Ref. [23], identifying

corners with twists uniﬁes encoding schemes relying

on boundaries with twist encoding schemes. This of-

fers a new understanding of lattice surgery used in

recent proposals for eﬃcient fault-tolerant quantum

computation schemes [89, 90]. Lastly, identifying cor-

ners as twists can help in the search for hardware ef-

ﬁcient quantum error-correcting codes. For instance,

Accepted in Quantum 2018-10-09, click title to verify 20

the code presented in Ref. [22] may be understood as

a surface code where one corner twist is detached and

pulled into the center of the lattice. This lowers the

overhead of physical qubits needed to encode logical

qubits at a given code distance. Even more eﬃcient

codes based the same insight are presented in App. D

and in the following Sec. 8, where G twists in the cen-

ter of the code can be interpreted as detached corners.

8 Stellated color codes

We next present a new family of two-dimensional sta-

bilizer codes that make use of the twist defects of

the color code phase. The proposed codes reduce

the number of physical qubits required per code qubit

while preserving desirable properties such as code dis-

tance and geometric locality of stabilizer generators in

a 2D euclidean lattice.

For topological stabilizer codes on ﬂat manifolds,

the number of logical qubits k one can encode using n

physical qubits with code distance d is upper bounded

by the holographic bound [63] for codes deﬁned by

geometrically local commuting constraints embedded

in 2D Euclidean space

k ≤ c

. (21)

Here, c is a constant which depends only on the lo-

cality of the constraints deﬁning the error-correcting

code. One can compare these c-values of diﬀerent

codes, where a higher c roughly corresponds to a

more eﬃcient encoding scheme. For the surface code

with three-body boundary terms, as proposed by Ki-

taev [2], one obtains c =

. This value can be dou-

bled to c = 1 by rotating the code and using two-

body boundary terms, resulting in the rotated surface

code [20, 64, 91]. The same c-value can be achieved

by appropriately arranging punctures [65]. In surface

codes, a further increase of the c value is possible us-

ing twist defects. The triangular surface code [22]

and certain conﬁgurations of twists and holes in hy-

brid encoding schemes [23] achieve c =

. The same

value, c =

, is found for the triangular 6.6.6 color

code. Using a 4.8.8 lattice, the triangular color code

with color boundaries reaches c = 2 [56]. As far as we

are aware of, this is the highest c-value of any known

ﬂat two-dimensional topological error correction code

with bounded stabilizer weight w ≤ 8. Hyperbolic

codes [92–94] exceed the bound given in Eq. (21), but

are expected to come along with signiﬁcant experi-

mental challenges, as they can not be embedded in

two spatial dimensions while maintaining the locality

of stabilizers.

Inspired by the triangular surface code [22], we

start by presenting an analogous color code model

that achieves c =

. It can be obtained by fatten-

ing [95] the vertices of the triangular surface code to

quadratic faces. Alternatively, this process can be

Figure 29: Three members of the family of stellated 4.8.8

color codes. The three shown examples have a code distance

d = 9. Left: the s = 3 case is shown. This code encodes

two logical qubits. The middle and the right show the cases

s = 5 and s = 7, encoding four and six logical qubits, re-

spectively. The logical operators apply X, Y and Z on qubits

supported by strings connecting diﬀerent boundaries. Some

logical operators are shown in the middle code as examples.

understood in terms of code concatenation with the

J4, 2, 2K-code, as discussed in Ref. [68]. It leads to the

left most code shown in Fig. 29. In the ﬁgure, a single

qubit lies on each vertex of the lattice and each pla-

quette p supports two stabilizers, s

j∈∂p

and

j∈∂p

, where ∂p denotes the set of qubits

lying on the vertices adjacent to p. This is the same

as the standard color code stabilizers, see Eq. (9). We

leave a puncture with a green boundary at the center

of the model where no stabilizer is enforced.

The properties of the model we have described are

understood naturally from the perspective of non-

abelian twist defects. We ﬁrst point out that the

color symmetry is broken when we undergo a full mon-

odromy of the central puncture. As such, we ﬁnd a

single G-twist is condensed in the central puncture,

connected to the boundary with a domain wall. Fur-

ther, there are exactly three corner twists of type G

lying at the interfaces where red and blue boundaries

meet. The ground-state degeneracy then is equiva-

lent to the fusion space of four G-twists that are con-

strained to fuse to the vacuum. The central puncture

encodes a qubit, which we treat as a gauge qubit.

The described code is a representative of a whole

code family we call stellated color codes. Codes in this

family are parameterized by s, the order of rotational

symmetry of the code. The representatives with s =

3, 5 and 7 of the stellated 4.8.8 color code family are

shown in Fig. 29. Again, each code supports s+1 color

permuting twists for odd s. Note that in principle s

can be chosen to be even, in which case the trivial

twist, i.e. no twist, is featured in the center. The case

where s = 4, for example, is a rectangular 4.8.8 color

code featuring alternating red and blue boundaries.

For odd s, a stellated code encodes k = s − 1 logical

qubits and reaches a c-value of c = 4 −

. Stellated

codes with even s encode k = s −2 logical qubits and

we ﬁnd c = 4 −

. In both cases, c approaches 4 as

s grows. Allowing s to diverge doubles the highest

known values of c for ﬂat topological error-correcting

codes such that c → 4. The advantage is achieved

Accepted in Quantum 2018-10-09, click title to verify 21

Figure 30: Examples of stellated color codes on a 6.6.6 lat-

tice. These are the three smallest codes with s = 5, depicted

with code distances d = 3, 5, 7 from left to right. The shown

codes each encode two logical qubits. The encoding rate for

the 6.6.6 stellated color codes approaches c =

≈ 2.7 for

large s, which, to the best of our knowledge, is the high-

est encoding rate of any ﬂat topological quantum code with

w ≤ 6.

by locally changing the curvature around the central

plaquette which is known to enhance the encoding

rate in hyperbolic codes [93, 94].

Given that the measurement of high-weight sta-

bilizers can be challenging experimentally we also

present the stellated color codes based on the 6.6.6

honeycomb lattice. Their encoding rate is c =

−

(c =

−

) for odd (even) s. See Fig. 30 for three

s = 5 representatives. These codes are, to the best

of our knowledge, the most eﬃcient topological error

correction codes with stabilizer weights w ≤ 6 which

are embeddable in two spacial dimensions.

Interestingly, the discussed increase of s can also be

applied to the twisted surface code model [22]. The

encoding rates for these models halve compared to

the stellated 4.8.8 color codes. However, the stel-

lated surface codes we present may be of interest

as typically surface code based architectures have

higher fault-tolerance thresholds than those based on

the color code model [9]. To substantiate such a

claim, one would likely need to conduct numerical

analyses which goes beyond the scope of the present

work. We give a brief description of stellated surface

codes in App. D. One should also investigate diﬀer-

ent ways of performing logical operations on qubits

encoded in stellated codes. As a ﬁrst approach one

could take the description of these models in terms

of their twists and generalize the schemes presented

in, for instance, Refs. [8, 22, 27, 43, 90], where fault-

tolerant approaches to braid and fuse twist defects

are considered. We leave the problem of optimiz-

ing fault-tolerant protocols for quantum computation

with these models to future work.

9 Unfolding boundaries and twists

It is well established [33, 66–68] that we can locally

map the color code onto two copies of the toric code

model [2]. This is clear because the excitations of the

color code have the same braid statistics and fusion

rules of their analogous excitations in the mapped pic-

ture, and hence these two models are members of a

common quantum phase [96, 97]. So too then must

two copies of the toric code share the Lagrangian sub-

groups and excitation symmetries of the color code,

and thus give rise to analogous boundaries and twists.

In this section we concisely describe these analogous

objects. We study the diﬀerent boundaries of the

model, as well as a generating set of twist defects.

To brieﬂy review, all anyons in the toric code are

abelian and take labels 1, e, m and ε = e ×m, where

e and m are bosons and ε is a fermion. Braiding

two diﬀerent non-trivial toric code anyons results in a

phase, M

e,m

= M

e,ε

= M

m,ε

= −1. The Lagrangian

subgroups of the toric code are thus M

= {1, e} and

= {1, m} [3, 14, 98]. The boundaries that con-

dense the excitations M

) are more colloquially

known as rough (smooth) boundaries. Moreover, the

model has a single non-trivial domain wall which ex-

changes e and m excitation [10, 12, 14], and thus gives

rise to a single twist defect that resembles a Majorana

mode [10, 12, 21].

We specify the excitations of the system of two

copies of the toric code as a

for a = e, m, ε where

the ± symbol indicates which of the two copies the

excitation lies on. We can consistently map the color

code excitations onto the dual-layer toric code model

under the following correspondence

rx ↔ e

−

, rz ↔ m

bx ↔ e

, bz ↔ m

−

(22)

which succinctly describe a generating set for C under

fusion [66]. We can also populate Tab. 1 as shown in

Tab. 6.

r g b

−

Table 6: Color code anyons after unfolding the color code

into two toric code layers. e and m refer to electric and mag-

netic charges, the superscript indicates on which toric code

layer the anyon lives. Certain color code anyons correspond

to combinations of two toric code anyons. The shown corre-

spondence is found if the lattice is contracted on the blue and

red lattices to form the − and + layer toric codes respectively,

following the unfolding procedure laid out in Ref. [33]. Note

that this relabeling may be deﬁned up to an automorphism

ϕ of the anyon model.

We next investigate the boundaries of the color

code. The boundaries described by Lagrangian sub-

groups that correspond to the columns of Tab. 6 have

been studied in Ref. [33]. In particular, the system

boundary where the top copy of the toric code is rough

and the bottom copy is smooth corresponds to a red

boundary in the color code picture. Likewise, the

Accepted in Quantum 2018-10-09, click title to verify 22

Figure 31: Illustrated is the folding of a toric code to obtain

a triangular color code. The top, (b) - (e) row which starts

with a toric code with a regular boundary conﬁguration re-

sults in a triangular color code with color boundaries. The

bottom row (f) - (i) starts with a toric code with a more

unconventional domain wall arrangement and ends with a

triangular color code with Pauli boundaries. The square (a)

represents a patch of toric code with smooth boundaries. In

(b) and (f) we introduce twists at the corners, connected by

domain walls which change the boundary type from smooth

to rough, see (c) and (g). Now, the code is folded in half on

top of itself to get (d) and (h). Depending on the domain

wall conﬁguration, we get a code equivalent to a triangular

color code with color boundaries (e) or Pauli boundaries (i).

boundary where the top (bottom) copy of the toric

code is smooth (rough) corresponds to a blue bound-

ary. The green boundary corresponds to a fold where

× e

−

and m

× m

−

are absorbed. In Fig. 31 (c)

and (d), we show the planar code with two rough

boundaries and two smooth boundaries being folded.

In Ref. [33] it was argued that this model is analogous

to the standard color code with a red, blue and green

boundary [28].

Before describing the Pauli boundaries in this

model, it will be helpful to understand the toric code

model with rough and smooth boundaries as a model

with only smooth boundaries but where twist defects

lie at its corners. Indeed, in Ref. [23] it was sug-

gested that we should regard a rough boundary as

a smooth boundary with a non-trivial domain wall

stretched across its surface. Using this picture we see

that twist defects lie at the corners of the planar code.

In Fig. 31 (b) we show a conﬁguration of domain walls

that gives rise to the conventional planar code with

two rough boundaries and two smooth boundaries.

The color code with three diﬀerent Pauli bound-

aries can also be understood from a folded system. In

contrast, we can regard the color code with three dis-

tinct Pauli boundaries as the folded code model with a

domain wall conﬁguration shown in Fig. 31 (f). With

this domain wall conﬁguration we obtain through

folding boundaries where both the top and bottom

layers are both rough or both smooth, see Fig. 31 (h).

These boundaries correspond to x and z Pauli bound-

aries, respectively. Finally, the y Pauli boundary

correspond to a fold with a non-trivial domain wall

stretched across it. With this we complete the bound-

aries of the unfolded color code, as we have now explic-

Figure 32: A domain wall and twist in two layers of toric

code which corresponding to a D symmetry in the color code.

It consists of a toric code domain wall in the second layer.

All stabilizers are shown in the legend below.

itly realized boundaries that condense the Lagrangian

subgroups of the rows of Tab. 6.

We ﬁnally look at the domain walls and twist de-

fects of the unfolded color code. Again, we consider a

generating set of three domain walls. As discussed in

Sec. 6 it is suﬃcient to ﬁnd a domain wall that trans-

poses Tab. 6 together with two more domain walls

that permute the rows of the table.

The domain wall that exchanges the excitations ac-

cording to a transposition of Tab. 6 is the mapping

−

↔ m

−

. Physically, this domain wall is the well

studied domain wall of the toric code [12, 14] on the

bottom layer of the two-copy system. We show this

domain wall together with its termination twist defect

in Fig. 32.

Next, we discuss the domain wall corresponding to

the G symmetry. In the unfolded color code it corre-

sponds to a layer swap. This means toric code anyons

moved over this domain wall get mapped according to

↔ a

∓

, where a = e, m, ε. See Fig. 33 for a lattice

representation of the layer swapping domain wall and

its terminating twist.

Figure 33: A layer swapping domain wall and twist in two

layers of toric code. This corresponds to the G symmetry in

the color code. The stabilizers along the domain wall have

support on both layers, see the legend below.

Accepted in Quantum 2018-10-09, click title to verify 23

Figure 34: This domain wall and twist in two toric code layers

correspond to the Bsymmetry in the color code. Certain

anyons get cloned to the other layer when passing it.

The B domain wall in the picture of two toric codes

maps as follows, e

−

↔ e

−

and m

↔ m

−

. This

means certain anyons get copied over to the other

layer when passing the domain wall. On the lattice,

it can be realized as shown in Fig. 34. It is ﬁnally

worth remarking that the two domain walls analogous

to the G and B maps that we have just presented can

be realized in the two-copy system by, respectively,

applying the swap gate and CNOT gate transversally

between the two copies of the model on a region of

the lattice. We thus realize a closed domain wall at

the boundary of this region in the fashion of Ref. [45].

Finally, we discuss an equivalence between the

color code model and two copies of the three-fermion

model [69]. This connection is explored in App. B.

The mapped system gives particularly natural way to

view the symmetries, and thus the twist defects of the

color code model.

10 Conclusions and future work

In this work, we have identiﬁed all of the bound-

aries, domain walls, and twist defects of the color

code model, giving rise to a complete and compre-

hensive classiﬁcation. On one hand, this endeavor

has been motivated from the mindset of condensed

matter physics. Speciﬁcally, using an exactly solv-

able toy model, we have been able to introduce a con-

crete example to give more substance to the abstract

mathematical theory of phases. Not only this, but

we have also described the model of focus in terms

of other simpler phases to achieve a better grasp on

the physics of the system. It is the hope that our

study will encourage readers to explore other explicit

phases to help build our understanding of the theory

of phases.

In addition to the exploration of fundamental

condensed-matter physics, a second and equally im-

portant motivation of our work has been to aid the

discovery of new applications of topological phases in

quantum information science. Indeed, we have ap-

plied our analysis to the development of topological

quantum codes. Topological quantum computing re-

mains the most promising road towards realizing a

scalable quantum computer. In order to contribute

to the theory of quantum codes, we have introduced

a new small code, a family of stellated codes with

competitive encoding rates and improved methods of

performing code deformations. These new applica-

tions clearly exhibit the potential to use our catalog to

help ﬁnd new codes and protocols for processing quan-

tum information. However, the examples we have pre-

sented may only be the tip of the iceberg of quantum

computational schemes that are yet to be discovered

using the framework we have built for the color code

model. In particular, the present study invites more

research on how to devise fully fault-tolerant schemes

for quantum computing within such a framework as

well as on implementations.

To extend this research path further, we encour-

age the reader to experiment with the diﬀerent topo-

logical features we have laid out to ﬁnd other new

schemes for encoding and manipulating topologically

protected quantum information. In particular we be-

lieve taking a more systematic approach [65, 99] to

combine the diﬀerent objects we have described may

be a fruitful avenue towards new discoveries. We

hope that the present study stimulates such further

research.

Acknowledgments

The authors thank A. Bauer, H. Bomb´ın, C. Brell,

J. Bridgeman, T. Jochym-O’Connor, A. Kubica,

D. Litinski, H. Poulsen Nautrup, A. Nietner, R.

Raussendorf, N. Tarantino, and T. J. Yoder for help-

ful and encouraging conversations. BJB is also grate-

ful to T. Adair, S. Bartlett, K. Korzekwa, and S.

Roberts for discussions on code deformations. Fur-

ther, we thank Zhenghan Wang for pointing out the

connection between the color code and two copies of

the three-fermion model. MSK is supported by the

DFG (CRC183, project B02). FP was supported by

the Alexander von Humboldt Foundation. JE is sup-

ported by DFG (CRC 183, project B02, EI 519/14-

1, and EI 519/7-1), the ERC (TAQ), the Templeton

Foundation, and the BMBF (Q.com). BJB is sup-

ported by the University of Sydney Fellowship Pro-

gramme, the Australian Research Council via the

Centre of Excellence in Engineered Quantum Sys-

tems(EQuS) project number CE170100009, and the

Villum Foundation.

Accepted in Quantum 2018-10-09, click title to verify 24

A Fermions in the color code

In the main text, we have discussed the existence of

six fermionic anyons in the color code. As mentioned,

fermions are the result of fusion of bosonic anyons

which do not share either the color nor the Pauli label.

For example, we obtain a fermion which we call f

by fusing rz and bx, i.e. f

= rz × bx. Since we

can let rz decay using the rule rz = gz × bz, we can

also identify a triple of bosonic anyons with the same

fermion, f

= gz × bz × bx. And using bz × bx = by,

we obtain an alternative way of writing f

in terms of

two bosons, f

= gz × by. In fact, one can also write

in a third way as the combination of two bosonic

fermions, namely f

= ry × gx. Once again, we can

use the 3×3 grid of color code bosons to represent

fermions, see Tab. 1. Speciﬁcally, in Tab. 7 we place

a spot in each of the cells of the grid that represent

the two bosons that constitute the fermion.

•

Table 7: The fermion f

written in three diﬀerent ways as

the fusion of two bosonic anyons.

We can write all six color code fermions in three

ways as the fusion product of two bosons,

= rz × bx = gz × by = ry × gx,

= gy × bz = rz × gx = ry × bx,

= gy × bx = rx × gz = gy × bz,

= rx × gy = gz × bx = rz × by,

= rz × gy = gx × bz = rx × by,

= rx × bz = ry × gz = gx × by.

(23)

And again draw them in the 3×3 grid, see Tab. 8.

Here, diﬀerent symbols where used for the diﬀerent

ways of composing the fermions.

F •

F N

• N

N F

F •

N •

F •

N F

• N

• F

F N

• F

• N

• F

N F

N •

Table 8: The six fermionic anyons in the color code. Each

one can be created by fusion of two bosonic anyons. Each

table is the 3×3 grid introduced in Tab. 1. The diﬀerent

shapes distinguish the diﬀerent ways in which a fermion can

be created by fusing two bosons, as per Eq. (23).

Note, for fermions with even index, the bullets lie

on the diagonal, for fermions with odd index, they lie

on the anti-diagonal. Indeed, the distinction between

fermions with even and odd index goes deeper. Braid-

ing odd (even) fermions with even (odd) fermions

is trivial, whereas braiding them with distinct odd

(even) fermions results in a phase −1,

= (−1)

δ(i6=j)

(−1)

i−j

. (24)

Furthermore, fusion of two diﬀerent odd (even)

fermions results in the third odd (even) fermion.

Whereas fusing an odd fermion with an even fermion

results in a bosonic anyon.

Fermions can not condense at boundaries, but there

are twists which localize fermionic anyons. For exam-

ple, as mentioned in Sec. 7.3 and shown in Fig. 21,

the D twist can localize the f

anyon, and apart from

the trivial charge 1 only the f

anyon. The other ﬁve

fermions also have a corresponding twist which only

localizes them. These are the twists in the conjugacy

class G in Tab. 4.

As discussed in Sec. 9, the color code is equivalent

to two copies of the toric code model. In the toric

code, there exists a single fermionic anyon which is

the product of fusing an electric ﬂux e with a mag-

netic ﬂux m, i.e. ε = e × m. Two copies of the

toric code feature six fermions, as the fusion of a

fermion on one layer and a boson on the other layer

produces a fermion, see Eq. (6). Written as two

toric codes, the fermions of the color code are hence

−

, ε

−

, ε

−

, ε

−

, and ε

−

In App. B, we discuss the equivalence of two copies

of the three-fermion model and the color code. This

equivalence also makes the diﬀerence between what

we here called even and odd fermions very apparent,

as they correspond to the fermions on diﬀerent layers

of the three-fermion model.

B The three-fermion model and the

color code

In the main text in Sec. 9, we have explored the equiv-

alence between the color code (CC) features and fea-

tures in two copies of the toric code (TC). In this ap-

pendix, we address a similar equivalence. Namely the

equivalence between two copies of the three-fermion

model (3F) and the color code [69].

The three-fermion model is discussed in Ref. [70]

where it is called the (D

, 1) modular tensor category.

It can physically be obtained as low energy excita-

tions of a topological subsystem code, as discussed

in Ref [100]. In the three-fermion model, there are

four anyonic charges. Apart from the trivial charge

1, there are three fermionic particles, f

and f

Fermionic here means θ

= −1. The fusion rules

are the same as in the toric code, all particles are

Accepted in Quantum 2018-10-09, click title to verify 25

their own antiparticles and fusion of two distinct non-

trivial anyons results in the third non-trivial anyon,

× f

= f

for i 6= j 6= k 6= i. Braiding two diﬀer-

ent(identical) fermions results in a phase of −1(+1)

such that M

= (−1)

δ(i6=j)

. Note that braiding and

fusion in the three-fermion model is exactly the same

as for the toric code, only the spin of two particles is

changed.

Let us now turn towards the features of 3F, its

boundaries and symmetries. The three-fermion model

does not have any boundaries. Due to the absence

of any non-trivial bosons, the only Lagrangian sub-

group is the trivial group. On the other hand, it has

a richer symmetry compared to the toric code. All

permutations between the fermions are valid symme-

tries, thus its symmetry group is S

, the permutation

of a set of three elements. Hence it has |S

| = 6 do-

main walls and twists, including the trivial one. It

has three order-two symmetries which exchange two

fermions while leaving the third invariant, and two

order-three symmetries which cyclically permute the

three-fermions, one or the other way.

Two copies of the three-fermion model is equiva-

lent to the color code model, 3F ⊗ 3F = CC. To con-

vince ourselves this is true, we show that the anyons in

3F⊗3F are the same as the color code anyons. Braid-

ing and fusion of anyons between layers is trivial; on

a single layer, the braiding and fusion statistic is the

same in 3F as in TC. Since we know TC ⊗TC = CC,

we only need to show how the self-exchange statis-

tics in 3F ⊗ 3F is the same as in CC. The spin of

a joint particle c = a

−

× b

is given by the product

of the self-exchanges on each layer, θ

= θa

−

, as

per Eq. (6). In particular, the only way to obtain a

fermionic anyon is to fuse a fermion on one layer with

the trivial particle on the other layer. Hence, we ob-

tain six fermions, the 10 other non-trivial anyons are

bosonic, just as in the color code. One possible way of

writing the color code anyons in terms of 3F anyons

is shown in Tab. 9. Conﬁrming the known CC fusion

rules and braid statistics is left as an exercise to the

reader.

r g b

−

Table 9: An example of writing the 3×3 grid of color code

anyons (Tab. 1) in terms of particles from two copies of the

three-fermion model 3F. The superscript indicates the layer,

the subscript the label of the fermion within a 3F layer.

With the convention used in Tab. 9 is such that the

fermion labels are related to those in App. A such that

= f

2j−1

and f

−

= f

. It is also interesting that

the color label of a boson can be determined uniquely

as the sum modulo 3 of the indices of the two parti-

cles from the three fermion layers. Equivalently, the

Pauli label corresponds to the diﬀerence modulo three

between said indices.

Now, let us study the boundaries of the color code

when read as two copies of the three-fermion model.

As mentioned, the 3F model does not have a non-

trivial boundary, as it has not a single non-trivial

bosonic particle. Instead, all six color code bound-

aries correspond to foldings between the two layers

of the 3F model, potentially with additional domain

walls on one layer. The x Pauli boundary is just the

direct fold, where f

−

turns into f

and vice versa.

This means particles of the form f

−

can condense

at said boundary. The y Pauli boundary consists of a

fold with a domain wall on the − layer which permutes

the fermions cyclically, f

−

7→ f

−

i⊕1

, where ⊕ is addi-

tion modulo 3. The z Pauli boundary features the

inverse domain wall, f

−

7→ f

−

i1

. The color bound-

aries consists of folds together with domain walls of

order two. For the red boundary it is the domain

wall which exchanges fermions 2 and 3 in the − layer,

−

↔ f

−

. For the green and blue boundary the do-

main walls have the eﬀect f

−

↔ f

−

and f

−

↔ f

−

respectively.

The symmetries of the color code can be con-

structed using the 3F symmetries. Each 3F layer has

the symmetry group S

, this combines to S

× S

for two layers. Additionally, we can introduce a

layer swap symmetry, which acts as f

−

↔ f

the anyons. This

action on the group leads to

×S

) n

= S

, which as we know from the

main text is the color code symmetry group. Again,

this group can be generated by three elements, the

layer swap with the generators on one of the 3F lay-

ers for example.

Through the folding of the three-fermion model, we

can interpret the corners between color code bound-

aries as color code twist defects. This is a practical

example of the theory given in Ref. [74]. Introducing

a domain wall in the folded 3F model along the crease

line changes the boundary type. Let us start with the

color code x boundary which corresponds to the triv-

ial fold between the two 3F layers. Adding a domain

wall which exchanges the second and third fermion,

i.e. f

↔ f

, changes the x Pauli boundary to a red

color boundary. This corresponds to a D domain wall

in the color code, which as discussed in the main text,

changes an x Pauli boundary to a red color boundary.

C Pachner moves and color code

twists

The goal of this appendix is to show a systematic

way of deforming the color code lattice to create and

move color exchanging twist defects and the domain

walls they terminate. We will ﬁnd that this can be

achieved using a series of simple Pachner moves on

Accepted in Quantum 2018-10-09, click title to verify 26

Figure 35: The Pachner moves needed to create and move

color twists are shown on the dual lattice (left) and the primal

color code lattice (right). (a) is a 2-2 Pachner move. Such

a move changes the three-colorability of the lattice as shown

on the right. After the move is applied, two plaquettes with

the same color touch. (b) is a combination of a 1-3 and a 2-2

Pachner move. On the primal lattice, this move introduces

(bi) or removes (bii) a square, depending on its direction. In

either case, the three-colorability of the lattice is violated,

either because a fourth color is introduced, or because two

plaquettes of the same color share an edge.

the dual of the color code lattice. This construction

also leads to a graphical understanding of some of

the features of color twists we have seen in the main

text. In particular, the fusion between color twists,

the eﬀect color domain walls have on color twists and

condensation of color twists at color boundaries can

be visualized nicely in this framework.

Pachner moves [101, 102] are a set of transforma-

tions that generate the mapping of a triangulation of

a manifold onto any other triangulation of the same

manifold. As the color code lattice is trivalent, its

dual lattice is a triangulation [29, 30]. We ﬁnd that

in general, applying Pachner moves to said triangula-

tion often introduces colored twists to the model as

they typically break the three-colorability of the lat-

tice.

To create twist defects, we distinguish between

three Pachner moves, shown in Fig. 35. Shown in

Fig. 35(a) is a 2-2 Pachner move, transforming two

triangles joint at one edge into two other triangles by

redrawing the diagonal. Fig. 35(bi) is a 1-3 followed

by a 2-2 Pachner move, transforming two triangles

into four by ‘adding a second diagonal’. While this

is not typically among the generators of the Pachner

moves, we ﬁnd this move to be useful to discuss the

introduction of twists to the dual lattice. Fig. 35(bii)

also shows the inverse of (bi). The eﬀect of these

moves on the dual lattice are shown on the left in

Fig. 35, the eﬀect on the color code lattice is shown

on the right.

In the dual color code lattice, the triangular faces

correspond to physical qubits and the vertices to sta-

bilizers. Thus, the (a) Pachner move changes the

number of qubits on which the four involved stabi-

lizers have support by ±1. This will turn the usually

even plaquettes into odd plaquettes, which has the ef-

fect that x-type and z-type stabilizers no longer com-

Figure 36: Only two of the Pachner moves shown in Fig. 35

are possible in the 6.6.6 color code. Changed or added edges

are colored in purple on the dual lattice. The resulting dis-

location lines are shown here in the dual and primal lattice.

Note that the top lattice on the right side is rotated 30

◦

compared to the rest.

mute. To go back to to a commuting Hamiltonian,

one of the stabilizers has to be removed. If, however,

a series of (a) moves are applied along a line, as shown

in Fig. 36 (a) and Fig. 37 (a), the parity of vertices

along the line gets changed twice and is even in the

end. This means the only odd plaquettes lie at the

terminal points and constitute the twists of the cre-

ated domain wall.

The (b) Pacher move only changes two surrounding

plaquettes, where it either adds (bi) or removes (bii)

an edge. Additionally, it adds or removes a square

stabilizer. Again, these moves can be applied along

a line such that vertices along the line change parity

twice. This restores the parity of all plaquettes to be

even, except for the ones at the terminal points. The

single plaquette at each terminal point changes its

Figure 37: The three Pachner moves from Fig. 35 applied

on the 4.8.8 color code. The resulting dislocation lines are

shown in purple. Note, the dashed line in (bii) indicates the

removed edges. The lattice in the middle on the right side is

rotated 45

◦

compared to the rest.

Accepted in Quantum 2018-10-09, click title to verify 27

Figure 38: A conﬁguration of a branching domain wall ter-

minated by three twists is shown on the dual lattice. On the

left an R twist, on the right a B twist and in the top an

RB twist. Alternatively, this picture can be understood as

showing the fusion between an R and a B twist to form an

RB twist. Fig. 18 depicts the exact same conﬁguration on

the primal lattice.

support to an odd number of qubits and its stabilizers

need to be changed. See Fig. 36 (bi) and Fig. 37 (bi)

and (bii).

Extending and shortening the line along which the

Pachner moves are applied, allows for twists to be

moved and fused together. Consider as an example

Fig. 38, where three twists are shown, an R twist on

the left, a B twist on the right and an RB twist at

the top. This conﬁguration can be obtained by fus-

ing an R and a B twist together. Notice how at the

point where the three domain walls meet, not a single

stabilizer has odd weight. Hence, there is no non-

trivial twist located there. The same conﬁguration

was shown in the main text in Fig. 18.

The way color twists change when moved over color

domain walls can also be visualized using Pachner

moves. In Fig. 39, two crossing domain walls are de-

picted. We can understand the conﬁguration as a pair

of G twists located at the bottom, one of which was

moved over an R domain wall in the middle which

Figure 39: A G twist (bottom), when moved over an R

domain wall (middle) changes into a B twist (top). This

process can be understood by starting with an R domain

wall and then applying Pachner moves which move the G

twist over said R domain wall. Since the lattice along the

R domain wall hosts yellow square stabilizers, the series of

Pachner moves changes such, that after crossing the R do-

main wall, it is terminated by a B twist on a blue vertex.

Fig. 24 depicts the same conﬁguration of twists in the primal

lattice.

Figure 40: Adding a color domain wall changes the color

boundaries. (a) shows a common red color boundary in the

dual lattice. The dashed lines going upwards indicate the

deleted red vertex (plaquette in the primal lattice) that was

removed to create the red color boundary. In (b), a G domain

wall was added to the boundary by applying (bi) Pachner

moves. In (c), the in (b) introduces yellow plaquettes has

been changed to red. And ﬁnally, in (d) the lattice was a bit

deformed to make it apparent that the obtained boundary is

a blue boundary. We summarize, a G domain wall changes

a red to a blue color boundary.

transformed in to a B twist at the top. In doing so,

the series of Pachner moves changed from connect-

ing green vertices to connecting blue vertices as soon

as the twist was moved over the domain wall. This

shows, how a G twist changes to a B twist if moved

over an R domain wall. See Fig. 24 for the same con-

ﬁguration in the primal color code lattice.

Next, let study how color domain walls change the

color boundaries of the color code. Consider Fig. 40,

where we show how a red color boundary turns into a

blue color boundary when a G domain wall is added.

The red color boundary corresponds to a single red

vertex (plaquette in the primal lattice), whose stabi-

lizer generators were removed from the code Hamil-

tonian. This is indicated in the ﬁgure by the dashed

lines extending to a red vertex far outside the top of

the ﬁgure. Adding a G domain wall at the boundary

by applying (bi) Pachner moves, adds new yellow ver-

tices. This transforms the boundary to a blue color

boundary, as is made clear in the ﬁgure by ﬁrst re-

coloring the vertices and then slightly deforming the

lattice.

Figure 41: An R defect line condensed at a red color bound-

ary as drawn on the dual lattice. The color boundary on the

top corresponds to a removed red plaquette, indicated by the

dashed edges. An R domain wall introduced by (bi) Pachner

moves applied along a line can terminate at the red boundary.

Accepted in Quantum 2018-10-09, click title to verify 28

Lastly, let us turn to the condensation of a color

twist at a color boundary. Fig. 41 depicts an R do-

main wall which extends over a red color boundary.

As discussed, the red boundary corresponds to a re-

moved red vertex, as indicated by the dashed edges.

An R domain wall created by (bi) Pachner moves

can be extended such that it terminates at the re-

moved red vertex. This corresponds to the condensa-

tion of the terminating R twist at the red boundary.

Fig. 28 (a) depicts the same conﬁguration of twists in

the primal lattice.

We ﬁnd the connection between color permuting

twists and Pachner moves is illuminating as it visu-

alizes the algebraic structure of the color permuting

subgroup of the color code symmetry. It also im-

plicitly oﬀers a protocol to braid and condense color

twists through code deformation techniques. Addi-

tionally, the Pachner move construction likely gen-

eralizes to show extensive twists [103] in higher di-

mensional color codes [29–31, 95, 104], see also the

supplemental material of Ref. [105].

D Stellated surface codes

As mentioned in Sec. 8 where stellated color codes

were introduced, there also exist similar codes for

the toric code model. We call them stellated sur-

face codes. The triangular surface code introduced

in Ref. [22] can be interpreted as the s = 3 member

of this code family, where s denotes the rotational

symmetry of the code. Stellated surface codes with

odd symmetry s host a toric code twist in their cen-

ter, which is connected to the boundary via a domain

wall, shown in purple in Fig. 42. The c value increases

with increasing s. We ﬁnd c = 2 −

for odd s and

c = 2 −

for even s, both approaching 2 for large

values of s.

Figure 42: The three ﬁrst stellated surface codes with odd

s, shown with a code distance d = 7. Each plaquette hosts

one stabilizer which acts in the z-(x-)basis on the qubit if

the plaquette is colored light (dark) gray where it meets the

qubit. Along the domain wall, indicated by the dashed purple

line, stabilizers act in both bases on diﬀerent qubits. At the

center of the code a stabilizer acts in the y-basis on the qubit

which it meets with two colors. Left: The s = 3 stellated

surface code is the Yoder-Kim triangular surface code [22].

The s = 5 and the s = 7 stellated toric codes are shown in

the middle and on the right, respectively.

The stellated surface codes with the three smallest

odd s are shown in Fig. 42. We use the convention

of placing physical qubits at vertices and associating

each plaquette with one stabilizer [64]. Light gray

plaquettes correspond to a z stabilizer, which checks

the parity of qubits adjacent to the plaquette in the

z-basis. Similarly, dark gray plaquettes correspond

to x stabilizers. Plaquettes along the purple dashed

domain wall are colored in both shades, dark and light

gray. They also host a single stabilizer each which acts

in diﬀerent bases; qubits which lie on their light gray

side are measured in the z-basis, the ones on the dark

gray side in the x-basis, and the single qubit where

both shades of gray meet is measured in the y-basis.

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