Some Remarks on Computation in a

Topological Quantum Field Theory

Raeez Lorgat

May 2, 2017

Abstract

Disclaimer: this is a work in progress. We do not guarantee its coher-

ence, correctness or usefulness. We investigate the computational com-

plexity of the particle excitations of topological phases of matter in the

restricted model of a Topological Quantum Field Theory in the sense of

Turaev. The ideas presented are a mixture of new and old, with most avail-

able in one form or another in the mathematics and physics literature. Our

main questions are 1. how does the computational power of these excitations

change as a function of the genus of a ﬁxed 2-dimensional space-time? and 2.

independent of any particular space-time, what structural properties of a TQFT

govern its computational power?

When restricted to a space-time with space-like degrees of freedom rep-

resented by a smooth surface of genus g, we answer the ﬁrst question by

observing a q

g

-fold degeneracy in the ground state of the TQFT resulting

from the presence of abelian anyons with exchange statistics a q-th root of

unity. Such a resource is a topologically fault-tolerant quantum memory.

The abelian character of the emergent particle statistics leads us to answer

the second question by via an algebraic realization of non-abelian anyons

as unitary modular tensor categories.

1 Computation in a Topological Quantum Field The-

ory

1.1 Computing with Anyons

A characteristic property of quantum particles is the indistinguishability of

identical particles. All photons in the world share the same dynamical be-

haviour: for any system composed of multiple identical photons a permutation

of the positions of any two photons cannot have any effect on the dynamics of

the system as a whole. It follows that S

n

the symmetric group on n elements

acts on an ensemble of n identical particles as a symmetry of the composite

system; physically, we observe the system’s wave function remain unchanged

1

save for an anomalous “total phase”. In this way Unitarity of our physical the-

ories shows the resulting transform of the wave function is an element of the

group U(1) i.e. transforms by scalar multiplication by a complex phase e

iθ

.

It is well-known that particle exchanges in three-dimensional space transform

in precisely two ﬂavours, each corresponding to one of two 1-dimensional ir-

reducible representations of S

n

. If the particles are bosons (e.g. photons), then

an exchange of two particles is represented by the identity symmetry: the wave

function is invariant, and the particles obey Bose statistics. If the particles are

fermions (e.g. electrons in a metal), then the action of exchange is represented

on the wave function as a multiplication by -1 i.e. the wave function changes

sign. Correspondingly, the particles obey Fermi statistics.

However, in a space-time restricted to two spatial dimensions we instead

ﬁnd a wealth of exotic particle statistics. The restricted topology of a surface

imposes a topological change to the symmetry in exchanging two particles; the

exchange symmetry group is no longer modeled by the symmetric group S

n

,

but instead by the braid group B

n

(See Appendix A), as depicted in Figure 1.1.

Vertical displacement is in the time direction and the n particles are subjected

to movement exchanging their positions from a

i

to b

i

. The braids are formed

from the individual particle’s world-lines.

Figure 1: Particle world-lines tracing out a braid.

Contrary to S

n

, the inﬁnite group B

n

has inﬁnitely many one-dimensional

irreducible unitary representations, each corresponding to a choice of phase

e

iθ

. We see that for θ = 0 and θ = π we recover bosonic and fermionic par-

ticles, but for every other choice of θ we ﬁnd a new particle. As these parti-

cles can obtain any change of phase upon permutation they have been dubbed

anyons.

Furthermore, in certain topological quantum ﬁeld theories (henceforth TQFTs)

we ﬁnd that the higher dimensional representations of B

n

manifest themselves

in an even more exotic type of particle; in contrast to those anyons that trans-

form according to a one dimensional unitary representation i.e. according to a

phase e

iθ

in the abelian group U(1), these non-abelian particles transform ac-

cording to a representation V

d

of B

n

in some non-abelian unitary group U(d)

where d > 1 is the dimension of the corresponding representation. Fixing V

d

,

a pair of anyons of type V

d

with movement conﬁned to a surface trace out a

2

topologically non-trivial braid formed by the particle’s world-lines. This braid

then corresponds via the representation to an element U(d). These anyonic

particles carry an intrinsic computational resource that is purely topological

in nature. According to the unitary representation of B

n

that they are mod-

eled by, each particle carries internal degrees of freedom, and this state can

be transformed by appropriately braiding the particles by moving them in 2-

dimensional space-time. It is this computational resource of a TQFT that this

report investigates.

Remark (On our treatment of TQFTs). In order to not move too far aﬁeld, we

focus on the strictly 2-dimensional part of a 2 + 1-dimensional TQFT i.e. its

anyonic excitations and corresponding braiding statistics. Furthermore, since a

Unitary Modular Tensor Category (UMTC) in the sense of Tuarev corresponds

uniquely to a TQFT, we will instead work with UMTCs. Appendices B and C

contain an informal discussion of TQFT and UMTCs in general. For a complete

treatment, see [12] and [13].

The rest of this report is divided into two sections: a review of the structure

and classiﬁcation of 2+1-dimensional TQFTs as it relates to the computational

power of the TQFT, followed by a discussion of a fault-tolerant quantum mem-

ory in the topological degeneracy held by abelion anyons braided on a genus

g surface. Here we ﬁnd that changing the topology of the surface does not

change the range of computations possible, but instead adds q

g

degrees of

freedom not accessible via local operations.

2 The Computational Power of a TQFT

2.1 Universality and Density of the Braid Group Representa-

tions

Given that each UMTC C (see Appendix C) models a topological state of matter,

it follows that the computational power of the nonabelian statistics of an anyon

native to a given 2 + 1-dimensional TQFT modeled by C is governed by the al-

gebraic structure of C. Much like a choice of gate set in the standard Quantum

Circuit Model may or may not afford efﬁcient approximation of any quantum

algorithm, the choice of UMTC governs the availability of a non-abelian anyon

with braiding statistics supporting implementation of any quantum algorithm.

More precisely: given an object V in C, the structure of a UMTC affords a ho-

momorphism from the group algebra of the braid group

CB

n

→ End(V

⊗n

)

acting on braid group generators σ

i

(See Appendix A) as

σ

i

7→ Id

⊗i−1

V

⊗σ

V,V

⊗Id

⊗n−i−1

V

3

where σ

V,V

denotes the braiding homomorphism

V ⊗V → V ⊗V.

Furthermore, naturality of the braiding homomorphism implies compatability

of braiding with the Hilbert Space structure on End(V

⊗n

). It follows that the

left action of End(V

⊗n

) on itself induces a unitary representation

B

n

→ U(End(V

⊗n

)).

Thus for a given anyon type modeled by V, choosing m, n ∈ N anyons to

encode one and two qubits respectively yields braid group representations of

B

m

and B

n

in End(V

⊗m

) and End(V

⊗n

). If each of these representations have

their images dense in the special unitary group, then we can apply the Solovay-

Kitaev theorem to approximate any desired unitary to precision in a braid

word of length poly(log(

1

)).

Example 2.1.

Freedman, Larsen and Wang characterize the images of the Jones represen-

tations of the Braid group in [7], deriving as corollary in [8] that the UMTC

C(sl

2

, e

πi5

) consisting of highest weight representations obtained as a subquo-

tient of the representation category of the Quantum Group U

q

sl

2

specialized at

q = e

πi5

is universal. This particular UMTC corresponds to the SU(2)-Chern-

Simons-Witten TQFT at level 3, or equivalently the Fibonacci UMTC F de-

scribed in Appendix B.

Example 2.2. For an example lacking universality, by Jones in [10] the SU(2)-

Chern-Simons-Witten TQFT at level 2 has braid group representations with

image factoring through a ﬁnite group. In this case, the corresponding UMTC

is C(sl

2

, e

πi4

).

2.2 Universality of a Given UMTC

Due to the divergence in computational power of different UMTCs, it is natural

to ask for necessary and sufﬁcient conditions for universality of the computa-

tional in a TQFT arising from a UMTC. As of December 2015, this remains an

open problem. Recent progress by Etingof, Rowell and Wang have led to a

conjectural characterization of UMTCs with braid group representations hav-

ing exclusively ﬁnite image [5]:

Deﬁnition 1. A UMTC C has property F if the associated representations of B

n

on

the centralizer algebras End(V

⊗n

) have ﬁnite image for all objects V and all n ∈ Z.

In any UMTC C we can associate to objects X, Y ∈ Obj(C) and a map f :

X → Y a number called the trace of f denoted tr f. In particular, for the identity

map Id

V

: V → V we can deﬁne the dimension dim(V) = tr Id

V

. For example,

for any V ∈ V

k

, dim(X) ∈ N is the standard vector space dimension. Denot-

ing by dim(C) =

P

i

dim(V

i

)

2

the global quantum dimension of C, then the

conjecture, veriﬁed for all known examples, is

4

Conjecture 2.1. A UMTC C has property F if and only if dim(C) ∈ N.

Returning to our standard example:

Example 2.3. Recall that the Fibonacci UMTC F was shown to be universal in

[7]. Here we have two anyons 1 and τ, with dimensions

dim(1) = 1 and dim(τ) =

1 +

√

5

2

.

It follows that

dim(F) =

5 +

√

5

2

,

in accord with F being universal and hence not having property F.

2.3 Classiﬁcation of UMTCs of Low Rank

Recall the rank of a UMTC C is the number of simple objects in C. In [4] Wang

conjectured that UMTCs could be classiﬁed not only as representation cate-

gories of Quantum Groups, but also directly according to their rank. This

conjecture was recently proven in [1] and is now called the Rank Finiteness

Theorem:

Theorem 2.2. There are ﬁnitely many isomorphism classes of UMTCs of ﬁxed rank r.

The theorem is directly analogous to a classical result of Landau’s: for a

ﬁxed number n ∈ N, there are only ﬁnitely many ﬁnite groups G with exactly

n irreducible complex representations. The proof follows from analyzing the

class equation

|G| =

n

X

i=1

[G : C(g

i

)],

dividing by |G| to yield the diophantine equation

1 =

n

X

i=1

1

x

i

which has ﬁnitely many solutions in the positive integers thus implying a

bound on |G|; it follows there are only ﬁnitely many such G with n irreducible

complex representations. Similarly, for a rank n UMTC C, the proof of the rank

ﬁniteness theorem in [1] analyses the equation

dim(C) =

n

X

i=1

dim(V

i

)

2

in order to produce a bound on dim(C) implying a ﬁnite possible set of fusion

rules for C; a further analysis then reveals ﬁnitely many UMTCs having a ﬁxed

5

fusion rule, implying the result.

The Rank Finiteness Theorem suggests the feasability of a classiﬁcation of

UMTCs by rank. The process of classiﬁcation can be understood from the ax-

iomatic speciﬁcation of a UMTC: each axiom imposes a polynomial constraint

with Z-coefﬁcients, equating the classiﬁcation of UMTCs with counting points

on certain algebraic varieties (solution sets to polynomial equations). As of

December 2015, UMTCs of rank 4 and lower have been classiﬁed via the galois

groups associated to them, while in rank 5 all that is known is a list of pos-

sible fusion rules. The difﬁculty of the problem in rank 5 and greater can be

described in terms of the higher dimensionality of the algebraic and arithmetic

varieties involved [4].

According to the classiﬁcation in [4], there are 70 UMTCs of rank less than

or equal to 4, 10 of which are prime. The rest can be obtained from these 10 by

applications of (categorical) direct sum and symmetry transformations. Table

1 below describes these, noting that 9/10 of the prime UMTCs are known to

arise from a Quantum Group; here we label them according to the construction

in [11]. We also explicitly name two UMTCs discussed in this report: Fibonacci

and Toric Code. Column 3, row 4 is the only UMTC not known to arise as a

coset construction from the category of representations of a Quantum Group;

see [4] for details.

[5] Spin(16) Toric Code A

[1] Trivial A [2] SU(2)

1

A [2] SU(3)

1

A [4] SU(4)

1

A

[8] E8

2

NA [4] (G

2

)

1

×SU(2)

1

[U]

[2] (G

2

)

1

, also called FibonacciF NA [U] [2] (A

1

, 5)

1

2

NA [U] [2](G

2

)

2

NA [U]

[3](G

2

)

1

×(G

2

)

1

NA [U]

Table 1: The ith column classiﬁes rank i UMTCs. In each box we record: [n]

the number n of distinct theories in the isomorphism class; the lie group G

and level k specifying the TQFT G

k

by which the UMTC arises; whether the

anyons are abelian (A) or non-abelian (NA); and [U] the presence of a universal

braiding anyon.

An example of a topological phase of matter is seen fractional quantum

Hall liquids. These are electron systems conﬁned to a disk and subjected to

a perpendicular magnetic ﬁeld at extremely low temperatures. Electrons in

the disk, pictured classically as orbiting concentric annuli around the origin,

organize themselves into a topological order. In this way the classiﬁcation of

UMTCs is akin to obtaining a periodic table of elements for the topological

phases of matter.

6

3 Toric Code and Abelian Anyons

We now restrict our attention to abelion anyonic particle dynamics on a surface

of genus g. To do this we ﬁrst look at a model of qubits on a surface with genus

g = 1 i.e. a torus. We then generalize this case to arbitrary genus.

3.1 Toric Code

The toric code, introduced by Kitaev, is an example of a symplectic or stabi-

lizer code. Stabilizer codes are quantum analogues of classical linear codes.

In general such a code works by selecting a set of “check operators”, analo-

gous to checksums, and encoding qubits in states belonging to the stabilizer

space for each of the check operators. This allows for correction of a partic-

ular set of errors by ﬁrst measuring a “syndrome” to determine the type of

error—essentially checking to see if the state is still in the stabilizer space. By

measuring the syndrome, and not the information in the bits directly, we can

reconstruct the error and reverse it without damaging our state.

Consider an r ×r lattice embedded on a torus. On each edge of the lattice

we place a qubit (or a spin-

1

2

degree of freedom). We then have two types of

check operators. The ﬁrst type we will call “star operators”. These are opera-

tors

A

x

s

=

Y

j∈star(s)

σ

x

j

where star(s) for some vertex s is the set of edges, or equivalently qubits, ad-

jacent to that vertex. The second type we will call “face operators”. These are

deﬁned by

A

z

u

=

Y

j∈boundary(u)

σ

z

j

where boundary(u) is the set of qubits around a given face u. For the purposes

of a stabilizer code, we now want to ﬁnd what the stabilizer space of these

operators looks like. We have 2r

2

qubits, and 2r

2

check operators. However,

we have two relations among the check operators i.e.

Y

s

A

x

s

= I =

Y

u

A

z

u

.

We conclude that the stabilizer space has dimension 2

2

= 4.

Furthermore, each operator only acts on 4 qubits, and each qubit is only acted

on by 4 operators. In addition, because the operators all commute (star oper-

ators and face operators commute because a boundary and a star either share

0 or 2 qubits) we can apply the syndrome measurement in a constant depth

circuit. We can check that the errors undetectable by this code are only those

errors that encompass an entire cycle i.e. an element of the homology group of

the torus. We can show this either by working with the state space directly or

by understanding the toric code in terms of anyonic particles.

7

3.2 Anyons and The Toric Code

Consider the same system as above—an r × r lattice with a qubit on each edge

embedded on a torus. When we view this as a physical system we are led to

the Hamiltonian

H =

X

s

(I − A

x

s

) +

X

u

(I − A

z

u

).

The null space of this operator is exactly the stabilizer code space of the toric

code. Because the operator is nonnegative, we then see that the stabilizer space

of the code is precisely the space of minimal energy for this Hamiltonian. It

follows that the excited states, having non-minimal energy, then form the or-

thogonal complement.

Now suppose we have the ground state |ξi. We want to consider some

minimal excited state |νi, and to this end we can consider the excited states as

ordered by the number of conditions they violate. For the state |ξiwe have that

for all faces and vertices A

x

s

|ξi = A

z

u

|ξi = |ξi. Due to the relations described

above, the number of violated conditions of a given type (either face or star)

must be even. Let us consider the simplest case of two violated conditions at

vertices s and p. Then we have at these vertices A

x

s

|νi = −|νi and the same

for p. It is standard terminology to say that we have two quasiparticles at s

and p. First, we note that we can generate the state |νi from the state |ξi. We

simply apply a product of σ

x

j

for all j along a path from s to p. We can also

move quasiparticles along a path by applying products of σ

x

j

along that path.

Similarly, we can talk about quasiparticles on faces by considering violated

conditions at faces. Again these can be created by products of σ

z

j

for paths

along the “dual lattice” i.e. the lattice given by thinking of faces as vertices,

and connecting those vertices that correspond to adjacent faces.

We now consider what happens when these two types of particles interact.

Suppose we have two vertex quasiparticles at s and p, and two face quasiparti-

cles at u and v. We consider the action of moving one of our face quasiparticles

around a closed loop containing the vertex p. Moving this quasparticle around

a closed loop is an application of star operators at every vertex inside the loop.

All of these act as the identity except for the star operator at vertex p which

ﬂips the sign of our state. Moving one particle around another results in a

phase change, even though the particles don’t directly interact!

Finally, because we are working on a torus and not a plane, there is another

action we can consider. Consider 4 operators, Z

1

, Z

2

, X

1

, X

2

, where the Z

1

, Z

2

operators are products of σ

z

around the two different nontrivial loops of the

torus, and X

1

, X

2

are products of σ

x

along nontrivial loops. The commutation

relations between these operators are

X

1

X

2

= X

2

X

1

Z

1

Z

2

= Z

2

Z

1

X

1

Z

1

= Z

1

X

1

X

2

Z

2

= Z

2

X

2

X

2

Z

1

= −Z

1

X

2

X

1

Z

2

= −Z

2

X

1

.

8

The operators X

1

, Z

2

and Z

2

, X

1

can each be thought of as acting as σ

x

and

σ

z

operators on the two different qubits embedded by the toric code—in other

words they act on the 4-dimensional space of ground states of the Hamiltonian.

One way to think about this is to consider creating a pair of vertex and a pair of

face particles. If we move one of the vertex particles around a nontrivial cycle,

and then annihilate it with the other vertex particle, essentially performing X

1

we get a phase change, since we have also moved the particle in a closed path

around a face particle. However, X

1

has to commute with the operators that

create face particles, and so if we ﬁrst perform X

1

and then create a pair of face

particles, we should get an already-ﬂipped state. In other words, the ground

state “remembers” the topological behavior of past anyons.

3.3 Abelian Anyons on a Surface: A Generalization

What we have described above is simply a pair of particles such that when

one is moved around the other the state vector acquires a phase ﬂip of −1. It

follows that these particles can be viewed as anyons with exchange statistics

which are a fourth root of unity. Now consider a more general case: anyons on

a torus with exchange statistics given by some e

iθ

where θ = π

p

q

. These are

also known as rational anyons, since the phase change given by their exchange

is a rational multiple of π. We will restrict our study to rational anyons.

Rather than looking at operators of particles moving around one another,

we instead look at two operators C

1

and C

2

, which are analogous to our X

i

and Z

i

operators above. The operator C

1

is given by the creation of a pair of

anyons, moving one around one nontrivial cycle of the torus, and then fusing

the anyons and annihilating them. The operator C

2

is the analogous opera-

tor for the other non-trivial loop. As above, these will act nontrivially on the

ground state of the torus, and again such action is a topological behavior. We

consider the commutator [C

1

, C

2

] = C

−1

2

C

−1

1

C

2

C

1

. Topologically what we

have done is to move one particle in a loop around another, and this is equiva-

lent to two exchanges. It follows that

[C

1

, C

2

] = e

i2θ

.

Consider some eigenvector |ψi of C

1

with eigenvalue e

iα

(since C

i

’s are uni-

tary their eigenvalues have this form). Then we have the relation

[C

1

, C

2

]|ψi = e

i2θ

|ψi

C

2

e

iα

|ψi = C

1

C

2

e

i2θ

|ψi.

Rearranging terms we can get

C

1

(C

2

|ψi) = e

i(α−2θ)

(C

2

|ψi).

In other words C

2

acts as a shift operator on eigenvalues of C

1

. Furthermore

we see

[C

1

, C

2

]

q

= e

qi2π

p

q

= 1.

9

In summary, moving abelion anyons on a torus yields an additional q-fold

degeneracy that does not exist when quasiparticles with the same exchange

statistics move on a plane.

There is another way we can generalize this picture, and that is passing

to surfaces of higher genus. Suppose then that the spatial degrees of freedom

form a surface of genus g. The above analysis depends only upon the existence

of 2 nontrivial loops on the torus. A genus g surface has 2g non-trivial loops.

We organize these loops into pairs—one pair for each hole in our surface (a

genus g surface can be thought of as a connected sum of g tori, or a surface

with g holes). To each of these loops we associate an operator C

j

i

where the

i = 1, 2 and j ranges from 1 to g. Operators corresponding to loops in different

pairs commute, and operators corresponding to the same loops commute. The

only nontrivial commutator is [C

j

1

, C

j

2

] = e

i2θ

. We conclude then that on a sur-

face of genus g, we have g copies of the q-fold degeneracy from the topology

of the torus. In other words our ground state space has a q

g

-fold degeneracy

above and beyond what is present when abelians are braided on a genus 0 sur-

face.

As the analysis shows, these abelian anyons on a genus g surface may have

a ﬁnite dimensional topological degeneracy, but their braiding does not allow

the implementation of arbitrary unitary transforms. At best we ﬁnd a stable

ground state protected from local errors.

4 Conclusion

This report discussed the computational power of a topological quantum ﬁeld

theory. We reviewed two basic questions: (i) How does the toplogy of space-

time affect the computational power of a TQFT and (ii) independent of a par-

ticular space-time, what intrinsic structural properties of a TQFT govern its

computational power? The observation that abelian anyons on a genus g sur-

face generalizes the toric code answers the ﬁrst question: a non-trivial 2-D

space-time topology provides an exponential (in the genus g) degeneracy in

the ground state of the TQFT, a resource that in principle could serve as a

quantum storage robust to local error, but cannot be used to implement any

transformation not already accessible to anyons with comparable statistics on

a genus g = 0 surface. To answer the second question, we appealed to the

algebraic model of anyonic quantum computation via the Tensor Category for-

malism. By relying on our intuition from classical group theory to guide our

understanding of their quantum generilizations, we have seen how the cor-

respondence between UMTCs and 3-dimensional TQFTs, affords an algebraic

and number theoretic analysis of the computational regime modeled by any

TQFT. Recent deep results in this direction such as the Rank Finiteness The-

orem and classiﬁcation of UMTCs of low rank are the fruits of this approach:

we are able to derive an explicit periodic table of elements for topological states

10

of matter, indexed by the nature of the anyonic excitations provided.

Several open questions present themselves: we still do not have a good charac-

terization of when a UMTC/TQFT has a universal computational model. Fur-

thermore, restricting to those cases where the computational model is not uni-

versal, we do not have a good characterization of what computational power

they do provide: at best we have a conjectural characterization of when ar-

bitrary braiding of anyonic excitations implements at most a ﬁnite subgroup

of the unitary group. Basic obstructions to answering these questions are a

lack of understanding the relationships between the objects involved. For ex-

ample, it’s expected—but not yet known—that there are UMTCs that do not

arise from Quantum Groups; in practice all known UMTCs can be constructed

via some procedure from a Quantum Group. This could be rephrased as a

gap in our understanding of the representation theory of a general Quantum

Group. Given how basic computational questions directly translate into open

problems concerning these relatively young ﬁelds, we expect new advances to

directly contribute towards a deeper understanding of quantum computation

arising from a TQFT in the years to come.

Appendix A The Braid Group

The braid group B

n

on n strands, is a generalization of the symmetric group S

n

on n symbols. It has an intuitive presentation as braids of n strings. Formally

it is presented by n − 1 generators σ

i

which corresponds to crossing the i

th

strand over the i + 1

th

strand. For example, the braid

is given by generators

σ

1

σ

2

σ

−1

1

.

Such a sequence of generators is called a braid word. This set of generators has

a set of relations:

σ

i

σ

j

= σ

j

σ

i

σ

i

σ

i+1

σ

i

= σ

i+1

σ

i

σ

i+1

where |i − j| > 2. Such relations are easy to see if you draw out the correspond-

ing diagrams.

Appendix B: The Fibonacci Theory F

We now introduce one of the simplest non-trivial TQFTs which we denote F,

known as the Fibonacci theory in the literature (See pp. 21 in [6]). The purpose

11

of explicitly describing F here is to provide a concrete example to help moti-

vate the abstract language of UMTCs used in this report.

We specify F by the structure of the anyonic excitations it supports in any 2

dimensional slice of space-time, as well as the resulting exchange statistics. In

this model, there are only two different types of anyons, the vacuum (or ab-

sence of an anyon) denoted 1 and the non-abelian anyon τ. Part of the data

specifying a TQFT is what is known as a fusion rule, which speciﬁes what

happens when two particles are brought together. Fusion can be considered as

identifying the two anyons as a composite particle and identifying the result-

ing statistical behaviour of the ensemble. For example, fusing two fermions

results in a boson. In the case of F:

τ ×τ = 1 + τ,

τ ×1 = τ,

1 ×τ = τ,

1 ×1 = 1.

These rules should be read, for example in the ﬁrst row, as stating that fus-

ing two τ particles results in a superposition of two outcome states: the ﬁrst

state being the annihilation of both particles (the outcome being the vacuum)

and the second state being the formation of another τ particle. The fact that this

fusion space has dimension higher than 1 and so allows for superpositions is

equivalent to τ being a non-abelian anyon, a fact that has only recently been

proved; see [2].

Figure 2: Fusion of multiple τ particles.

The computational power of F is now made apparent in the process of fus-

ing n anyons of type τ. Consider a line of τ particles as depicted in ﬁgure 2

and proceed to fuse these particles in a step-wise fashion. We begin by fus-

ing the ﬁrst two particles, and then continue by fusing the outcome with the

remaining particles incrementally. To each step i we assign an index e

i

that

indicates the outcome of the fusion at that step as being either 1 or τ. The

states |e

1

, e

2

, . . . , e

n−3

i belong to what is called the fusion Hilbert space of the

τ anyons, denoted H

n

. In principle, there are 2

n−3

possible outcomes of e

i

’s,

12

but not all are allowed by the fusion rules. For n = 1, we deal with the impos-

sible case of the vacuum turning into a τ anyon, so

dim(H

1

) = 0.

For n = 2 we see τ as input and output going through a trivial process so

dim(H

2

) = 1.

Next, the possible outcomes are 1 or τ, giving

dim(H

3

) = 1,

but at the very next step we see that there are two possible ways of yielding a

τ from two different processes and so

dim(H

4

) = 2.

Continuing this way we see that

dim(H

5

) = 3,

and so on, yielding the sequence

0, 1, 1, 2, 3, 5, 8, 13, . . .

which grows proportionally to φ

n

where φ =

1+

√

5

2

. Thus the fusion state

space H

n

of internal degrees of freedom of n anyons of type τ grows exponen-

tially with n. In order for this to be a viable computational resource, we ideally

want to be able to implement arbitrary unitary transforms on H

n

. As men-

tioned in the previous section, this is accomplished by braiding two τ anyons

around one another as in Figure 2.3.

Figure 3: Braiding of two τ particles.

The process of doing so is a unitary transform on those basis vectors in the

fusion space spanning possible fusion outcomes c according to the fusion rules

governing a and b in the particular TQFT. Hence, this braiding matrix is part

of what speciﬁes a TQFT. In the case of F again, we ﬁnd that exchanging any

two τ anyons acts as the matrix

R

τ,τ

=

e

4πi

5

0

0 −e

2πi

5

!

.

13

Given an exponential state space, rules for braiding, and rules for fusing, we

can now see that in order to encode a logical qubit as in the standard Quan-

tum Circuit Model, we might employ four τ anyons. There are two distin-

guishable ways these anyons can be fused that can encode the qubit states:

|0i = |τ, τ → 1i and |1i = |τ, τ → τi. The possible logic gates are accomplished

by forming arbitrary braids amongst these four qubits, with behaviour gov-

erned by R

τ,τ

. Fusing and reading the outcome then correspond to a measure-

ment in the QCM.

As our goal is to study the general computational power of a given TQFT, we

ﬁnd it fruitful to abstract the data presented here in the form of an algebraic

model of anyons given by Tensor Categories as described in [12]. As described

in Section 2, not only does the language of tensor categories simplify the de-

scription of a TQFT, but it also affords us algebraic and number-theoretic tools

to analyse the computational power carried by any given TQFT.

Appendix C: TQFTs and Unitary Modular Tensor Cat-

egories

In the example furnished in Appendix B, a model of computation arose from a

TQFT after specifying:

• a set of objects corresponding to the available types of anyons

• rules for fusing these objects pairwise

• rules for braiding these objects pairwise

We now capture this data in an algebraic structure known as a Unitary Mod-

ular Tensor Category. We include the concise deﬁnition for readers familiar

with tensor categories, but for those not, immediately after we turn to recast

the deﬁnition in terms of two familiar objects: i) the category V

k

of ﬁnite di-

mensional vector spaces over the ﬁeld k and ii) the TQFT F.

Deﬁnition 2. A Unitary Modular Tensor Category (abbreviated UMTC) is a semi-

simple ribbon category C satisfying the following properties:

• C has only a ﬁnite number of isomorphism classes of simple objects.

• C is modular i.e. has non-degenerate S-matrix.

We refer the interested reader to [12] for a detailed exposition of the termi-

nology used here. Modularity, though important for the dictionary between

TQFTs and UMTCs, will not play an immediate technical role in any of our

subsequent discussions, so we omit covering it here. Instead, for our purposes

we now quickly describe the structure of two UMTCs: V

k

and F.

14

C is a Semi-Simple Ribbon Category with ﬁnitely many isomor-

phism classes of simple objects

That C is a category means that C can be thought of as a collection of objects

Obj(C). For example, in the category V

k

of ﬁnite dimensional vector spaces

over the ﬁeld k, Obj(V

k

) is the set of isomorphism classes of vector spaces

over k of ﬁnite dimension; i.e. one object for each n ∈ N. In the case of F,

= {1, τ} ⊂ Obj(F), but the full collection of objects includes additional objects

that can be constructed from these two, as soon shall be seen.

That C is a Semi-Simple Ribbon Category means that C has natural duals, nat-

ural tensor products, and a natural braiding structure on Obj(C). In the case

of V

k

, we see for any V, W ∈ Obj(V

k

) these constructions are the familiar dual

vector spaces:

V → V

∗

tensor product

V ⊗W

and braiding map coinciding with commutativity of tensor product

V ⊗W = W ⊗V.

In F the natural duals on Obj(F) are particle-to-antiparticle correlation

1 =

¯

1

and

τ = ¯τ

i.e. both 1 and τ are their own antiparticle (i.e. are self-dual). Tensor product

then corresponds to fusion e.g.

τ ×τ ∈ Obj(F)

is the composite particle with behaviour governed by the collective statistical

behaviour of two τ particles, and ﬁnally a natural braiding map given by

R

τ,τ

: τ ×τ → τ ×τ.

Finally, that C is semi-simple means each tensor product of objects can be de-

composed into a direct sum of a distinguished class of simple objects. Here

simple should be taken in the sense of an irreducible representation, prime

number etc: a simple object is one that cannot be decomposed into a direct

sum of smaller objects. For example, in V

k

, any vector spaces V and W of

dimensions m and n decomposes as

V ⊗W =

mn

M

i=1

k.

15

Analogously, the fusion rules of F specify the decomposition of the fusion of

two τ particles:

τ ×τ = 1 + τ.

The simple objects of V

k

clearly consist exclusively of the one dimensional vec-

tor space k, while the simple objects of F are {1, τ}.

Braid Group Representations and non-triviality of UMTCs

The preceeding section should have conferred the sense that UMTCs are in a

sense categories that behave like the category of vector spaces, while at the

same time UMTCs capture the basic data of anyonic statistics in a TQFT. While

both of these form examples of a UMTC, they can be seen to lie on opposite

ends of a vast spectrum of complexity in a UMTC: the former is in a certain

sense trivial, while the latter is very interesting. To make this precise, consider

the braiding maps in V

k

V ⊗W → W ⊗V

given by the familiar map a ⊗b → b ⊗a, then this map squares to the identity.

It follows that if in our TQFT we think of V and W as anyonic excitations, each

with their worldlines (See Figure 4), then V ⊗ W would model the fused parti-

cle i.e. the composite subsystem consisting of the two particles. The braiding

Figure 4:

map then models the action on the fusion space when we twist one particle

about the other; following this logic through, if we try to represent the braid

group generator σ

V,W

that crosses the two strands corresponding to the world-

lines of V and W (see appendix A), since this braid map squares to the identity,

we have in effect limited ourselves to a highly constrained representation of

the braid group B

2

. It follows that a UMTC such as V

k

does not model any

interesting anyonic exchange statistics. Conversely, F has braid map R

τ,τ

that

yields a highly non-trivial braid group representation (as referenced in Section

2, this representation is computationally universal).

The above discussion illustrates that in order to ﬁnd UMTCs modeling non-

trivial TQFTs, we have to go outside the scope of classical algebra. One way

to realize this principle is via the correspondence between certain TQFT and

suitable categories of representations of a Quantum Group. For a survey of

ways to obtain UMTCs in this way, see [9].

16

Unitarity, Modularity and TQFTs

Together with Unitarity which guarantees a given hermitian structure on End(V)

for any object V ∈ C, the Modularity condition (which we have not treated

here) guarantees that to each UMTC we can uniquely associate a TQFT; the

precise construction of a ﬁeld theory from the abstract algebraic description of

a tensor category is beyond the scope of this report. We instead refer the inter-

ested reader to now-standard reference [12].

References

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http://arxiv.org/pdf/1310.7050.pdf, 2015

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http://arxiv.org/pdf/1508.04793.pdf, 2015

[3] Bruillard, NG, Rowell, and Wang On classiﬁcation of modular categories by

rank http://arxiv.org/pdf/1507.05139.pdf, 2015

[4] Rowell, Stong and Wang, On classiﬁcation of Modular Tensor Categories

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[5] Naidu and Rowell, A ﬁniteness property for braided fusion categories

http://arxiv.org/pdf/0903.4157.pdf, 2009

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[8] Freedman, Larsen and Wang, A Modular Functor Which is Uni-

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[11] Freed, Hopkins, Lurie and Teleman, Topological Quantum Field Theories

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http://arxiv.org/pdf/0905.0731v2.pdf, 2009

17

[13] V. Tuarev, Quantum Invariants of Knots and 3-Manifolds, 2nd. Edition, De

Gruyter Studies in Mathematics 18, April 20

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[15] arXiv:0902.3275

18