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
.
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
. 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
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
i1
V
σ
V,V
Id
ni1
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
jstar(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
jboundary(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
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
(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
|ψi = C
1
C
2
e
i2θ
|ψi.
Rearranging terms we can get
C
1
(C
2
|ψi) = e
i(α)
(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
n3
i belong to what is called the fusion Hilbert space of the
τ anyons, denoted H
n
. In principle, there are 2
n3
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].
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17
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