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Part III Essay on
Topological Quantum Computing
by Philip Zupancic
DAMTP, University of Cambridge
supervised by Prof Richard Josza
X
=
(Hormozi et al., 2007)
Contents
1 Introduction
2 Topological States
2.1 A Phenomenology of Quantum Hall Physics . . . . . . .
2.2 Particle Statistics in 2D . . . . . . . . . . . . . . . . . .
2.3 The Braid Group Bn and its Representations . . . . . .
2.4 The Bigger Picture: Topological Quantum Field Theory
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3 Topological Quantum Computation with Anyons
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3.1 Anyon Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Universality of Quantum Computation via Fibonacci Anyons . . 17
3.3 Fault Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Conclusion
23
References
25
1
Introduction
Calculating the time evolution of an interacting quantum many-body system is
an NP-hard problem (Kitaev et al., 2002). Turning this around, one could say:
this physical system efficiently solves an NP-hard problem — its own evolution.
If we could encode other problems onto such a quantum system, we would be able
to speed-up computations considerably. This idea of quantum computation has
been around for more than thirty years (Feynman, 1982). The complexity class
associated with quantum circuits is BQP. While its relation to QMA and QMA’s
classical analogue NP is an open question, certain quantum algorithms promise
an exponential improvement over the best known classical ones (Nielsen and
Chuang, 2010). A famous example is Shor’s factoring algorithm (Shor, 1994).
The reason we still rely on public-key cryptography and do not design the
structure of room temperature superconductors on a commercial quantum computer is that building such a device is very difficult. For this, we need to be
able to encode information onto a physical qubit, let it interact with others in
a precisely defined manner, store it, and read it. After solving these issues conceptually, there remains the problem of decoherence, i.e. interactions with the
environment that cause a degradation of the qubit state and thus loss of information. As Asher Peres put it, “Quantum phenomena do not occur in Hilbert
space. They occur in a laboratory.”
There exist a number of error correcting codes that can work against the
effects of decoherence (Shor, 1996; Kitaev et al., 2002; Nielsen and Chuang,
2010). Those are effective if decoherence is sufficiently weak, but they require
a certain threshold fidelity of the quantum gates. Estimates of the required
precision range from 10−6 (Knill et al., 1996) to 1/300 (Zalka, 1996), which is
hard to achieve experimentally.
A topological quantum computer follows a different approach. It “does not
try to make [the system] noiseless, but instead make it deaf” (Sarma et al., 2006).
The idea due to Kitaev (2003, submitted 1997) is to make the logical qubit
inaccessible to local noise sources by hiding it in a topological Hilbert space,
encoded as a homotopy invariant. Though presenting additional challenges,
the still theoretical concept of a topological quantum computer relaxes the high
requirements on low noise in physical systems for reliable quantum computation.
Writing a short but non-trivial introduction to the vast topic of topological
quantum computing is a challenging task, since general and specific aspects as
well as the involved mathematical structures fill entire books. My goal is to
write a coherent story, from the emergence of topological phases of matter in
the form of anyons up to an investigation of their applicability for computation.
The reader is expected to be knowledgeable in concepts of quantum computing
on the level of an introductory class. Basic familiarity with quantum field theory
might also be helpful.
The essay is comprised of two major sections. In the first, I will motivate
the concept of anyons, a kind of quasiparticle that obeys neither Bose nor Fermi
statistics, from its emergence in a real physical system in the quantum Hall effect
(2.1), before deriving some of their elementary properties from their symmetry
group (2.2 and 2.3). The first section concludes with a brief outline of topological
quantum field theories and their relation to the mathematics of knots, which
1
provides a bit of context to the behavior of anyons (2.4).
The longer second section develops a particular anyon model that is simple
enough to fit on a few pages and complex enough to be useful for quantum
computing (3.1). How useful exactly is evaluated by considering the possibility
to simulate a conventional quantum circuit with anyons and vice versa (3.2).
Finally, the advantages of topological quantum computers over conventional
ones are analyzed more thoroughly (3.3).
2
2.1
Topological States
A Phenomenology of Quantum Hall Physics
It is an empirical fact that all elementary and composite particles are either
fermions or bosons. These labels distinguish between behaviors of identical
particles under exchange. The wave function describing a system of indistinguishable bosons remains the same when any two particles are swapped while
the wave function of a system of identical fermions acquires a minus sign. Properties and dynamics of such a system are insensitive to global phases, so the
physical observables are not affected by this sign change. The behavior under
an exchange operation — called the statistics — is linked to the spin of a particle by the spin-statistics theorem which states that bosons have integer spins
(0, 1, ...), while fermions have half-integer spins (1/2, 3/2, ...).
Protons and electrons carry a charge e — the elementary charge. Quarks
carry fractional charges of 1/3e or −2/3e, but they are forced by color confinement into composite particles such as protons and neutrons, which have a
total charge of an integer multiple of e. Thus there are no free particles with
charges that are a fraction of e, and the elementary charge really deserves its
name. However, this view was challenged at the end of the twentieth century
by experiments on the quantum Hall effect.
When an electric current I~ = Ix~ex is sent through a two-dimensional conduc~ = B0~ez applied, the Lorentz
tor (x-y-plane) with a transverse magnetic field B
~
~
force F = −e~v × B acts on the conducting electrons (where ~v = v0~ex denotes
their velocity). This leads to a redistribution of charges along the y-axis and
thus to a voltage difference Vy . Simple calculations reveal that Vy ∝ B0 and
thus
Vy
ρxy =
∝ B0
(1)
Ix
where ρxy denotes the Hall resistivity. This is called the (classical) Hall effect
after its discoverer Edwin Hall in 1879.
About a hundred years later, Klaus von Klitzing turned to a two-dimensional
electron gas (2DEG, which can be found in semiconductor devices) at low temperatures and exposed to high magnetic fields. New quantum physics was expected to be found in two-dimensional systems, i.e. systems where particles are
confined to a length on the order of 10nm in one dimension (Fowler et al., 1966).
Von Klitzing discovered a quantization of the Hall resistivity (Englert and von
Klitzing, 1978):
1 h
(2)
ρxy =
ν e2
2
Figure 1: Integer Quantum Hall effect. At low temperatures, the electrons in
a 2DEG can only occupy states in narrow bands, the so-called Landau levels. The
magnetic field strength takes the role of the chemical potential which defines the
threshold up to which energy these bands are filled. When the chemical potential
µ(B0 ) lies within a Landau level, this band is not filled completely and excitations can
occur within this band. These excitations are necessary for longitudinal conductance
and correspond to peaks in ρxx . The transverse conductance, however, is given by the
number of filled bands. The Hall resistivity ρxy rises steeply when µ(B0 ) lies within a
Landau level, while a chemical potential between the ν th and (ν + 1)th Landau levels
gives rise to a plateau at h/(νe2 ) (Gross and Marx, 2008). Figure from von Klitzing
(1986).
where ν can be any integer (fig. 1). The discovery earned von Klitzing the Nobel
Prize in Physics in 1985.
A few years later, Daniel C. Tsui and Horst L. Störmer made a discovery
that brought them a Nobel Prize (1998) and the community of physicists some
riddles: in their experimental setup, ν did not only assume integer, but also
fractional values, which is called the Fractional Quantum Hall Effect (FQHE)
(Tsui et al., 1982; Willett et al., 1987). A measurement of the Hall resistivity
is shown in fig. 2.
The emergence of a fractional ν was explained partly by Robert B. Laughlin,
who shares Tsui’s and Störmer’s Nobel Prize. In a nutshell, he explained that
the FQHE is mediated by quasiparticles (collective excitations with particle-like
character) with fractional charges. Even more surprisingly, these FQHE states
were also shown to carry fractional spin and obey fractional statistics (Arovas
et al., 1984; Laughlin, 1983; Wilczek, 1990).
3
Figure 2: Fractional Quantum Hall Effect. In the FQHE, maxima in the longitudinal resistivity ρxx occur at fractional values of ν with corresponding rises in the
transverse resistivity ρxy . This effect is mediated by collective excitations of fractional
charge (Gross and Marx, 2008). Figure from Willett et al. (1987).
2.2
Particle Statistics in 2D
In quantum mechanics, identical particles are indistinguishable. Thus, the dynamics of a system of identical particles has to be invariant under an exchange
of any number of those particles. This means that the Hamiltonian governing
the evolution of the particles commutes with the swapping operator.
In introductory classes to quantum mechanics, students learn that particles
can be either fermions or bosons. The product wavefunction of two identical
bosons is constant under the swapping operation, while the product wave function of two indistinguishable fermions changes sign:
B/F
Ψ1
B/F
(x1 )Ψ2
B/F
(x2 ) = ±Ψ1
B/F
(x2 )Ψ2
(x1 ).
(3)
(The behavior of a system is insensitive to overall phases, so that the negative
sign appearing after the fermion exchange does not affect the physics.)
A particle exchange can be represented by a (counterclockwise) rotation (in
space time) of both particles around their common center by π:
SO(3) is the Lie Group of proper rotations in 3D and can thus be connected to
4
Figure 3: The Lie group SO(3) is isomorphic to a three-dimensional ball with radius
π, where opposite points on the boundary are identified. A closed loop extending from
one point on the surface to its antipodal point twice (• → ×, × → and identified
with the top ) covers a rotation by 4π, but can be reduced to a point.
π
–π
Figure 4: In SO(2), a closed loop connecting 0 and 4π cannot be reduced to a point.
particle exchange. The fact that the swapping operation σ with
B/F
B/F
B/F
B/F
σ Ψ1 (x1 )Ψ2 (x2 ) = Ψ1 (x2 )Ψ2 (x1 )
B/F
B/F
= ± Ψ1 (x1 )Ψ2 (x2 )
(4)
only has the eigenvalues ±1 is due to the fact that the symmetry group SO(3)
of a three-dimensional vector space has the simply connected covering group
SU (2).
SO(3) is parameterized by three angles, two of them defining a rotation axis
and one the angle to be rotated by. The parameter space of the rotation axis
is the 2-sphere S 2 while the rotation angle φ ∈ [−π, π] defines the radius of
the sphere. Noticing that a rotation by π is the same as a rotation by −π, we
conclude that the manifold underlying SO(3) is isomorphic to a ball of radius
π in R3 with opposite points of the boundary identified (e.g. Manton, 2013).
Now, a rotation by 2π is identified with the identity operator (no rotation
at all), but the two operators are still not equivalent. This is to say that a
curve connecting two antipodal group elements of SO(3) cannot be smoothly
contracted to a point, though they are identified. A rotation by 4π, however,
can be contracted to a point, so that a rotation by 4π is the identity operation
(fig. 3). Thus, exchanging twice leads back to the original configuration,
σ 2 (Ψ1 (x1 )Ψ2 (x2 )) = Ψ1 (x1 )Ψ2 (x2 )
(5)
Eq. 4 follows directly.
In two dimensions, however, things are different. SO(2), the Lie group of
proper rotations in 2D, does not have a simply connected covering group. The
underlying manifold is the interval (−π, π] with the ends of the range identified.
Lacking the “extra dimensions”, no smooth transformation exists that could lift
the discrete jump (fig. 4). Thus, in 2D the major constraint σ 2 = I on exchange
statistics is lifted.
We should clarify what we mean when we talk about particles in two dimensions. Clearly, our world is three-dimensional, and even if we confine the
5
movements of particles to a two-dimensional plane, they will still “live” in three
dimensions. However, excitations in quantum many-body systems are usually
collective modes that are localized enough to be considered particles — so called
quasiparticles. These inherit the dimensionality of their environment, which can
well be two-dimensional, as is the case with the 2D electron gas exhibiting fractional spins in the FQHE, which was mentioned earlier.
What do we know about particle exchange in 2D? Restricting ourselves to
one-dimensional representations of SO(2) for now, we have
σ (Ψ1 (x1 )Ψ2 (x2 )) = Ψ1 (x2 )Ψ2 (x1 ) = eıθ Ψ1 (x1 )Ψ2 (x2 )
(6)
where the statistical angle θ = 0 describes bos-ons, θ = π fermi-ons and any
other value of θ describes any-ons.
Rotations R by an angle φ are generated by angular momentum,
ˆ
R(φ) = eıJφ ,
(7)
where Jˆ is the total angular momentum operator composed of the spins Ŝ of
the individual particles, Jˆ = 2Ŝ. For the swapping operation we get
σ (Ψ1 (x1 )Ψ2 (x2 )) = R(π) (Ψ1 (x1 )Ψ2 (x2 ))
ˆ
= eıπJ (Ψ1 (x1 )Ψ2 (x2 )) = eı2πs Ψ1 (x1 )Ψ2 (x2 )
(8)
where s is the spin of the particle. Comparing eqns. 6 and 8, we see that
2πs = θ
mod 2π.
(9)
We have seen in eq. 6 that θ = 0 (π) corresponds to bosons (fermions), so we
recover the spin-statistics theorem stating that bosons have integer spins while
fermions have half-integer spins. Furthermore, we notice that anyons can have
any spin.
As it turned out, the emergence of anyons is sufficient to explain some peaks
in the FQHE (Laughlin, 1983). In particular, FQHE states with ν = 1/k can be
ascribed to anyons with statistical angle θ = π/k, fractional spin s = 1/k and
fractional charge 1/k (Arovas et al., 1984). Schemes that allow those anyons to
condense into more elaborate quantized states give a satisfactory explanation of
the other values of ν (Haldane, 1983; Halperin, 1984; Prange and Girvin, 1990).
Since anyons carry fractional spin and angular momentum is a conserved
quantity, single anyons cannot be created from vacuum or annihilate into it.
They can, however, be created in anyon-antianyon pairs and fuse back together. Similar to fermions, which have a (−1)F superselection rule with F
being the fermion number, anyons of statistical angle θ have a θ-superselection
rule (Preskill, 2004).
2.3
The Braid Group Bn and its Representations
The key difference between two and three dimensions is that the condition
σ 2 = I is lifted in 2D, i.e.
6=
6
(10)
As a consequence, σ ∈
/ Pn where Pn denotes the permutation group of n elements. Instead, σ is an element of the braid group Bn .
The name “braid group” makes sense intuitively if you imagine N indistinguishable anyons at positions (x1 , ..., xN ) at a time t = 0 and the same anyons
at the positions (x1 , ..., xN ) at a time t = T . In between t = 0 and t = T ,
the world lines of the particles can wind around each other a number of times
before ending at their final position, creating a braid:
t=T
t=0
x1 x2 x3 x4 x5 x6 x7 x8
The braid group is generated by elements σi , which exchange particles i and
i + 1 in a counterclockwise manner:
7→
σi :
i–2
i–1
i
i+1 i+2 i+3
(11)
i–2
i–1
i
i+1 i+2 i+3
Under certain conditions (which will be discussed later), braiding is the only
non-trivial operation acting on the anyons. The topological class of a braid does
not depend on the specific particle trajectories, thus the final state is invariant
under smooth transformations of paths, such as noise in a physical system:
∼
=
i–2
i–1
i
i+1 i+2 i+3
(12)
i–2
i–1
i
i+1 i+2 i+3
Eqn. 6 is an example of a one-dimensional representation of the braid group.
Since the action is restricted to a complex phase, this representation is abelian,
i.e. different swapping operators commute as their phases simply add up. But
even this representation is infinite, since e.g. σin and σim are topologically distinct
for any n 6= m. The braiding operator is given by
R1−dim (σi ) : Ψ 7→ eıθi Ψ
(13)
with the multi-particle state Ψ = ψ(x1 )...ψ(xn ), where I assume that all particles are in the ground state ψ.
However, if the particles’ ground state space is g-fold degenerate with a basis
{ψ1 , .., ψg }, R can mediate between different states of this space and becomes a
g × g matrix
 




x1
R11 · · · R1g
x1



..  ·  ..  .
..
R :  ...  7→  ...
(14)
.
.   . 
xg
Rg1
···
Rgg
xg
Elements of this higher-dimensional representation generally do not commute.
These non-abelian anyons give rise to interesting braiding behavior, which shall
be investigated in the next section after a brief overview of the broader theory
of topological quantum field theories.
7
2.4
The Bigger Picture:
Topological Quantum Field Theory
This subsection is intended to provide a little context to the anyons introduced
before and the mathematical structures we will encounter in the following section.
Anyons and their braiding characteristics are an emergent feature of topological phases of matter. These phases are subject and object of a topological
quantum field theory (TQFT), i.e. a field theory in which transition amplitudes
are completely determined by topological properties. In analogy to functional
field integrals that emerge in non-topological field theories, which depend on
space-time parameters such as velocities or distances, amplitudes in TQFT are
of the kind
Z
Z = D{configurations}eıS[configuration]
(15)
where Z denotes the partition function and S is the action of a particular
configuration. A configuration is a topological class of particle trajectories from
the initial to the final state (Simon, 2012).
One particular formulation of a TQFT in 2 + 1 dimensions (2 space dimensions plus time) is Chern-Simons theory, which is defined by the action integral
Z
2
k
tr a ∧ da + a ∧ a ∧ a
(16)
SCS [a] =
4π M
3
where a is the field of the theory, M is the underlying manifold and k is an
integer defining the level of the theory. The Chern-Simons theory of level k has
a gauge group SU (2)k . Here, SU (2)k denotes a “deformation” of SU (2) that
has a level k cutoff, i.e. the only allowed quantum numbers are 0, 1/2, 1, ..., k/2
(SU (2) ≡ SU (2)∞ ). For k > 2, this group is non-abelian (Nayak et al., 2008).
I will discuss a model associated with the SU (2)3 theory later, which is a candidate for topological quantum computation and is believed to be found in the
ν = 12/5 FQHE state.
To find out how amplitudes in a TQFT can be calculated, we need to take
a look at the mathematical theory of knots. Knot theory is concerned with the
classification of knots into topologically distinct groups.
A knot invariant is a mathematical object that is different for topologically
distinct knots and the same for those which can be transformed into each other
by a diffeomorphism. Examples are the Jones polynomial (Jones et al., 1985)
and the related but simpler Kauffman invariant (Kauffman, 1987).
The Kauffman invariant is defined by the relations
=A·
+ A−1 ·
(17a)
=A·
+ A−1 ·
(17b)
= −A2 − A−2 = d.
(17c)
where A is some number (Simon, 2012). (The second rule is just a rotated
version of the first one.) These rules tell us to substitute crossings by arcs until
8
we are left with circles. To demonstrate them, we can take a look at a simplistic
example:
+ A−1 ·
=A·






= A
A ·



+ A−1 ·
 "








 + A−1 A ·









#2
+ A−1
= A A

 "
 + A−1 A










+ A−1 ·
#3
"
#2 
+ A−1
= A Ad2 + A−1 d + A−1 Ad3 + A−1 d2
= d + d3 + A2 + A−2 d2
{z
}
|
−d
2
= −A − A
−2
,
which should not come as a surprise since the pictured knot can be transformed
into — and is thus equivalent to — a circle, in contrast to e.g. the distinct
diagram (notice the orientation in the crossings)
= −A8 − A4 − A−4
by a similar calculation. With each crossing, the number of terms doubles. It is
apparent that knot invariants like this one are exponentially hard to compute
(Jaeger et al., 1990).
Fields Medalist Edward Witten (the only physicist who was ever awarded
this prize) discovered that the amplitudes of a TQFT are really given by Jones
polynomials describing the intertwining of the particle world lines (Witten,
1989). This important link between the unitary evolution of a many-body system on the one hand and topological classes of knots on the other hand should
give a good motivation for using the braid group to simulate the unitary action
of a quantum circuit.
Matters of interest for topological quantum computing, such as the effect of
braiding on a multi-particle state, can be derived from the full Chern-Simons
9

theory (Nayak et al., 2008). However, this will not be necessary for the simple
example considered in the following section.
3
Topological Quantum Computation
with Anyons
The previous section gave an introduction to particular aspects of topological
quantum field theories describing (non-abelian) anyons. However, our goal is to
build a quantum computer, and this means being able to (DiVincenzo, 2000)
1. define physical qubits,
2. prepare a particular initial state of the multi-qubit system,
3. have long coherence times (compared to gate operation times),
4. implement a universal set of gates, and
5. measure the final state.
A general topological quantum computation starts with creating a set of
anyon-antianyon pairs from the vacuum as an initial state. This state is then
acted on by unitary transformations that are implemented via braiding (since we
have said that braiding is the only non-trivial operation acting on the Hilbert
space of the anyons). The anyons are finally fused back together and it is
measured if they annihilate to vacuum or not.
In this section, I will fill in some details on those steps, in particular on
the mapping of quantum information onto anyons, the universality of the braid
group for quantum gates, state measurement through fusion, and fault tolerance
of the setup. These topics cover the afore-mentioned DiVincenzo criteria and
can be considered necessary conditions for universal quantum computing.
3.1
Anyon Models
There is a number of physical systems that non-abelian anyons can (theoretically) live in, some of which will be mentioned later. In order to analyze computational models of anyons, however, the physical realization is not of primary
importance. All we need to know are some general properties:
• What types of anyons can occur in this theory?
• What are the rules for creation/splitting and fusion?
• What is the effect of braiding?
The answers to those questions define an anyon model (or, more technically,
a topological field theory in 2 + 1 dimensions as well as a unitary topological
modular functor — see e.g. Freedman et al. (2002a) for a definition). The
presentation in this section follows the structure in Preskill (2004).
10
Particle Types
An anyon is a quasiparticle carrying a locally conserved (topological) charge,
such as the fractional electric charge in the FQHE. The type of the anyon is a
label associated with this charge. There can be any number of types {a, b, c, ...}
in an anyon model, as long as two conditions are fulfilled:
Firstly, the trivial anyon 1 must be in the set. It carries zero charge and has
trivial quantum numbers, so that it can be associated with the vacuum state.
Thus, it has trivial braiding effect and behaves under braiding like no anyon at
all.
Secondly, since the charge is conserved, for each type a there must be a
conjugate type ā, the antiparticle, with opposite quantum numbers. Particles
can be self-conjugate, e.g. 1 = 1̄.
I will assume that our system allows just two particle types, denoted by 1
and τ .
Fusion
Fusion rules describe what happens when two anyons a and b are combined to
form a new anyon c:
X
c
a×b=
Nab
c
(18)
c
c
Nab
c
If
= 0, a and b cannot fuse to c. On the other hand, if Nab
> 0, they can
c
fuse to c, and if Nab = 1, the fusion channel is unique.
I denote
a
|a, b; c, µi =
b
(19)
µ
c
c
as the fusion state of a and b fusing into c, where µ ∈ {1, ..., Nab
}. These states
c
span the space Vab . (This should look familiar from representation theory: E.g.
for irreducible representations d(j) of su(2) with spin j, we have
j ⊗ j 0 = |j − j 0 | ⊕ ... ⊕ (j + j 0 ).
(20)
An example of a non-unique fusion channel in this picture is the product of two
octet representations of SU (3)
8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27
(21)
8
so that N88
= 2 (e.g. Fuchs and Schweigert, 2003).)
For the example model with anyons 1 and τ there shall be only one nontrivial
fusion rule
τ ×τ =1+τ
(22)
in addition to the trivial ones, 1 × a = a (which is trivial since 1 behaves like
1
c
no particle at all), so e.g. Nτττ = Nτ1τ = 1, but N1τ
= 0. Since Nab
> 1 does not
c
occur for any a, b and c, the vector space Vab has dimension zero or one, and we
can drop the index µ from its basis states,
a
b
|a, b; c, µ = 1i ≡ |a, b; ci =
.
c
11
(23)
This diagrammatic representation of fusion rules, with arrows following the
direction of time describing particles, motivates to represent antiparticles by
arrows in the opposite direction, similar to Feynman diagrams in quantum field
theory. By moving the legs of the diagrams around,
a
b
a
c
c
∼
=
∼
=
c
b
(24)
a
b
we get the identities
c ∼
b ∼
Vab
= Vac
= Vcab ∼
= ...
(25)
This confirms that only pairs of opposite charge can be created from vacuum,
a
a ∼
since 1 × a = a, so that N1a
= 1 and V1a
= V1aa .
The direct sum
M
c
Vab
(26)
c
is called topological Hilbert space of a pair (ab). Its dimension is
!
M
X
c
c
dab = dim
Vab =
Nab
.
c
(27)
c
If, for any a and b, dab > 1, i.e. the fusion is not unique, the anyon model is
non-abelian. For the 1-τ -model,
X
dτ τ =
Nτcτ = Nτ1τ + Nτττ = 1 + 1 = 2,
(28)
c∈{1,τ }
so it is a non-abelian anyon model (and actually the simplest one).
Multi-particle basis. When more than two particles fuse, we need to keep track
of the order in which they do so. The total charge is an intrinsic property of a
group of anyons and should not depend on this order, i.e. fusion is associative:
(a × b) × c = a × (b × c).
(29)
For the fusion of three particles a, b and c with total charge d, we get the fusion
space
M
M
d ∼
e
d ∼
d
e0
Vabc
Vab
⊗ Vec
Vae
(30)
0 ⊗ Vbc .
=
=
e0
e
Here the direct sum over e on the left-hand side describes the case in which
a and b first fuse to e and then e and c fuse to d. The vector space has an
orthonormal basis |(ab)c → d; ei ≡ |ab; ei ⊗ |ec; di. Because of the associativity
of fusion, it is isomorphic to the direct sum over e0 on the right hand side of eq.
30, in which b and c fuse to e0 before e0 and a fuse to d. This latter vector space
has an ONB |a(bc) → d; e0 i ≡ |ae0 ; di ⊗ |bc; e0 i.
Since the two spaces are isomorphic, the basis elements can be transformed
into each other by
|(ab)c → d; ei =
X
d
Fabc
e0
12
e 0
e
|a(bc) → d; e0 i
(31)
a
b
c
a
X
=
e
d
Fabc
b
c
e 0
e’
e
e0
d
d
d
Fabc
where
is called the F-matrix in the literature.
This scheme can be generalized to more particles. Having n anyons a1 , ...,
an of total charge c ordered along a line, the standard basis shall be the one in
which particles fuse from the left to the right, i.e. a1 × a2 → b1 , b1 × a3 → b2 ,
..., bn−2 × an → c. The fusion space is given by
M
Vac1 ...an =
Vab11a2 ⊗ Vbb12a3 ⊗ ... ⊗ Vbcn−2 an
(32)
b1 ,...,bn−2
with basis elements
|a1 ...an → c; b1 ...bn−2 i ≡ |a1 a2 ; b1 i ⊗ |b1 a2 ; b2 i ⊗ ... ⊗ |bn−2 an ; ci
a1
a2
=
...
a3
b1 b
2
(33)
an–1 an
.
...
bn–2
c
This particular fusion ordering can be turned into any other by the action of
the F -matrix. Again, the fusion result, i.e. the total charge, does not depend
on the ordering, but the basis state does.
There is no natural decomposition of the space Vac1 ...an into a tensor product
of subsystems.
Fibonacci anyons. Let us return to the 1-τ -model. The basis of the fusion
c
space Vτ...τ
= Vτcn of dimension Nnc is labeled by n − 2 binary elements bi (eq. 33
with bi ∈ {1, τ }). However, since 1 × τ = τ , there cannot be two subsequent 1s
in this “bitstring” b. If the first two anyons fuse to 1, then the remaining n − 2
c
anyons can fuse to c in Nn−2
different ways, while if the first two anyons fuse
c
to τ , this τ can fuse with the remaining n − 2 anyons in Nn−1
different ways:
τ
τ
n τ fusing
τ τ ...
b1 b
2
...
τ
τ
τ
τ
=
c
1 τ
τ
...
τ
...
(n–2) τ fusing
τ
+
c
τ
τ
τ
b2
τ
...
...
(n–1) τ fusing
τ
(34)
c
Thus, the dimension of the fusion space is given by the recursion relation
c
c
Nnc = Nn−1
+ Nn−2
(35)
which defines the Fibonacci numbers Fn ∈ {0, 1, 1, 2, 3, 5, 8, 13, ...}. For this
reason, the 1-τ -model of anyons is often called Fibonacci model.
Braiding
For two particles a and b with a total charge c, braiding is an isomorphism
R:
c
c
Vba
→ Vab
13
c
c
|ba; ci = Rab
|ab; ci = eıθab |ab; ci
a
b
a
=
(36)
b
c
Rab
c
(37)
c
where the R-matrix is just a phase factor since we restrict ourselves to onec
c µ0
dimensional fusion spaces here (but Rab
could be any unitary matrix (Rab
)µ if
we had not dropped the index µ in eq. 19).
The action of braiding on a pair of definite total charge is thus not particularly interesting. The situation is different for a multi-particle fusion state.
Given a state |a1 ...an → c; b1 ...bn−2 i in the standard basis, we need to figure out
the effect of an exchange of the j th with (j + 1)th particle. This can be achieved
by applying the F -matrix to get to a basis in which the R-matrix is diagonal,
i.e. a state where the j th and the (j +1)th do have definite total charge, applying
R, and returning to the standard basis via F −1 :
a
b
c
a
b
F
c
a
b
a
b
c
F –1
R
d
c
d
d
d
d
Let’s take the fusion space Vacb
and calculate the effect of exchanging b with c.
The braiding operator B in the standard basis acts like
X
f
d
B |(ac)b → d; ei =
Facb
B |a(cb) → d; f i
e
f
=
X
f
d
Rbc
Facb
f
e
|a(bc) → d; f i
f
=
X
g f
f
d
(F −1 )dacb f Rbc
Facb
|(ab)c → d; f i
e
(38)
f,g
so that we get the B-matrix
g −1 d g f
f
d
d
Babc
= (F )acb f Rbc Facb
.
e
e
(39)
Moore-Seiberg Polynomial Equations. In a general anyon model, the operators
F , R and B need to be derived from the topological quantum field theory.
However, the simple Fibonacci model is completely determined by two algebraic
consistency equations, the Moore-Seiberg polynomial equations.
These consistency conditions are identities between different paths that apply F-moves and R-moves to get from one fusion space to another (Trebst et al.,
2009). They can best be understood diagrammatically (fig. 5). The pentagon
equation (fig. 5a) says that the state
X
d 5 c
5
|12; ai |a3; bi |b4; 5i =
F12c
Fa34 b |1d; 5i |2c; di |34; ci
(40)
a
c,d
obtained via the upper path in the figure should equal the state
X
c 5 d b e
d
|12; ai |a3; bi |b4; 5i =
F234
F1e4 b F123 a |1d; 5i |2c; di |34; ci ,
e
c,d,e
14
(41)
1
a
4
3
a
c
final state
2 3 4
F
F
initial state
1 2 3 4
2
5
1
d
b
5
c
5
F
F
1
2
4
3
1
e
b
2
4
3
e
F
d
5
5
(a) “Pentagon” equation.
1
2
3
1
F
R
3
c
4
R
a
initial state
1 2 3
2
4
final state
1 2 3
c
a
4
4
1
2
3
1
2
3
1
2
3
F
F
b
=
b
4
R
b
4
4
(b) “Hexagon” equation.
Figure 5: The Moore-Seiberg relations are identities between different paths of F moves and R-moves leading to the same final state. They act as consistency equations
that completely determine the F - and R-matrices in the Fibonacci model (Trebst
et al., 2009).
which we get on the lower path. Since we expect all incoming anyons to be of
kind τ , we get
Fτ5τ c
d
a
5
Faτ
τ
c
b
=
X
Fτdτ τ
c
e
Fτ5eτ
d
b
Fτbτ τ
e
a
(42)
e
τ
τ
τ
The fusion spaces Vτ1τ τ , V1τ
τ , Vτ 1τ and Vτ τ 1 are one-dimensional, i.e. there is a
τ
unique fusion channel, and thus we have some trivial F -matrices Fτ1τ τ = F1τ
τ =
τ
τ
Fτ 1τ = Fτ τ 1 = 1 (up to negligible phase factor). The solution to eq. 42 is then
Fτ1τ τ = 1
on the space spanned by |(τ τ )τ → 1; τ i and
−1
φ
φ−1/2
τ
Fτ τ τ =
φ−1/2 −φ−1
15
(43)
(44)
√
on the space spanned by {|(τ τ )τ → τ ; 1i , |(τ τ )τ → τ ; τ i} where φ = (1 + 5)/2
is the golden ratio (Trebst et al., 2009).
Similarly, the hexagon equation (fig. 5b) is based on the observation that
X
a
4
c
|13; ai |a2; 4i =
R13
(F132
)ca R23
|1c; 4i |23; ci
(45)
c
(upper path) must be equivalent to
X
4
4
4
|13; ai |a2; 4i =
(F312
)ba Rb3
(F123
)cb |1c; 4i |23; ci
(46)
b
(lower path). This leads to
c
Rτc τ (Fτττ τ )a Rτaτ =
X
c
b
(Fτττ τ )b Rττ b (Fτττ τ )a
(47)
b
with the solution
R=
eı4π/5
0
0
−eı2π/5
(48)
acting on the space spanned by {|τ τ → 1i , |τ τ → τ i} (Trebst et al., 2009).
Qubit encoding. We know that braiding two anyons only adds a phase factor
to the fusion state (eq. 48). However, the braiding operation gets non-trivial
when more particles are added. An anyon triplet has three fusion states: |0i ≡
|(τ τ )τ → τ ; 1i, |1i ≡ |(τ τ )τ → τ ; τ i and |NCi ≡ |(τ τ )τ → 1; τ i (reminder: the
last label in the ket describes the intermediate fusion result). These triplets
can be used to encode qubits. The third anyon is labeled non-computational
because it is not used for information storage (but it is essential for non-trivial
braiding).
The braid group of a triplet is generated by the elementary counterclockwise
exchanges σ1 (swapping particles 1 and 2), σ2 (swapping particles 2 and 3)
and their clockwise inverses. Their action on the fusion states in the basis
{|0i , |1i , |NCi} is (in generalization of eqns. 44 and 48)

 −ı4π/5
0
e
0
 |Ψi
0
−e−ı2π/5
0
σ1 : |Ψi 7→ R |Ψi = 
(49)
−ı2π/5
0
0
−e
and
σ2 : |Ψi 7→ B |Ψi
= F −1 RF |Ψi

−e−ıπ/5 /φ
√

−e−ıπ/10 / φ
=
0

√
−e−ıπ/10 / φ
0
 |Ψi
−1/φ
0
−ı2π/5
0
−e
(Nayak et al., 2008).
16
(50)
Relevance of the Fibonacci model. The Fibonacci model is a neat example of
a topological quantum field theory because it is very simple but still exhibits
non-abelian statistics. But apart from that, it is also of importance because a
close relative of it is believed to be found in the ν = 12/5 FQHE state.
This state is described by the so called Z3 parafermion model, which has a
gauge group SU (2)3 /U (1) at its core (Read and Rezayi, 1999). Again, SU (2)k
is SU (2) with a level k cutoff, i.e. the only allowed quantum numbers are
0, 1/2, 1, ..., k/2, so that the SU (2) fusion rules are truncated, e.g. in SU (2)3
we have 1 × 1 = 0 + 1 while we get 1 × 1 = 0 + 1 + 2 in SU (2) ≡ SU (2)∞ .
The SU (2)3 fusion rule should look familiar from the Fibonacci model. While
SU (2)3 also contains the half-integer labels 1/2 and 3/2, the reduction to the
(integer) Fibonacci subspace is unproblematic since the half-integer subspace is
inaccessible from within the integer one anyway due to the fusion rules (Nayak
et al., 2008).
The ν = 12/5 FQHE state has been studied experimentally (Xia et al.,
2004). However, there has been no direct evidence of the emergence of nonabelian statistics in this or any other FQHE state yet.
3.2
Universality of Quantum Computation via Fibonacci
Anyons
Now that the Fibonacci model has been introduced and the action of elementary
braiding operations has been examined, it is time to prove that it is as powerful
as the quantum circuit model. This will be done in two steps: First, I will show
that any quantum circuit can be efficiently simulated using Fibonacci anyons,
i.e. this kind of topological quantum computer is at least as powerful as nontopological ones. In a second step, I will demonstrate that the reverse is true as
well, i.e. anyon braiding can be simulated on a conventional quantum computer.
This leads to the result that the topological and the non-topological models for
quantum computing are equally powerful.
In order to show universality, we will need the Solovay-Kitaev theorem.
The Solovay-Kitaev Theorem
The definition of a universal quantum computer is that it can perform any
unitary operation on the Hilbert space of its n qubits H = C2n . Obviously,
a physical computer cannot have the uncountable set of SU (2n) operations
as elementary building blocks at its disposal. Instead, the condition can be
watered down slightly by only requiring that any operation U ∈ SU (2n) can
be approximated to arbitrary precision by some countable (possibly finite)
universal set G of transformations.
More technically, let S be the product of l elements in G. S is an approximation to U if d(U, S) ≡ kU − Sk ≡ supkψk=1 k(U − S)ψk < . Saying
that for any U and there exists such an S, is to say that the image of G is
dense in SU (2n).
However, the scaling of the length of the sequence S as a function of the precision, l(), is of paramount importance. As an example, take Grover’s search
algorithm, which brings a quadratic speed up over classical algorithms (Nielsen
and Chuang, 2010). Assuming that the m gates U1 , ..., Um it is comprised of are
not in the set G and our target precision is , we can allow an error of /m per
17
gate. If the scaling of l was l = O(1/), which might be a reasonable guess, we
would need a total of m · O(m/) = O(m2 /) gates. Thus, the advantage of the
quantum algorithm over its classical counterpart would be lost. Luckily, this is
not the case:
Solovay-Kitaev Theorem. Let G be a finite set of elements in SU (d)
containing its own inverses, such that the image of G is dense in SU (d), and
let a desired accuracy be given. There exists a constant c such that for any
U ∈ SU (d) there exists a finite sequence S of gates in G of length O(logc (1/))
with d(U, S) < (Dawson and Nielsen, 2005).
In simpler terms, the number of gates needed to approximate a unitary
transformation to precision scales polylogarithmically in 1/. The proof of
the theorem, which can be found in the cited paper, consists of an iterative
algorithm which explicitly constructs the element S (and, as a matter of fact,
this algorithm runs on a classical computer in polylogarithmic time as well).
(Quantum compiling is an active field of research. The value of c depends
on both the compiling algorithm and the chosen basis set. For conventional
quantum computing, the algorithm in Dawson and Nielsen (2005) gives c = 3.97.
Harrow et al. (2002) have proven that c = 1 can theoretically be achieved for an
optimally chosen basis set, though it is not clear how to construct such a set.)
Simulating Quantum Circuits with Anyons
The elementary braiding matrices σ1 and σ2 from the Fibonacci model generate
a representation of the braid group that is indeed dense in SU (2) (Freedman
et al., 2002b). (The proof is very technical and its mathematical framework
goes beyond the scope of this introduction, so I will not outline it here, though
it is an important link in the argument.) The instruction set G = {σ1 , σ2 , σ1−1 ,
σ2−1 } ∈ SU (2) thus fulfills the requirements of the Solovay-Kitaev theorem (the
inverse operators correspond to clockwise quasiparticle exchanges), so that any
one-qubit operations can be approximated efficiently by braiding operations.
Since the dimension D = Fn of the topological Hilbert space of n Fibonacci
anyons grows exponentially, this generalizes to multi-qubit systems. E.g. the
Hilbert space of six anyons (corresponding to 2 qubits) has dimension F6 = 5.
Similar to the one-qubit case, the braiding matrices turn out to be dense in
SU (F6 ) = SU (5). This clearly includes SU (4), so that arbitrary two-qubit
transformations can be applied (e.g. Nayak et al., 2008).
It is well known that a generating set of one-qubit operations and a two-qubit
entanglement gate such as the CNOT-gate are universal (Nielsen and Chuang,
2010). The Solovay-Kitaev algorithm can be applied to construct such a set
from G = {σ1 , σ2 , σ1−1 , σ2−1 }. E.g., the NOT-gate


0 i 0
 i 0 0 
(51)
0 0 1
acting on (|0i , |1i , |NCi) is approximated to ≈ 8.5 × 10−4 by (Bonesteel et al.,
2005)
σ1−2 σ2−4 σ14 σ2−2 σ12 σ22 σ1−2 σ24 σ1−2 σ24 σ12 σ2−4 σ12 σ2−2 σ12 σ2−2 σ12 σ2−2 σ1−2
18
(52)
1
(53)
(remember that an arbitrarily lower can be achieved by longer sequences). In
this figure, the world lines of anyons 1 and 3 are “strained” so that it becomes
apparent that this braid is effectively created by “weaving” anyon 2 through
the world lines of the other two. In fact, it is a general theorem that any braid
operation can be accomplished by weaving a single particle (Simon et al., 2006).
This (K)NOT-gate can be extended in a very clever way. Weaving a τ -anyon
around two others in the way described acts as a NOT-gate. Weaving a 1-anyon
around the other two does not do anything, since a 1-anyon is like no particle
at all. This means that the action of the braid depends on the particle type at
anyon 2. Now, there are only τ -anyons in the system, but a pair of anyons can
have a total charge of either 1 or τ . Thus, building this braid with a pair of
particles in place of anyon 2, a controlled NOT operation is performed.
In order to get to the original two-qubit CNOT gate, we need to take anyons
2 and 3 of the control qubit and braid them with anyons 2 and 3 of the target
qubit. But since those anyons are no direct neighbors, we first need a braid
that swaps the anyons around while acting trivially on their states. This can be
accomplished by an “injection weave” (Bonesteel et al., 2005)
σ23 σ1−2 σ2−4 σ12 σ24 σ12 σ2−2 σ1−2 σ2−4 σ1−4 σ2−2 σ14 σ22 σ1−2 σ22 σ12 σ2−2 σ13
1
(54)
1
1
1
that moves one particle down two positions while acting as


1 0 0
 0 1 0 
0 0 1
(55)
with a precision of ≈ 1.5 × 10−3 . Combing the injection weave (eq. 54) with
the NOT operation (eq. 52) and an inverse injection, modified trivially to braid
two anyons instead of one, we get a fully functional two-qubit CNOT-gate with
a total accuracy of at least ≈ 1.8 × 10−3 (again Bonesteel et al., 2005):
1
injection
NOT
inverse injection
1
Concluding, the elementary braiding operators σ1 and σ2 and their inverses
constitute a universal set of unitary operations. Since any quantum circuit
can be simulated by anyon braiding, the complexity class associated with its
computational power is at least BQP.
19
Simulating Anyons with Quantum Circuits
Now I will show that the topological and non-topological models of quantum
computation are in fact equally powerful by illuminating how to simulate anyons
with quantum circuits. The argument is a modified version of the proof in
Preskill (2004) (which itself is inspired by Freedman et al. (2002a)).
The mapping of anyons to qubits is a little less obvious than the other way
round, since the topological Hilbert space of n anyons of total charge 1 (the
total topological charge has to be 1 since all anyons are created in pairs from
vacuum)
M
bn−2
1
1
Vτ...τ
Vτbτ1 ⊗ Vbb12τ ⊗ ... ⊗ Vbn−3
≡
τ ⊗ Vbn−2 τ
b1 ,b2 ,...,bn−2
b
M
=
=τ
n−2
Vτbτ1 ⊗ Vbb12τ ⊗ ... ⊗ Vbn−3
τ ,
(56)
b1 ,b2 ,...,bn−3
does not have a natural decomposition into a tensor product of subsystems.
This is apparent e.g. from the dimension of this space,
1
dim Vτ...τ
= Fn ∝ φn for large n
(57)
√
which assigns an irrational dimension (the golden ration φ = (1+ 5)/2 ≈ 1.618)
to each anyon. The task will be to explain a mapping of anyons to qubits, to
show how braiding can be implemented in the qubit space, and to get an idea
of how fusion outcomes can be measured in the qubit basis.
Encoding The braiding operation acting on the topological Hilbert space in eq.
56 swaps around labels, but does not change the general structure of the vector
space. Each summand in eq. 56 consists of n − 2 fusion spaces
O
b
d
Vaτ
∈
Vcτ
≡ Hd ,
(58)
c,d
so that for each summand from eq. 56
⊗(n−2)
.
Vτbτ1 ⊗ Vbb12τ ⊗ ... ⊗ Vbτn−3 τ ∈ (Hd )
(59)
The dimension of Hd is given by
d=
X
d
Ncτ
.
(60)
c,d
Thus n − 2 d-qudits can encode an n-anyon state. For Fibonacci anyons, Hd is
spanned by {|τ τ ; 1i , |τ τ ; τ i , |1τ ; τ i}, so d = 3. (A change to an m-qubit system
is easily done via C2dd(n−2)/2e = C2m , i.e. m = dd(n − 2)/2e, but the qudit
representation is more convenient for now.)
Braiding The effect of braiding in the anyon standard basis is completely determined by the B-matrix:
a
τ
τ
a
=
c
b
X
b
Baτ
τ
d
d
c
·
τ
τ
d
b
20
|aτ ; ci |cτ ; bi =
|
{z
}
X
b
Baτ
τ
d
c
d
∈(Hd ⊗Hd )
|aτ ; di |dτ ; bi
|
{z
}
(61)
∈(Hd ⊗Hd )
Thus, the action of braiding on the qudit space is given by a unitary transformation
B : Hd ⊗ Hd = C2d → C2d ,
(62)
so that braiding can be implemented by a d2 × d2 unitary matrix.
Fusion To calculate the outcome of fusing two neighboring anyons in the context of a multi-anyon state in the standard basis, we need to apply the F -matrix:
a
τ
τ
a
=
c
b
Faτ
τ
d
c
d
b
|aτ ; ci |cτ ; bi =
|
{z
}
∈(Hd ⊗Hd )
X
X
τ
·
τ
d
b
b
Faτ
τ
d
b
= Faτ
τ
d
c
|ad; bi |τ τ ; di
|
{z
}
∈(Hd ⊗Hd )
1
c
b
|a1; bi |τ τ ; 1i + Faτ
τ
τ
c
|aτ ; bi |τ τ ; τ i .
(63)
Fusion can be simulated in the qudit system by applying the F -matrix (another
unitary d2 ×d2 -matrix), and performing a projective measurement on the second
qubit in the {|τ τ ; 1i , |τ τ ; τ i} basis and sampling the probabilities for getting 1
and τ .
In the previous section, I had demonstrated that anyonic quantum computing
is at least as powerful as “conventional” quantum computing. Having in place
a scheme for braiding and fusing anyons in a qubit representation, we see that
the topological model of computation is also not more powerful than the circuit
model. This completes the proof of the conjecture that both models dwell in
the same complexity class BQP.
3.3
Fault Tolerance
Now that we know how topological quantum computers work in general, and
that they can be just as powerful as non-topological ones, it is time to return
to the original motivation why people (Alexei Kitaev in particular, 2003) came
up with the idea. As already stated in the introduction, “conventional” quantum computers suffer from decoherence, i.e. interaction with the environment
that decreases the lifetime of qubits. There exist a number of error correcting
codes that can work against the effects of decoherence (Shor, 1996; Nielsen and
Chuang, 2010), including Kitaev’s toric code which also hides logical information in topological states (Kitaev et al., 2002). Those are effective, but they
require a certain threshold fidelity of the quantum gates. Estimates of the required precision range from 10−6 (Knill et al., 1996) to 1/300 (Zalka, 1996),
which is hard to achieve experimentally.
Topological quantum computers on the other hand offer an alternative solution to the problem. Quantum information is not encoded in local properties
of particles (such as spins), instead the information is stored in topological, i.e.
21
non-local, features of a system of particles. This leads to an essential immunity
of the quantum information from local perturbations in the physical system —
the associated Hilbert space is protected (Preskill, 2004).
Errors are not only introduced by decoherence but also by inexact gate
operation. Can we be sure that the braiding generator σ is a rotation by π and
not π + δ, i.e. do particles return to their original position or is the winding
number changed slightly? Another beautiful aspect of a topological quantum
computer is that we do not need to care about this. Since pairs are created from
vacuum and re-annihilate to it, all world lines are closed loops — all operations
are truly topological.
In a physical system with an energy gap ∆ above the degenerate ground
state, excitations from the ground state are suppressed by the Boltzmann factor
e−∆/(kT ) where k is the Boltzmann constant and T denotes temperature. Any
local interactions (e.g. electron-phonon-interaction) has trivial matrix-elements
within the ground state subspace (Nayak et al., 2008). Thus, at sufficiently low
temperatures, local interactions and noise do not disturb the Hilbert space of
the anyons. Thus the only non-trivial unitary evolution a system of anyons can
undergo is by braiding. (So, clearly, a topological quantum computer should
keep good track of its particle positions to avoid unintentional braiding.) But
what about unintentional anyons existing in the system?
Anyon-antianyon pairs can be created from vacuum by thermal fluctuations
or virtually by quantum fluctuations. They can travel through the system and
re-annihilate. These fluctuations are again suppressed by the Boltzmann factor
e−∆/(kT ) for thermal processes and by e−∆t/h for virtual processes (temperature
T , quasiparticle energy gap ∆, time t for which the virtual excitation exists).
Taking a look at the parameters, a topological quantum computer should operate at low temperatures and on a physical system with a big energy gap to
minimize thermal processes. Virtual processes however only depend on the distance l = vt traveled by the pair before re-annihilation (with some characteristic
velocity v) (Nayak et al., 2008).
When a particle-antiparticle pair bb is created close to the world line of a
computational anyon pair aa0 , either thermally or virtually, there are two main
processes by which the pair can interact with it (Nayak et al., 2008):
• Firstly, b (b) could wind around a and re-annihilate with b (b) (fig. 6a).
Since the pair bb is created from vacuum, its total charge is 1. Let the
charge of the computational pair be j. Then, the total charge of the 4
particles upon creation (time t1 ) is also j. One virtual particle winds
around one anyon and the two virtual particles annihilate, which means
that their total charge is still 1. Thus, the charge j remains with the
computational pair, so that the process has a trivial effect.
• The alternative process is that a virtual pair is created next to a computational pair of charge j and one virtual particle annihilates a computational
one while the other virtual particle replaces the lost anyon of the computational pair (fig. 6b). However, reasoning similar to the previous case
reveals that the state of the new computational pair will be indistinguishable from the original state.
This shows that any braiding and fusing involving a single computational
anyon has a trivial effect on the topological Hilbert space. Braiding processes
22
j
j
t2
1
1
time
1
1
t1
(a)
j
(b)
j
Figure 6: The only virtual processes involving just a single computational pair do
not cause any errors, since the topological charge of this pair is not affected. Processes
involving more anyons are exponentially suppressed in the anyon spacing. Figure
reproduced from Nayak et al. (2008).
that do induce errors must act on at least two anyons, so that those created by
quantum fluctuations occur with a probability ∝ e−2∆L/v where L is the spacing
of the physical anyons in the system. Thus, errors of this kind are suppressed
exponentially in the distance of anyons.
(There is another advantage in having large quasiparticle spacings. The
degeneracy of the ground state space that is necessary for non-abelian statistics
is often an approximate degeneracy. For example, the degeneracy can be lifted
by Coulomb or spin-spin interactions of the particles involved. Since these
interactions are local, the degeneracy is better approximated for larger spacings
(Nayak et al., 2008).)
A further error source is disorder in the physical system which creates localized quasiparticles that can interfere with braiding operations (Nayak et al.,
2008). This puts high constraints on the quality of the used material.
4
Conclusion
I have given a short description of topological quantum computing by first
demonstrating the emergence of anyons with non-abelian exchange statistics in
two-dimensional systems, and then showing that a particular anyon model, the
Fibonacci model, is capable of carrying out universal quantum computations.
Because of its topological nature, an anyonic quantum computer is inherently
insusceptible to decoherence and noise, which are the Achilles’ heel of conventional quantum computers.
I have chosen to present the Fibonacci model because of its simplicity and
(suspected) implementation in a FQHE state, but it is by far not unique. ChernSimons theories at other levels k > 2 can be treated in the same way by defining
fusion rules and constructing R-matrices and F -matrices. Some, but not all, of
these alternative models have a representation of the braid group which is dense
in SU (n), which enables them to simulate quantum circuits (Nayak et al., 2008).
Topological quantum computation is a relatively new field, having been
started by Kitaev’s paper in 1997, and there are no devices yet that can actually carry it out. Furthermore, there is not even direct evidence of the existence of non-abelian anyons. But there are theoretical papers proposing a
wide range of physical systems that could host those quasiparticles, including
23
but not restricted to the already discussed FQHE states, super conductors (Sau
et al., 2010), rotating Bose-Einstein condensates (Viefers, 2008), atom lattices
(Tewari et al., 2007) and Josephson-junction arrays (Ioffe et al., 2002). Nayak
et al. (2008) includes an overview of some of these systems. Several experiments
are being built to realize those proposals. Finding topological phases of matter
is a non-trivial task on the theoretical side as well. Different models lack a common footing and there is no mathematical classification of topological phases
(Nayak et al., 2008).
Topological phases of matter are intriguing subjects and their application
for quantum computing is a promising idea. The next years should bring forth
interesting experimental insights into its feasibility.
Anyon, anyon, where do you roam?
Braid for a while before you go home.
Though you’re condemned just to slide on a table,
A life in 2D also means that you’re able
To be of a type neither Fermi nor Bose
And to know left from right — that’s a kick, I suppose.
You and your buddy were made in a pair
Then wandered around, braiding here, braiding there.
You’ll fuse back together when braiding is through
Well bid you adieu as you vanish from view.
No one can say, not at this early juncture
If someday we’ll store quantum data in punctures
With quantum states hidden where no one can see,
Protected from damage through topology.
Anyon, anyon, where do you roam?
Braid for a while before you go home.
——— John Preskill
24
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