Download Topological Quantum Computation from non-abelian anyons

Topological Quantum Computation
from non-abelian anyons
Paul Fendley
Experimental and theoretical successes have made us take a close look at
quantum physics in two spatial dimensions.
We have now found (mostly theoretically, but also experimentally) behavior unseen in 3+1
dimensions. The mathematics describing the physics also can be quite different.
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One potential application of this novel physics is a topological quantum computer.
This idea relies on one particularly unusual property possible only in two dimensions:
non-abelian statistics.
There are several possibilities for realizing non-abelian statistics experimentally:
• a two-dimensional electron gas in a large magnetic field (the fractional quantum Hall
effect);
• rapidly rotating Bose-Einstein condensates
• frustrated magnets.
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Outline:
1. What are non-abelian statistics?
2. What does this have to do with quantum computation?
3. What does this have to do with mathematics?
The Potts model and the BMW algebra: 2002, with N. Read
Quantum Potts nets: 2005, with E. Fradkin
New classical loop models: 2006; 2008 with J. Jacobsen
Quantum Potts loop models: 2007; 2008
The BMW algebra, the chromatic algebra, and the golden identity: 2007, with V. Krushkal
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The fundamentally weird thing about quantum mechanics: The quantum two-state
system is really an infinite number of states: any linear combination of two quantum
states is also a quantum state.
Currently-fashionable buzzword: Entanglement
A conventional computer uses bits, which are classical two-state systems. A quantum
computer uses qubits, which are quantum two-state systems:
| ↑↑↓↑↓↓↓↑↓↑ . . . i
A quantum computation is done by a unitary transformation on the initial quantum state.
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A quantum computer is massively parallel: the initial state can be a superposition of every
phone number in the Manhattan phone book.
A single unitary transformation acts on all of these at once: could do reverse lookups in
order
√
N time as opposed to order N .
Numbers could be prime-factored in polynomial time.
But with every silver lining.... . .
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Errors!
In a classical computer, just be redundant.
In a quantum computer, errors are continuous, not just bit flips. Moreover, measurement
destroys the superposition. How do we know errors have occurred?!? Need to preserve
+ α
It’s non-trivial to show that error correction is even possible for quantum computers.
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The idea behind a topological quantum computer to exploit systems with quantum
redundancy.
In particular, in 2+1 dimensions, identical particles can have some very striking
properties.
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In three dimensions, the only statistics for identical particles are bosonic and fermionic:
ψ(x1 , x2 , . . . ) = ±ψ(x2 , x1 , . . . )
Exchanging twice is a closed loop in configuration space.
1
0
0
1
x1
1
0
0
1
x2
In three dimensions, this can be adiabatically deformed to the identity.
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In two dimensions, it cannot!
Identical particles can pick up an arbitrary phase:
ψ(x1 , x2 ) = eiα ψ(x2 , x1 )
Particles with α
6= 0, π are called anyons.
They have been observed in the fractional quantum Hall effect
(these have fractional charge as well).
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In 2+1 dimensions, the statistics comes from behavior under braiding. Each particle
traces out a worldline in time, and these braid around each other.
figure from Skjeltorp et al
Braiding is a purely topological property.
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Even more spectacular phenomena than anyons can occur under braiding.
Particles can change quantum numbers!
ψa (x1 , x2 . . . ) =
X
Bab ψb (x2 , x1 . . . )
When the matrix B is diagonal with entries ±1, particles are bosons and fermions.
When B is diagonal with entries eiα , particles are anyons.
When B is not diagonal, can get non-abelian statistics.
With non-abelian statistics, the probability amplitude changes
depending on the order in which the particles are braided.
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Braiding particles with non-abelian statistics entangles them!
This is the fundamental idea behind topological quantum
computation.
Kitaev; excellent review in Preskill’s lecture notes
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We need to understand fusion: what are the statistics of a pair of particles?
This is non-trivial even for abelian anyons. Say we have two identical particles which pick
up a phase eiα when exchanged, or equivalently, e2iα under a 2π rotation of one around
the other. Then a pair of these picks a phase e8iα when exchanged with different pair.
In the non-abelian case, fusing two particles can give a linear combination of states,
analogous to
1 1
⊗ =0 + 1
2 2
in SU (2).
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The simplest kind of non-abelian statistics is in the braiding of Fibonacci anyons.
Fusing two Fibonacci anyons gives a linear combination of a single Fibonacci anyon
(denoted φ) and a state with trivial statistics (denoted 1). They satisfy the fusion rule
φ×φ∼1 + φ
This is like the tensor product of two spin-1 representations if we throw out the spin-2
piece.
The consistency rules governing braiding and fusing of anyons are identical to those of
2d rational conformal field theory. These are known as the Moore-Seiberg axioms for a
universal modular tensor category.
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φ×φ=I +φ
Precise meaning in field theory: The operator product expansion of two φ fields contains
both the identity field I and the field φ itself .
Precise meaning in conformal field theory: the “fusion coefficients” for the primary fields I
I
and φ are Nφφ
φ
= Nφφ
= 1.
Precise meaning for this talk: There are two possible quantum states for two Fibonacci
anyons. One has trivial statistics, the other has the braiding properties of a single
Fibonacci anyon.
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This fusion algebra allows us to count the number of quantum states for 2N
quasiparticles:
1:
I
1:
φ
2:
φ×φ=I +φ
3:
φ×φ×φ=I +φ+φ
5:
φ × φ × φ × φ = φ + 2(I + φ)
8:
φ × φ × φ × φ × φ = 3I + 5φ
The number of states for n particles is the nth Fibonacci number.
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To make a qubit requires 4 Fibonacci anyons. There are two different ways of fusing
these four into an overall identity channel – the two states of a qubit!
Namely, when the overall channel is I , when the combination of anyons 1 and 2 fuses to
state φ, then anyons 3 and 4 must also be in the state φ. If anyons 1 and 2 fuse to I ,
then anyons 3 and 4 must also fuse to I .
The system can be in any linear combination of these two states. We can change the
state, by braiding particle 2 around particle 3.
This entanglement is possible even if none of the particles are near each other!
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This is why non-abelian statistics allow a fault-tolerant quantum computer.
The braiding of particle 2 with particle 3 allows us to do quantum computation. The result
of this braiding depends only on the topological properties of the system: these particles
can be very far from each other.
No local perturbation will change the results of the braiding.
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One can find the braiding and fusing matrices from standard conformal field theory
techniques.
For example, for four Fibonacci anyons in overall channel I ,

1  e4πi/5
B=
τ −e2πi/5 √τ
where τ
= 2 cos(π/5) = (1 +
√
−e
√
2πi/5

τ

−1
5)/2 is the golden mean.
However, a much more intuitive and instructive way to study non-abelian braiding in depth
is to use algebras and draw pictures!
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We project the world lines of the particles onto the plane. Then the braids become
overcrossings and undercrossings
The braids must satisfy the consistency condition
=
which in closely related contexts is called the Yang-Baxter equation.
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A simple way of satisfying the consistency conditions leads to the Jones polynomial in
knot theory. Replace the braid with the linear combination
=q
1/2
q 1/2
= q 1/2
q 1/2
q is a parameter which is a root of unity in the cases of
interest: the Fibonacci case corresponds to q = eiπ/5 .
so that the lines no longer cross.
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This gives a representation of the braid group if the resulting loops satisfy d-isotopy.
• isotopy: Configurations related by deforming without making any lines cross receive
the same weight.
• d: A configuration with a closed loop receives weight
d = q + q −1
relative to the configuration without the loop.
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These relations can all be treated in terms of the Temperley-Lieb algebra, which
graphically is
= d
=
This is usually written in terms of generators ei acting on the ith and i + 1st stands:
e2i = dei
ei ei±1 ei = ei
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To ensure finite-dimensional representations of this algebra, need
d = 2 cos[π/(k + 2)]
i.e.
q = eiπ/(k+2)
with k a positive integer.
This is related to SU (2)k Chern-Simons theory. In this topological field theory, all
physical quantities follow from topological data (how the loops link with each other).
Witten
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The problem: find a quantum Hamiltonian acting on a two-dimensional Hilbert space
which has the above properties.
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One answer:
Fig. 1, Pan et al
0.5
T e ~ 4 mK
2
7/3
2
Rxy (h/e )
Sample
5/2
0.4
8/3
3
5/2
1.0
Rxx (kΩ)
0.3
7/3
8/3
12/5
13/5
16/7
19/7
3.2
3.4
3.6
3.8
4.0
MAGNETIC FIELD [T]
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0.0
4.2
The leading candidate wavefunction for the ν
= 5/2 plateau is the Moore-Read Pfaffian,
√
which contains non-abelian anyons with d = 2.
A closely-related theory describes the superconductor with px
+ ipy pairing, believed to
be realized in strontium ruthenate. The vortices here are non-abelian anyons.
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Another idea:
Much effort has been devoted to constructing quantum loop models whose quasiparticles
have abelian and non-abelian braiding.
Such models are closely related to frustrated magnets, and there is hope that they can be
realized in the lab, in what are often known as spin-liquid phases.
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A (hopefully not unique) procedure to construct such a quantum loop model is
1. find a 2d classical loop model which has a critical point
2. use each loop configuration as a basis element of the quantum Hilbert space
3. use a Rokhsar-Kivelson Hamiltonian to make the quantum ground state a sum over
loop configurations with the appropriate weighting.
Kitaev; Moessner and Sondhi; Freedman
The inner product must be topological, i.e. ha|bi depends only on topological data.
Levin; Fendley
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To realize this form of braiding, each loop in the ground state gets a weight d (=
Fibonacci)
i.e. the ground state is the sum over all loop configurations
|g.s.i =
X
dnL |Li
L
where nL is the number of loops in configuration L.
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τ for
The excitations with non-abelian braiding are defects in the sea of loops.
When the defects are deconfined, they will braid with each other like the loops in the
ground state.
Kitaev; Freedman
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To find a quantum loop model which has Fibonacci anyons, one needs to overcome
several technical obstacles: the d
=
√
2 barrier, and finding a topological inner product.
These both end up being solved by allowing “loops” to branch, so the degrees of freedom
are best called nets.
A very exciting mathematical byproduct is that using quantum topology we find new
algebraic proofs of the golden identity for the chromatic polynomial. We also find many
generalizations.
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These nets can be described using the SO(3) Birman-Murakami-Wenzl algebra:
= q
=
1
q
q 1
q
Here the lines are “spin-1” versions of the “spin-1/2” Temperley-Lieb lines. Note there are
3 possibilities on the right, i.e. 1 ⊗ 1
= 0 ⊕ 1 ⊕ 2.
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The generalization of d-isotopy for spin-1 lines depends on a parameter
Q = q + q −1 + 1. Graphically, it is
= Q
1
= (Q 1)
= (Q 2)
+
plus a three-strand relation.
To get Fibonacci anyons, one still has q
= eπi/5 , so closed loops still get weight
Q − 1 = q + q −1 = τ .
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Such loops satisfy a property understood long before BMW, or even Temperley-Lieb.
Weights obeying this generalization of d-isotopy have have exactly the same properties
as the chromatic polynomial χQ .
When Q is an integer, χQ is the number of coloring each region with Q colors such that
adjacent regions have different colors. For Q not an integer, the chromatic polynomial
χQ is defined recursively via the contraction-deletion relation.
1
2
3
3
3
3
2
3
2
2
Think of these loops as domain walls in the low-temperature expansion of the Q-state
Potts model.
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The chromatic algebra satisfies
Fendley and Krushkal
1.
d-isotopy with closed non-intersecting loops getting a weight Q − 1.
2. Tadpoles get vanishing weight.
=0
3. Crossings can be rewritten in terms of pairs of trivalent vertices (or vice versa) via
=
+
=
+
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We find models with deconfined Fibonacci anyons by exploiting a miracle called the
golden identity.
Tutte 1969
For branching lines on the sphere with only trivalent vertices:
χτ +2
¢2
τ + 2 3Ntri /2 ¡
=
τ
χτ +1
4
τ
where Ntri is the number of trivalent vertices.
Incidentally, this identity gives a 3.618.. color theorem.
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Graphically,
= τ 3/2
The two types of blue line (solid and dashed) are independent.
Physically, this means that the weight of each domain-wall configuration in the the
low-temperature expansion of the Potts model with Q
√
= τ + 2 = (5 + 5)/2 is the
square of the weight of the same configuration in the Potts model with Q = τ + 1 (and
times an extra weight per vertex).
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This allows construction of a quantum “loop” model with non-abelian braiding, the
quantum Potts net.
Fendley and Fradkin; Fidkowski, Freedman, Nayak, Walker and Wang; see also Levin and Wen
The degrees of freedom are branching nets with only trivalent vertices allowed, e.g.
One creates two Fibonacci anyons by cutting one of the strands, i.e. each anyon
corresponds to the end of a strand.
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The new proof of the golden identity using quantum topology exploits the chromatic
algebra plus two special properties valid at special values of Q.
1. The Jones-Wenzl projector exists when q is a root of unity. When q
Q = τ + 1, setting it to zero gives
τ
=
+
This can be viewed as an identity for chromatic polynomials.
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= eiπ/5 , i.e.
2. Level-rank duality: Here we exploit O(3)4
↔ O(4)3 , and the fact that as algebras
O(4) = O(3) × O(3).
=
=
τ+1
=
τ
+ τ
=
τ2
−
−
Each type (solid and dashed) of blue line behaves as a independent non-crossing
loop! Each loop receives a weight d
= τ.
The golden identity can now be proved by drawing pictures!
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Conclusions
• New identities for the chromatic polynomial can be derived using the Jones-Wenzl
projector when Q = 2 + 2 cos(2πj/(k + 2)) and j ≤ k are positive integers. For
j = 1, these are the Beraha numbers.
• Models with non-abelian braiding have been constructed. Because they are gapped,
there is hope that they might be realized in frustrated magnets.
• Topological quantum computation is not likely in the near future, but you never know...
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