Topological Quantum Computation II
Eric Rowell
October 2015
QuantumFest 2015
Topological Quantum Computation
Math
Physics
TQC
Computer
Science
Everything Else
Fundamental Questions
I
Distinguish: indecomposable anyons
I
Classify: (models) by number of colors.
I
Detect: non-abelian anyons
I
Detect: universal anyons
I
3-dimensional generalizations?
Wilson Loops
Define Sab for a, b ∈ L:
C
a
b
C
I
Sa,b ∈ C
I
det(S) 6= 0
I
columns ⊥,
distinguishes anyon
types.
Fusion rules
The dimension N(a, b, c) of
H(P; a, b, c):
If b
a = a, Na symmetric
a
b
Example
(Fibonacci)
N0 =
c
1 0
0 1
0 1
N1 =
1 1
provides fusion matrices:
(Na )c,b := N(a, b, c) for particles a ∈ L
Classify/Constraints
Question
How many fusion rules with r = |L|?
Use constraints, for example:
d
d
=
m
a
b
c
I
H(P; a, b, c) = CN(a,b,c)
I
Braiding:
N(a, b, c) = N(b, a, c)
I
compute dimensions:
gluing+ disjoint union
I
Na N c = Nc Na
m
a
b
c
Rank-Finiteness
Theorem (Bruillard, Ng, R, Wang: J. Amer. Math. Soc.)
There are finitely many models with fixed r = |L|.
Proof is by algebra and number theory.
Complete classification known up to |L| = 5.
|L|
Models
1
Vec
2
Fib, semion
3
Z3 , PSU(2)7 , Ising
4
products, Z4 , PSU(2)9
5 Z5 , PSU(2)11 , SU(3)4 /Z3 , SU(2)4
Topological Spin
Each anyon has a topological spin, which may distinguish them
θa
a
a
where θa = e 2πiha with ha ∈ Q.
Bosonic: ha = 0; Fermionic: ha = 1/2; anyonic: any ha . Related
to Dehn twist on torus.
Examples
Example
Example
Fibonacci: L = {1, f }
f ×f =1+f.
S=
1√
1+ 5
2
and θf = e 4πi/5 .
√ !
1+ 5
2
−1
Ising: L = {1, σ, ψ}
σ × σ = 1 + ψ, σ × ψ = σ,
ψ × ψ = 1.
√


2
1
√1
√
S = 2
0
− 2
√
1 − 2
1
and θσ = e πi/8 , θψ = −1.
Algebraic Constraints
Set Tij = δij θi , define dj := S0j , D 2 :=
(S, T ) satisfy
P
j
dj2 , p± :=
P
j
dj2 θj±1 .
t
1. S = S t , SS = D 2 Id, T diagonal, ord(T ) = N < ∞
2N
2. (ST )3 = p+ S 2 , p+ p− = D 2 , pp−+
=1
P S S S
3. Nijk := a iaD 2jada ka ∈ N
P
4. θi θj Sij = a Nik∗ j dk θk where Nii0∗ uniquely defines i ∗ .
n
P
5. νn (k) := D12 i,j Nijk di dj θθji
satisfies: ν2 (k) ∈ {0, ±1}
6. Q(S) ⊂ Q(T ), AutQ (Q(S)) ⊂ Sr , AutQ(S) (Q(T )) ∼
= (Z2 )k .
7. Prime (ideal) divisors of hD 2 i and hNi coincide in Z[ζN ].
Braid group representations
Bn acts on state spaces:
I
I
Fix anyons a, b
Braid group acts linearly:
Bn y H(D 2 \ {zi }; a, · · · , a, b)
by particle exchange
i
i+1
i
Non-abelian Anyons
Definition
a ∈ L is a non-abelian anyon if particle exchange on
H(D 2 ; a, a, a, i) (for some i) is a non-abelian group.
a
a
a
i
Quantum Dimensions
Definition
Let dim(a) be the maximal eigenvalue of Na . Alternatively,
dim(a) = S0,a .
Fact
1. dim(a) ∈ R
2. dim(a) ≥ 1
3. dim(a) dim(b) =
P
c
N(a, b, c) dim(c)
4. dim(a) > 1 implies Degeneracy: dim H(D 2 ; a, a, a, i) > 1 (for
some i).
Degeneracy implies Non-Abelian Statistics
If dim(X ) > 1 there is a Y 6= 1 with N(X , X , Y ) 6= 0.
Y
IF σ1 σ2 σ1−1 σ2−1 = Id then
X
Y
Y
X
X
Y
6= 0
= α
= γ
X
1
= 0
1
X
1
X
X
X
Universal Anyons
Question
When does an anyon a provide universal computation models?
This means: simulate QCM. Mathematically: when does particle
exchange on H(D 2 , a, . . . , a, i) simulate a universal gate set?
Example
√
Fibonacci dim(a) = 1+2 5 is
universal: braid group Bn image
is dense in SU(Fn ) × SU(Fn−1 )
Example
√
Ising dim(a) = 2 is not
universal: braid group Bn image
is a finite group.
Property F conjecture
2-dimensional B3
Fibonacci:
" −4iπ/5
e
σ1 7→
0
" 3iπ/5
e
σ2 7→
−e 3iπ/5
rep from
e −4iπ/5
#
e 3iπ/5
0
,
#
e −4iπ/5
Ising: localized
with

1
0 0

0
1 1
R = √12 
 0 −1 1
−1 0 0
related to Bell states.

1
0

0
1
Conjecture
Anyon a is universal if, and only if, dim(a)2 6∈ Z.
Remark
We expect: Universal anyons have “hard” classical computational
complexity, whereas non-universal anyons have “easy” classical
computational complexity. (assuming P 6= NP...)
3-dimensional materials
I
Point-like particles in R3
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Point-like particles in R3
loop-like particles?
…
Two operations:
Loop interchange
si :
↔
and Leapfrogging (read upwards):
σi :
···
σi =
1
···
i
···
si =
1
n
i +1
···
i
i +1
n
The Loop Braid Group LB n is generated by
s1 , . . . , sn−1 , σ1 , . . . , σn−1 satisfying:
Braid relations:
(R1) σi σi+1 σi = σi+1 σi σi+1
(R2) σi σj = σj σi if |i − j| > 1
Symmetric Group relations:
(S1) si si+1 si = si+1 si si+1
(S2) si sj = sj si if |i − j| > 1
(S3) si2 = 1
Mixed relations:
(M1) σi σi+1 si = si+1 σi σi+1
(M2) si si+1 σi = σi+1 si si+1
(M3) σi sj = sj σi if |i − j| > 1
PRL 113, 080403 (2014)
PHYSICAL REVIEW LETTERS
week ending
22 AUGUST 2014
Braiding Statistics of Loop Excitations in Three Dimensions
Chenjie Wang and Michael Levin
James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
(Received 31 March 2014; published 19 August 2014)
While it is well known that three dimensional quantum many-body systems can support nontrivial
braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we
argue that a more fundamental quantity is the statistical phase associated with braiding one loop α around
another loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context of
ðZN ÞK gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model
with ðZN ÞK symmetry. We find that different short-range entangled bosonic states with the same ðZN ÞK
symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop
braiding statistics.
DOI: 10.1103/PhysRevLett.113.080403
Introduction.—A powerful way to characterize the topological properties of two dimensional gapped quantum
many-body systems is to examine their quasiparticle braiding statistics [1]. Thus, it is natural to wonder: what is the
analogous quantity that characterizes three dimensional
(3D) systems? The simplest candidate—3D quasiparticle
statistics—is of limited use since 3D systems can only
support bosonic and fermionic quasiparticles. On the other
hand, 3D systems can support much richer braiding statistics
between particlelike excitations and looplike excitations
[2–4] or between two looplike excitations [5–7]. Thus,
one might guess that particle-loop and loop-loop braiding
statistics are the natural generalizations of quasiparticle
statistics to three dimensions.
In this Letter, we argue that this guess is incorrect:
particle-loop and loop-loop braiding statistics do not fully
capture the topological structure of 3D many-body systems.
Instead, more complete information can be obtained by
considering a three-loop braiding process in which a loop α
is braided around another loop β, while both are linked with
a third loop γ (Fig. 1). We believe that three-loop braiding
statistics is one of the basic pieces of topological data that
describe 3D gapped many-body systems, and much of this
Letter is devoted to understanding the general properties of
this quantity. Also, as an application, we show that threeloop statistics can be used to distinguish different shortrange entangled many-body states with the same (unitary)
symmetry—i.e., different symmetry-protected topological
(SPT) phases [8–10]. The latter result shows that the
braiding statistics approach to SPT phases, outlined in
Ref. [11], can be extended to three dimensions.
Discrete gauge theories.—For concreteness, we focus
our analysis on a simple 3D system with looplike excitations, namely lattice ðZN ÞK gauge theory [12]. More
specifically, we consider a 3D lattice boson model built
out of K different species of bosons, where the number of
PACS numbers: 05.30.Pr, 03.75.Lm, 11.15.Ha
system has a ðZN ÞK symmetry. We suppose that the ground
state of the boson model is gapped and short-range
entangled—that is, it can be transformed into a product
state by a local unitary transformation [13]. We then
imagine coupling such a lattice boson model to a ðZN ÞK
lattice gauge field [14].
In general, these gauge theories contain two types of
excitations: pointlike “charge” excitations which carry
gauge charge, and stringlike “vortex loop” excitations
which carry gauge flux. The most general charge excitations can carry gauge charge q ¼ ðq1 ; …; qK Þ where each
component qm is an integer defined modulo N. The most
general vortex loop can carry gauge flux ϕ ¼ ðϕ1 ; …; ϕK Þ
where ϕm is a multiple of 2π=N. In fact, since we can
always attach a charge to a vortex loop to obtain another
vortex loop, a general vortex loop excitation carries both
flux and charge.
Let us try to understand the braiding statistics of these
excitations. In general, there are three types of braiding
processes we can consider: processes involving two
charges, processes involving a charge and a loop, and
processes involving multiple loops. Clearly, the first type of
process cannot give any statistical phase since the charges
are excitations of the short-range entangled boson model
and, therefore, must be bosons. On the other hand, the
FIG. 1. (a) Three-loop braiding process. The gray curves show
the paths of two points on the moving loop α. (b) A top view of
the braiding process within the plane that γ lies in. (c) A torus Ωα
is swept out by α during the braiding. Loop β (dashed circle) is
Question
Do such materials exist in nature?
Maybe...
Thank you!
References
I
Chang,R.,Plavnik,Sun,Bruillard,Hong: 1508.00005 (J. Math.
Phys.)
I
Kadar,Martin,R.,Wang: 1411.3768 (Glasgow J. Math.)
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R.,Wang: 1508.04793 (preprint)
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Bruillard,Ng,R.,Wang: 1310.7050 (J. Amer. Math. Soc.)
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Naidu,R.: 0903.4157 (J. Alg. Rep. Theory)