Topological Quantum Computation IIrowell/MtrryTectalk2.pdf · Topological Quantum Computation TQC Math Physics Computer Science Everything Else

Topological Quantum Computation II

Eric Rowell

October 2015QuantumFest 2015

Topological Quantum Computation

TQC

Math Physics

Computer

ScienceEverything 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:

a b

C

C

I Sa,b ∈ CI det(S) 6= 0

I columns ⊥,distinguishes anyontypes.

Fusion rules

The dimension N(a, b, c) ofH(P; a, b, c):

a

c

b

provides fusion matrices:(Na)c,b := N(a, b, c) for particles a ∈ L

If a = a, Na symmetric

Example (Fibonacci)

N0 =

(1 00 1

)N1 =

(0 11 1

)

Classify/Constraints

QuestionHow many fusion rules with r = |L|?Use constraints, for example:

a a

m

d

c b

m

b c

d

=

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 NaNc = NcNa

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)95 Z5,PSU(2)11, SU(3)4/Z3,SU(2)4

Topological Spin

Each anyon has a topological spin, which may distinguish them

a

θa

a

where θa = e2πiha with ha ∈ Q.Bosonic: ha = 0; Fermionic: ha = 1/2; anyonic: any ha. Relatedto Dehn twist on torus.

Examples

Example

Fibonacci: L = {1, f }f × f = 1 + f .

S =

(1 1+

√5

21+√5

2 −1

)

and θf = e4πi/5.

Example

Ising: L = {1, σ, ψ}σ × σ = 1 + ψ, σ × ψ = σ,ψ × ψ = 1.

S =

1√

2 1√2 0 −

√2

1 −√

2 1

and θσ = eπi/8, θψ = −1.

Algebraic Constraints

Set Tij = δijθi , define dj := S0j , D2 :=∑

j d2j , p± :=

∑j d2

j θ±1j .

(S ,T ) satisfy

1. S = S t , SSt

= D2Id , T diagonal, ord(T ) = N <∞

2. (ST )3 = p+S2, p+p− = D2,(

p+p−

)2N= 1

3. Nkij :=

∑aSiaSjaSkaD2da

∈ N

4. θiθjSij =∑

a Nki∗jdkθk where N0

ii∗ uniquely defines i∗.

5. νn(k) := 1D2

∑i ,j Nk

ij didj

(θiθj

)nsatisfies: ν2(k) ∈ {0,±1}

6. Q(S) ⊂ Q(T ), AutQ(Q(S)) ⊂ Sr , AutQ(S)(Q(T )) ∼= (Z2)k .

7. Prime (ideal) divisors of 〈D2〉 and 〈N〉 coincide in Z[ζN ].

Braid group representations

Bn acts on state spaces:

I Fix anyons a, bI Braid group acts linearly:

Bn y H(D2 \ {zi}; a, · · · , a, b)

by particle exchange

i i+1 i

Non-abelian Anyons

Definitiona ∈ L is a non-abelian anyon if particle exchange onH(D2; a, a, a, i) (for some i) is a non-abelian group.

i

a a a

Quantum Dimensions

DefinitionLet dim(a) be the maximal eigenvalue of Na. Alternatively,dim(a) = S0,a.

Fact

1. dim(a) ∈ R2. dim(a) ≥ 1

3. dim(a) dim(b) =∑

c N(a, b, c) dim(c)

4. dim(a) > 1 implies Degeneracy: dimH(D2; a, a, a, i) > 1 (forsome i).

Degeneracy implies Non-Abelian Statistics

If dim(X ) > 1 there is a Y 6= 1 with N(X ,X ,Y ) 6= 0.

1 X

Y X

= α

X

Y X

6= 0

IF σ1σ2σ−11 σ−12 = Id then

1 X

Y X

= γ

1 X

Y X

= 0

Universal Anyons

QuestionWhen does an anyon a provide universal computation models?This means: simulate QCM. Mathematically: when does particleexchange on H(D2, a, . . . , a, i) simulate a universal gate set?

Example

Fibonacci dim(a) = 1+√5

2 isuniversal: braid group Bn imageis dense in SU(Fn)× SU(Fn−1)

Example

Ising dim(a) =√

2 is notuniversal: braid group Bn imageis a finite group.

Property F conjecture

2-dimensional B3 rep fromFibonacci:

σ1 7→

[e−4iπ/5 e−4iπ/5

0 e3iπ/5

],

σ2 7→

[e3iπ/5 0

−e3iπ/5 e−4iπ/5

]Ising: localized with

R = 1√2

1 0 0 10 1 1 00 −1 1 0−1 0 0 1

related to Bell states.

Conjecture

Anyon a is universal if, and only if, dim(a)2 6∈ Z.

RemarkWe expect: Universal anyons have “hard” classical computationalcomplexity, whereas non-universal anyons have “easy” classicalcomputational complexity. (assuming P 6= NP...)

3-dimensional materials

I Point-like particles in R3

I Point-like particles in R3

loop-like particles?

…

Two operations:

Loop interchange si : ©↔©and Leapfrogging (read upwards):

σi :

σi =

1

· · ·

i i + 1

· · ·

n

si =

1

· · ·

i i + 1

· · ·

n

The Loop Braid Group LBn is generated bys1, . . . , sn−1, σ1, . . . , σn−1 satisfying:Braid relations:

(R1) σiσi+1σi = σi+1σiσi+1

(R2) σiσj = σjσi if |i − j | > 1

Symmetric Group relations:

(S1) si si+1si = si+1si si+1

(S2) si sj = sjsi if |i − j | > 1

(S3) s2i = 1

Mixed relations:

(M1) σiσi+1si = si+1σiσi+1

(M2) si si+1σi = σi+1si si+1

(M3) σi sj = sjσi if |i − j | > 1

Braiding Statistics of Loop Excitations in Three Dimensions

Chenjie Wang and Michael LevinJames 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 nontrivialbraiding statistics between particlelike and looplike excitations, or between two looplike excitations, weargue that a more fundamental quantity is the statistical phase associated with braiding one loop α aroundanother loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context ofðZNÞK gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson modelwith ðZNÞK symmetry. We find that different short-range entangled bosonic states with the same ðZNÞKsymmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loopbraiding statistics.

DOI: 10.1103/PhysRevLett.113.080403 PACS numbers: 05.30.Pr, 03.75.Lm, 11.15.Ha

Introduction.—A powerful way to characterize the topo-logical properties of two dimensional gapped quantummany-body systems is to examine their quasiparticle braid-ing statistics [1]. Thus, it is natural to wonder: what is theanalogous quantity that characterizes three dimensional(3D) systems? The simplest candidate—3D quasiparticlestatistics—is of limited use since 3D systems can onlysupport bosonic and fermionic quasiparticles. On the otherhand, 3D systems can support much richer braiding statisticsbetween particlelike excitations and looplike excitations[2–4] or between two looplike excitations [5–7]. Thus,one might guess that particle-loop and loop-loop braidingstatistics are the natural generalizations of quasiparticlestatistics to three dimensions.In this Letter, we argue that this guess is incorrect:

particle-loop and loop-loop braiding statistics do not fullycapture the topological structure of 3Dmany-body systems.Instead, more complete information can be obtained byconsidering a three-loop braiding process in which a loop αis braided around another loop β, while both are linked witha third loop γ (Fig. 1). We believe that three-loop braidingstatistics is one of the basic pieces of topological data thatdescribe 3D gapped many-body systems, and much of thisLetter is devoted to understanding the general properties ofthis quantity. Also, as an application, we show that three-loop statistics can be used to distinguish different short-range entangled many-body states with the same (unitary)symmetry—i.e., different symmetry-protected topological(SPT) phases [8–10]. The latter result shows that thebraiding statistics approach to SPT phases, outlined inRef. [11], can be extended to three dimensions.Discrete gauge theories.—For concreteness, we focus

our analysis on a simple 3D system with looplike excita-tions, namely lattice ðZNÞK gauge theory [12]. Morespecifically, we consider a 3D lattice boson model builtout of K different species of bosons, where the number ofbosons in each species is conserved modulo N so that the

system has a ðZNÞK symmetry. We suppose that the groundstate of the boson model is gapped and short-rangeentangled—that is, it can be transformed into a productstate by a local unitary transformation [13]. We thenimagine coupling such a lattice boson model to a ðZNÞKlattice gauge field [14].In general, these gauge theories contain two types of

excitations: pointlike “charge” excitations which carrygauge charge, and stringlike “vortex loop” excitationswhich carry gauge flux. The most general charge excita-tions can carry gauge charge q ¼ ðq1;…; qKÞ where eachcomponent qm is an integer defined modulo N. The mostgeneral vortex loop can carry gauge flux ϕ ¼ ðϕ1;…;ϕKÞwhere ϕm is a multiple of 2π=N. In fact, since we canalways attach a charge to a vortex loop to obtain anothervortex loop, a general vortex loop excitation carries bothflux and charge.Let us try to understand the braiding statistics of these

excitations. In general, there are three types of braidingprocesses we can consider: processes involving twocharges, processes involving a charge and a loop, andprocesses involving multiple loops. Clearly, the first type ofprocess cannot give any statistical phase since the chargesare excitations of the short-range entangled boson modeland, therefore, must be bosons. On the other hand, the

FIG. 1. (a) Three-loop braiding process. The gray curves showthe paths of two points on the moving loop α. (b) A top view ofthe 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) isenclosed by Ωα.

PRL 113, 080403 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

22 AUGUST 2014

0031-9007=14=113(8)=080403(5) 080403-1 © 2014 American Physical Society

QuestionDo 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.)

I R.,Wang: 1508.04793 (preprint)

I Bruillard,Ng,R.,Wang: 1310.7050 (J. Amer. Math. Soc.)

I Naidu,R.: 0903.4157 (J. Alg. Rep. Theory)