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C6.1
INTRODUCTION TO TOPOLOGICAL
QUANTUM COMPUTING
A. J. Leggett
Department of PhysicsUniversity of Illinois at Urbana-Champaign
CIFAR Winter School
Whistler, BC
April 5, 2011
based on work supported by the National Science Foundation
under grant no. NSF-DMR09-06921
4/1/2011
C6.2
TOPOLOGICAL QUANTUM MEMORY/COMPUTING
Qubit basis. | â â©, | â â©
|Κ⩠= α| â â© + ÎČ | â â©
To preserve, need (for ârestingâ qubit)
ËË 1 H â in | â â©, | â â© basis
12 1 11 22 2Ë Ë Ë( 0 " ": " ")H T H H T= â â â = â â â
on the other hand, to perform (single-qubit) operations, need to impose nontrivial
âwe must be able to do something Nature canât.
(ex: trapped ions: we have laser, Nature doesnât!)
Topological protection:
would like to find dâ(>1) dimensional Hilbert space within which (in absence of intervention)
Ë .H
/ËË 1( .) ( ) Lconst o eH Οâ+= iiii
size ofsystem
microscopic lengthHow to find degeneracy?
Suppose â two operators 1 2Ë Ë, s.t.Ω Ω
2
1 2 1
1 21
Ë Ë Ë Ë[ , ] [ , ] 0
Ë Ë Ë[ but
Ë Ë(and , commutes with b.c's)
, ] 0 (and | > ) 0
H H
Ï
Ω = Ω =
Ω Ω â
Ω
Ω â
Ω
C6.3
EXAMPLE OF TOPOLOGICALLY PROTECTED STATE:
FQH SYSTEM ON TORUS (Wen and Niu, PR B 41, 9377 (1990))
Reminders regarding QHE:2D system of electrons, B â„ planeArea per flux quantum = (h/eB) â df.
âFilling fractionâ ⥠no. of electrons/flux quantum âĄ ÎœâFQHâ when Îœ = p/q incommensurate integers
Argument for degeneracy: (does not need knowledge of w.f.)can define operators of âmagnetic translationsâ
1/ 2( / )M
eB⥠â âmagnetic lengthâ
( 100 for B = 10 T)M A âŒ
(⥠translations of all electrons through a(b) à appropriate phase factors). In general
In particular, if we choose no. of flux quanta
then commute with b.c.âs and moreover
Ë Ë( ), ( )x y
T Ta b
Ë Ë[ ( ), ( )] 0x y
T T â a b
2
1 21 2 ( / 2 )/ , /s s L LN N Ï== = a L b L
1 2 2 1Ë Ë Ë Ë exp 2T T T T iÎœÏ= â
1 2Ë Ë,T T
1 2
1 2
Ë Ë Ë Ë[ , ] [ , ] 0 (*)Ë Ëso since [ , ] 0
T H T H
T T
= =
â
must â more than 1 GS (actually q).
Corrections to (*): suppose typical range of (e.g.) external potential V(r) is o, then since |Ï>âs oscillate on scale osc,
1 2Ë| | ~ exp / ~ exp /
Ë( const. 1)o oscH LÏ Ï ÎŸâ â
+
But the o. of m. of a and b is ·( /L)« , and â 0 for Lââ. Hence to a very good approximation,
C6.4TOPOLOGICAL PROTECTION AND ANYONS
Anyons (df): exist only in 2D
Nonabelian statistics* is a sufficient condition for topological protection:(a) state containing n anyons, n â„ 3:
(bosons: α = 1, fermions: α = œ)
abelian if
12(1,2) exp(2 ) (2,1) Ë (1,2)TiÏ Î±Îš = Κ ⥠Κ
12 23 23 12Ë Ë Ë Ë (ex: FQHE)T T T T=
nonabelian if 12 23 23 12 , Ë Ë i.e., iË fËT T T Tâ
1
2
3
1 2
("braiding statistics")Ï Ïâ
1Ï2Ï
[not necessary, cf. FQHE on torus]
12 23
12 23
Ë Ë Ë Ë[ , ] [ , ] 0
Ë Ë[ , ] 0
T H T H
T T
= =
â
â space must be more than 1D.
(b) groundstate:
GS GS
create anyons annihilate anyons
annihilation process inverse of creation â
GS also degenerate. *plus gap for anyon creation
C6.5
THE BRAID GROUP
i+2
i+1
i
i-1
â
=
(1) and (2) may be taken as the definition of the braid group
C6.6
SPECIFIC MODELS WITH TOPOLOGICAL PROTECTION
1. FQHE on torus
Obvious problems:
(a) QHE needs GaAsâAlGaAs or Si MOSFET: how to âbendâinto toroidal geometry?
QHE observed in (planar) graphene (but not obviously âfractionalâ!): bend C nanotubes?
(b) Magnetic field should everywhere have large compt â„ to surface: but div B = 0 (Maxwell)!
2. Spin Models (Kitaev et al.) (adv: exactly soluble)
(a) âToric codeâ model
âLinksâ with spin Âœ on lattice
s
p
ËË Ës p
s p
H A B= â ââ â
Ë ËË Ë,s j p j
j s j p
x zA BΔ Δ
Ï Ï⥠Π⥠Î
Ë Ë(so [ , ] 0 in general)s p
A B â
Problems:
(a) toroidal geometry required (as in FQHE)
(b) apparently v. difficult to generate Hamn physically
(c) does not permit topological quantum computation(only protection)
C6.7
subl. Asubl.
B
y x
z
(b) Kitaev âhoneycombâ model
Particles of spin œ onhoneycomb lattice(2 inequivalent sublattices,A and B)
SPIN MODELS (cont.)
Ë Ë Ë Ë Ë Ë Ëx x y y z z
x j k y j k z j ky linkx l zi linksks sn
H J J JÏ Ï Ï Ï Ï Ïâââ
= â â ââ â â
| | | | | |, | | | | | |,| | | | | | and 0x y z y z x
z x y
J J J J J J
J J J
†+ †+†+ â H
nb: spin and space axes independent
Strongly frustrated model, but exactly soluble.*
Sustains nonabelian anyons with gap provided
(in opposite case anyons are abelian + gapped)
Advantages for implementation:
(a) plane geometry (with boundaries) is OK
(b) bilinear in nearest-neighbor spins
(c) permits partially protected TQC
H
* A. Yu Kitaev, Ann. Phys. 321,2 (2006)H-D. Chen and Z. Nussinov, cond-mat/070363 (2007)
magnetic field
C6.8
Some suggested implementations of the Kitaev
honeycomb model:
(1) Optical lattices (Duan et al. PRL 91, 090402
(2003))
Can create 2D honeycomb lattice by suitable
arrangements of lattices. By appropriate
adjustments of laser polarizations, can make both t
and U dependent on âspinâ (hyperfine index). Then
to 2nd order in t (Mott) get spin-dependent n.n.
interaction.
Obvious problems:
(a) need for background trapping potential
(b) ultralong spin relaxn. times
(2) Polar molecules in optical lattices
(3) Josephson junction arrays
(4) Other solid-state lattice systems âŠ
Problems: mostly massive âengineeringâ
So: different setup, occurring ânaturallyâ?
t λ/2 U
C6.9
The Μ = 5/2 STATE
First seen in 1987: to date the only even-denom. FQHE
state reliably established (some ev. for Μ = 19/8). Quite robust:
ÎŁxy /(e2/h) =5/2 to high accuracy, excluding e.g. odd-
denominator values Îœ = 32/13 or 33/13, and ÎŁxy vanishes
within exptl. accuracy. The gap â ~ 500 mK.
WHAT IS IT?
If spin - polarized (probable), it is the n = 1 analog of Μ = 1/2.
However, the actual Μ = 1/2 state does not correspond to a
FQHE plateau. In fact the CF (composite fermion) approach
predicts that for this Μ
N Ïeff = NÏ - 2Ne = 0
and hence the CFâs behave as a Fermi liquid: this seems to be consistent with expt. If LLL â,â both filled, this argt. should apply equally to Îœ = 5/2 (since (Ne/NÏ)eff = Âœ).
So what has gone wrong?One obvious possibility â :Cooper pairing of composite fermions!since spins Ç, must pair in odd â l state, e.g. p-state.
C6.10
THE âPFAFFIANâ ANSATZ
Consider the Laughlin ansatz formally corresponding to
Μ = 1/2:
Ï(L) N = | |i<j (zi â zj)
2 exp-ΣI | zi|2/fl2m (zi ⥠x + iy= electron coord.)
This cannot be correct as it is symmetric under i j.
So must multiply it by an antisymmetric function. On the
other hand, do not want to âspoilâ the exponent 2 in
numerator, as this controls the relation between the LL
states and the filling.
Inspired guess (Moore & Read, Greiter et. Al): (N = even)
Pf(f(ij)) ⥠f(12)f(34)âŠ-f(13)f(24). ⊠+ âŠ. (⥠Pfaffian)
âantisymmetric under ij
Is this consistent with expt.?
Κ
Κ Pf
!"
C6.11
MOST PROMISING CANDIDATE FOR TQC IN
âNATURALâ SYSTEMS: MAJORANA FERMIONS
System: 2D âp + ipâ Fermi syperfluid
(? 3He-A thin slabs, Sr2RuO4�)
In such a system, half-quantum vorticesmay occur.
Near one of these, BdG (Bogoliubovâde Gennes) eqns
possess a single solution with the properties
E = 0, u(r) = Μ*(r)
Such solultions are called Majorana fermions*
Crudely speaking,
2 Majorana fermions (on 2 separated vortices) â 1 ârealâ (âDirac-Bogoliubovâ) fermion.
TQC can be achieved by âbraidingâ vortices
with/without M.F.âs
*also believed to occur in s-wave superconductor next
to topological insulator. Related excitations in # $%â
QH state.
C6.12
BRIEF REVIEW OF âESTABLISHED WISDOMâON p + ip FERMI SUPERFLUIDS
For a general spin-1/2 Fermi superfluid, OP df. by
p + ip:
thus if âα taken as z-axis, F. T. is
order parameter
â â "anomal( , ) ( ) ( ) ous average"F r r r rαÎČ Î± ÎČÏ ÏâČ âČ⥠â
( , ) ( , )F r r F r rαÎČ Î±ÎČ Î±ÎŽâČ âČ= (ESP)
âequal spin pairingâ
( , ) ( , ) ( ) ( )F r r F R F R fαÎČ Î± α Î±Ï ÏâČ âĄ â
COM rel. coord.
( ) sin expf iα α Î±Ï Îž ÏâĄ
breaks TRI
( ) â â x y
F p p i pp ipα = â+ +
âα Ïα
Ξα
C6.13
,z x iy= +
Standard ansatz for MBWF (COMâs of â, â at rest):
where, if z-axis chosen along âα.
/2( ) | vacN
k k kc a a
α α
αα
+ +â
â â
Κ ⥠â©
Κ = Κ
â
Κ
k
( ) ex(| | p)x y kk
c f ik ikkα Ï= â +
âŒ
1 21 2 3 4 5 6
1 1 1( ... ) const. ....
N Nz z z
z z z z z z
Κ = â â
â â â A
Cf MR ansatz for 5 / 2 stateQHΜ =
âPfaffianâ
Moore-Read
Note dependence on Ïk extends to whole Fermi sea
Some properties of âstandardâ ansatz:
1. Ang. Momentum alongeven in limit
2. For 2D (âplanarâ) case ( â„ plane), put then for all coord-space MBWF is of form
( )/ 0( /2 )F
N Eαα â â â
0 or .c
T Tâ â â
α,
i jz z Οâ
pair radius
(ESP)
(p + ip)
C6.14
Strontium Ruthenate: Sr2RuO4
History
Superconductivity in cuprates up to ~ 150K
Typical (original) cuprate:
La2-x Bax CuO4 (TC ~ 40K)
Quasi -2D CuO2 planes appear to be essential to high â TC
superconductivity. How essential is the Cu? Try replacing it: Ag, AuâŠâŠ doesnât work, but:
Cu (Z = 29 ) : [Ar + 3d104s1 â 3d9
Ru (Z = 44 ) : [Kr] + 4d75s1 â 4d4
Normal-state props quite dissimilar
La2-x Bax CuO4 has TC ~ 40K
Sr2RuO4 has TC of ?
Side view
Cu O
La(Ba)
Top view
Side view
Ru O
Sr
Top view
C6.15
Experimental properties of Sr2RuO4*Normalphase
Below~25K,appearstobehaveasstronglyanisotropicFermi
liquid(nb:cuprates quitedifferent)
CV ~ÎłT +ÎČT3 Ï ~const.
electrons phonons
Ï ~A+BT2
bothinab-planeandÏ ~A+BT2
alongc-axis(char.ofcoherent
(Blochwave)transportlimitedbye- - e-
Umklapp scattering).av.small(~1”Ω cm)
âsamplesverypure.
However, Ïc/Ïab ~103 (comparabletocuprates)
Bandstructure:
Expt.(dHvA,Shubnikov-DeHaas)andtheory(LDA)agree:
Fermisurfaceconsistsof3strongly2Dsheets:α (hole-like),
ÎČ,Îł (electronlike)
kF (Ă -1) 0.3 0.6 0.75
m*/m 3.3 7.0 16.0
m*/mb 3.0 3.5 5.5
Ï ~const.insupD. stateâ triplet,ESP
*Mackenzie&Maeno,RMP75757575,1(2003)
a b plane
c
indication of strong correlations
Deviation from perfectcylinder ~ 0.1 â 1%
C6.16
Sr2RuO4
Al
is indeed of ESP form: thenLM N:OO% â Q N (ââ ST ââ)
orbital wf of pairs
o
h
e
C6.17
ESTABLISHED WISDOM (cont.)
Half-quantum vortices (âHQVâ)Should occur in any ESP Fermi superfluid, provided
coupling between ĂĂ and ââ sufficiently weak. e.g. (neutral case):vortex in ââ components, nothing in ââ component, i.e.âââ â exp iΊ , âââ â const. (âhalf-quantumâ vortex)Note, however, that quantization condition for ââ pair velocityis still
Can tolerate Majorana fermions.
What about charged case?
At current only in ââ component â total j â 0.However, for âââsare involved :
Total trapped flux = Ί0/2 = nh/4e.(Hence, vortex with Dirac (non-Majorana) fermion circlinganother picks up phase Ï/2, not Ï as for BCS)Note: HQV in charged system carries circulating spin current as
rââ â energetically disadvantaged relative to simple (h/2e) vortex.
,Lr λ
20
~/ 2
sK h mdl
Ï Ï â
⥠â =â«
,Lr λ>
0
0tot
j j
j
â ââ =
=
â1/2
C6.18
J. Jang et al. Science 331, 186 (2011)
C6.19
MAJORANA FERMIONS ON HQVâs
single Dirac-Bogoliubov fermion with E=0
First important point:
impossible to tell, by any local meast. whether DC
fermion is present or absent.
Second important point:
by appropriate braiding of vortices, can realize (for 2
vortices) nontrivial relative phase factors of the âoccupiedâ
and âunoccupiedâ states, and (for N>2 vortices) nonabelian
operations.
For 2 vortices effect of 1 2 (interchange)
= effect of (encirclement) provided we ignore
change of OP at position of 1, i.e. consider only âÏs = 2Ïfor 2.
For a DB quasiparticle,
â Ï = Âœ âÏs = Ï
For an M.F., only half is localized on 2 ââÏ= ÂŒ âÏs = Ï/2
âchange in phase of
supercond *. OP
MF1 MF2
1 2
â
2-1
C6.20
MAJORANA FERMIONS ON HQVâs
BRAIDING OF ANYONS
From the arguments of Ivanov*, if an E = 0 fermion is
âsharedâ by 2 vortices and they are exchanged, MBWF changed by a factor of exp iÏ/2 ⥠i, while if there is no
fermion, this factor is 1. Hence, for the single-qubit system
formed by 2 vortices
Case of 4 vortices:
At first sight, 2 qubits, e.g. associated with (1,2) and
(3,4). â 4D Hilbert space. Then:
but
However this is a bit misleading, because all operations
preserve parity of state. (as do all âreal-lifeâ physical
operations). Hence, preferable to fix the parity and regard 4-
anyon system as single qubit associated with e.g. anyons 1
and 2: e.g. for odd N
*PRL 86 (2001)
XY% 1 00 Z ST exp Z [4 O]
XY% 1 00 Z
1Y%, XY^_1Y 1 00 Z ^_
XY%^ 12
1 00 1
0 `1`1 0
0 `1`Z 0
1 00 1
121 4O]aO]b%entangling
|0d = |no MF on (1,2), MF on (3,4)d|1d = MF on (1,2), no MF on (3,4)d
C6.21
Case of 4 HQVâs
4-dime. E = 0 manifold, corresponding occupation or
nonoccupation of (1,2) and (3,4)
But (claim!) can equally well pair differently, e.g.
If the occupation states in this basis are single
superpositions of those in original basis, then this just
corresponds to a basic change in the 4D manifold.
We know this in original basis
so to find effect of
write out in (13)=(24) basis and transform.
1 2
3
4
1
3
2 4
1
2
3
4
1 2
occ un
âč Z 00 1 occun g% h1
Y^_
(1,2)â (3,4) basis
(1,2)â (3,4) basis