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Quantum Topology, Quantum Physics and Quantum Computing. Zhenghan Wang Microsoft & Indiana Univ. (visiting KITP/CNSI & UCSB) http://www.tqc.iu.edu. Collaborators:. Michael Freedman (MS) Alexei Kitaev (MS & Caltech) Chetan Nayak (MS & UCLA) Kevin Walker (MS) ( Station Q )
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Zhenghan Wang
Microsoft & Indiana Univ.
(visiting KITP/CNSI & UCSB)
http://www.tqc.iu.edu
Michael Freedman (MS)
Alexei Kitaev (MS & Caltech)
Chetan Nayak (MS & UCLA)
Kevin Walker (MS)
(Station Q)
Michael Larsen (Indiana)
Richard Stong (Rice)
Eric Rowell (Indiana)
……
Given W4, c2(TWC)=p1(W)2 H4(W,Z)
• Characteristic Forms and Geometric
Invariants Ann. Math. (1974)
This work, originally announced in [4], grew out of an attempt to derive a purely combinatorial formula for the first Pontrjagin number of a 4manifold. … This process got stuck by the emergence of a boundary term which did not yield to a simple combinatorial analysis. The boundary term seemed interesting in its own right and it and its generalizations are the subject of this paper.
3dim cs form: Tr(AÆ dA+⅔ A3)
The hope was that by integrating the characteristic curvature form (with respect to some Riemannian metric) simplex by simplex, and replacing the integral over each interior by another on the boundary, one could evaluate these boundary integrals, add up over the triangulation, and have the geometry wash out, leaving the sought after combinatorial formula
W4 closed, CraneYetter state sum invariants:
e[2 i (6r2r2)/24r]p1(W), r=3,4,…
Topology Algebra
4dim W4 Integer
3dim M3 Complex number
2dim 2 Vector space
1dim X1 Category
0dim pts 2category
SU(2)bundle over Y$f: Y! BSU(2)
BSU(2)'* [ e4 [ e8 [
dim Y=4, deg(f)2 Z f: Y! S4
indep of a connection A
dim Y=3, pick a W4 s.t. W=Y, a connection
A and an extension A’
cs(A,W,A’)2 R, depending on A’ and W,
but cs(A,W1,A’)cs(A,W2,A’’)2 Z
cs: connections A ! R mod 1
Using path integral (Witten) or quantum groups (ReshetikhinTuraev),
define Zk(M3)=sA e2 i k cs(A)DA
What is Zk()?
A vector space, a typical vector looks like a 3mfd M s.t. M=
(Atiyah, Segal, Turaev, Walker,…)
Atiyah’s Axioms of (2+1)TQFT:
(TQFT w/o excitations and central charge=0)
Surface 2 vector space V()
3manifold M3 a vector Z(M3)2 V(M3)
● V(;) C
● V(1t2) V(1) V(2)
● V(*) V*()
● Z(£ I)=IdV()
● Z(M1[ M2)=Z(M1)¢Z(M2)
Given a closed surface and d2 C,
S()=vector space generated by
isotopy of classes of multicurves
Let V() be S() modulo
1. trivial loop=d
2. a local relation supported on a disk
A Local Relation:
Fix 2n points on the boundary of the disk, and {Di} all different n disjoint arcs connecting the 2n points.
A local relation is a formal equation: ii¢ Di=0.
Given a compact Lie group G, and
a level k, there is a TQFT (anomaly).
For surfaces with boundaries, each boundary component is marked by a label
Each TQFT gives rise to projective representations of the mapping class group of labeled surfaces.
When G=SU(2),=npunctured disk, the resulting reps of Bn are the Jones representations which lead to the Jones polynomial of knots.
● A quantum system whose low energy effective theory is described by a TQFT
●Some features:
Elementary excitations (called quasiparticles or particles) in a topological quantum system are anyons.
In general the vector space V() describes the ground states of a quantum system on , and the rep of the mapping class groups describes the evolutions.
In R3, particles are either bosons or fermions
Worldlines (curves in R3£R) exchanging two identical particles depend only on permutations
=
Statisitcs is : Sn! Z2
In R2, an exchange is of infinite order
Not equal
Braids form groups Bn
Statistics is : Bn! U(1)
If not 1 or 1, but ei, anyons
Suppose the ground states of n identical particles has a basis e1, e2, …, ek
Then after braiding two particles:
e1! a11e1+a21e2+…+ak1ek
.●
●
Particle statistics is : Bn! U(k)
Particles with k>1 are called nonabelian anyons
In general the statistics of a particle with configuration space X is n:1(Cn(X),p0)! U(kn)
E. H. Hall, 1879
On a new action of the magnet on electric currents
Am. J. Math. Vol 2, No.3, 287292
“It must be carefully remembered that the mechanical force which urges a conductor carrying across the lines of the magnetic force, acts, not on the electric current, but on the conductor which carries it”
Maxwell, Electricity and Magnetism
● 1980 K. von Klitzing IQHE
(1985 Nobel)
● 1982 H. Stormer, D. Tsui FQHE
R. Laughlin (1998 Nobel)
quasiparticle with 1/3 electron charge
and braiding statistics (anyons)
Electron system:
e

+
+

B
I
Hall resistance Rxy=1¢ h/e2, with precision 1010
( is the Landau filling fraction)
=1/3 or 2/3 SU(2) TQFT at r=3
(Laughlin)
=5/2 SU(2) TQFT at r=4
(Universal TQC)
=12/5 or 13/5 SU(2) TQFT at r=5
(Universal AQC)
Storage, processing and communicating
information using quantum systems.
Three milestones in QIS:
1.Shor's polytime factoring algorithm (1994)
2. Errorcorrecting code, thus
faulttolerant quantum computing (1996)
3. Security of private key exchange (BB84 protocol)
Given a Boolean map f: {0,1}n! {0,1}n,
for any x2 {0,1}n, represent x as a basis
x>2 (C2) n, then find a unitary matrix U so that U (x>) = f(x)>.
f(x)>
Basis of (C2) n is in11correspondence with nbit strings or 0,1,…,2n1
x>
Factoring is in BQP (Shor's algorithm), but not known in
FP (although Primality is in P).
Given an n bit integer N» 2n
Classically ~ ec n1/3 poly (log n)
Quantum mechanically ~ n2 poly (log n)
For N=2400, classically » billion years
Quantum computer » 1 second
BQP
Ф?
Pspace
♪
☻
P
NP
Can we build a large scale universal QC?
The obstacle is mistakes and errors (decoherence)
Error correction by simple redundancy
0!000, 1! 111
Not available due to the Nocloning theorem:
The cloning map > 0>! >> is not linear.
Faulttolerant quantum computation shows if hardware can be built up to the accuracy threshold ~104, then a scalable QC can be built.
SolutionQuantum Topology
Classifications of TQFTs or anyonic systems
Hamiltonianization of TQFTs (generalizing Jones’ Baxeterization of link invariants)
3. Physics
Search for topological phases of matter
Fix the number of quasiparticle types, there are essentially only finitely many TQFTs.
True for 1,2,3,4 (Rowell, Stong, W.)
Analogues:
2. (L. Bieberbach) Finitely crytallographic groups in each dimension n, n=3 230 crytals
A MTC is a ribbon category with finitely many isomorphism classes of simple objects and a nonsingular Smatrix (a ribbon category is a braided tensor category with compatible duality.)
Given a MTC, there associates a TQFT.
● Use RCFT to show there are only finitely many reps of SL(2,Z) from the S,T matrices
● Using number theory to show there are only finitely many possible twists T and S with the same rep of SL(2,Z)
A pair (V, (C2) n) is a (k, n)code
if for every klocal operator, the following composition is a scalar multiple of idV:
V! (C2) n!(C2) n! V
Given a TQFT, and a triangulation of a surface , then V() can be constructed as the ground states of a local Hamiltonian
on (C2) n which is a (k,n)code
Quantum medium
First Law of Physics (S. Girvin)
Whatever is not forbidden is compulsory
Noncommutative Chern classes???