Quantum Topology, Quantum Physics and Quantum Computing

Zhenghan WangMicrosoft & 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)

Michael Larsen (Indiana)Richard Stong (Rice)Eric Rowell (Indiana) ……

Chern-Simons Theory• Chern Classes: 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 4-manifold. … 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.

3-dim 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, Crane-Yetter state sum invariants:

e[-2 i (-6-r-2r2)/24r]p1(W), r=3,4,…

Chern-Simons Theory

Topology Algebra 4-dim W4 Integer 3-dim M3 Complex number 2-dim 2 Vector space 1-dim X1 Category 0-dim pts 2-category

Gauge Group G=SU(2)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

Quantum Chern-Simons Theory

Using path integral (Witten) or quantum groups (Reshetikhin-Turaev),

define Zk(M3)=sA e2 i k cs(A)DA

What is Zk()? A vector space, a typical vector looks like

a 3-mfd 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() 3-manifold M3 a vector Z(M3)2 V(M3)● V(;) C● V(1 t 2) V(1) ­ V(2) ● V(*) V*()● Z( £ I)=IdV()

● Z(M1[ M2)=Z(M1)¢Z(M2)

Examples of TQFTs

• Z2 homology: V()=C[H1(,Z2)]

= d

= dd=§ 1

Picture TQFTs: 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: i i ¢ Di=0.

Chern-Simons TQFTs

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

Reps of the MCGs

Each TQFT gives rise to projective representations of the mapping class group of labeled surfaces.

When G=SU(2), =n-punctured disk, the resulting reps of Bn are the Jones representations which lead to the Jones polynomial of knots.

Topological quantum system

● A quantum system whose low energy effective theory is described by a TQFT

● Some features: 1) Ground states degeneracy2) No continuous evolution3) Energy gap

Topological quantum system Elementary excitations (called quasi-

particles 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.

Hypothesis:

TQFTs describe the topological properties of quantum media in the thermodynamic limit

Applications: fault-tolerant quantum computers

Questions:

1. Classification of TQFTs

2. Find physical realizations of TQFTs, hence build quantum computers

Statistics of Particles

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

=

Braid statistics

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

Non-abelian anyonsSuppose 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 non-abelian anyons In general the statistics of a particle with

configuration space X is n: 1(Cn(X),p0)! U(kn)

Classical Hall effect E. H. Hall, 1879 On a new action of the magnet on electric currents Am. J. Math. Vol 2, No.3, 287--292 “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

Quantum Hall Effect● 1980 K. von Klitzing ---IQHE (1985 Nobel)● 1982 H. Stormer, D. Tsui ---FQHE R. Laughlin (1998 Nobel) quasi-particle with 1/3 electron charge and braiding statistics (anyons)

Electrons in a flatland

Electron system:

Hall resistance Rxy=-1¢ h/e2, with precision 10-10

( is the Landau filling fraction)

BI

e

+ -

-+

Magnetic field T30

3

h/e2

=4,3,2,1,2/3,3/5,4/7,2/5,1/3

Rxy

Rxx

Read-Rezayi conjecture:

=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)

Quantum information science:

---Storage, processing and communicating information using quantum systems.

Three milestones in QIS:

1. Shor's poly-time factoring algorithm (1994)

2. Error-correcting code, thus fault-tolerant quantum computing (1996)

3. Security of private key exchange (BB84 protocol)

How a quantum computer works

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)>.

|x>

|f(x)>

Basis of (C2)­ n is in1-1correspondence with n-bit strings or 0,1,…,2n-1

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 yearsQuantum computer » 1 second

PspaceNPP

BQP

♪ ☻Ф?

Can we build a large scale universal QC?

The obstacle is mistakes and errors (decoherence)

Error correction by simple redundancy0!000, 1! 111Not available due to the No-cloning theorem:

The cloning map |>­ |0>! |>­|> is not linear.

Fault-tolerant quantum computation shows if hardware can be built up to the accuracy threshold ~10-4, then a scalable QC can be built. Solution---Quantum Topology

A topological quantum computer

Creating anyons

Braiding anyons

Measurement=annihilating anyons

Work in progress:

1. Mathematics Classifications of TQFTs or anyonic systems

2. Mathematical Physics Hamiltonianization of TQFTs (generalizing

Jones’ Baxeterization of link invariants)

3. Physics Search for topological phases of matter

Conjecture:

Fix the number of quasi-particle types, there are essentially only finitely many TQFTs.

True for 1,2,3,4 (Rowell, Stong, W.)

Analogues: 1. (E. Landau) Finitely many finite groups with a

fixed number of irreps2. (L. Bieberbach) Finitely crytallographic groups

in each dimension n, n=3 230 crytals

Modular Tensor Category

A MTC is a ribbon category with finitely many isomorphism classes of simple objects and a non-singular S-matrix (a ribbon category is a braided tensor category with compatible duality.)

Given a MTC, there associates a TQFT.

Rep of SL(2,Z)

There is a trace on morphisms of a MTC:

=i

Xi

sij=1/D

=di

D2= di2

s S=(sij)

t T=(i ij)

General Strategy:

● 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)

Fault tolerance of TQFTsA pair (V, (C2)­ n) is a (k, n)-code if for every k-local 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

Why a believer

First Law of Physics (S. Girvin)

Whatever is not forbidden is compulsory

Non-commutative Chern classes???