1

The Topological approach to building a quantum computer.

Michael H. Freedman

Theory Group

Microsoft Research

2

Classical computers work with bits: {0,1}.

Quantum computers will store information in a superposition of

and , i.e. a vector in 2, a “qubit”.

The standard model for quantum computing:

• Local gates on 2, followed my measurement of the qubits.

| 0ñ |1ñ

Ä

3

Successes:• Shor's factoring algorithm • Grover’s search algorithm• great for simulating solid state physics• theoretical fault tolerance

But practical fault tolerance may require physical (not software) error correction inherent in topology.

¹

4

There is an equivalent model for quantum computation [FLW1,FKW2] based on braiding the excitations of a 2dimensional quantum media whose ground state space is the physical Hilbert space of a topological quantum field theory TQFT.

1. The two-eigenvalue problem and density of Jones representation of braid ground. Comm. Math. Phys.228 (2002),no.1,177 –199.

2. Simulation of topological field theories by quantum computers. Comm. Math.Phys.227 (2002), no.3, 587-603.

The Topological Model

5

tim

e

birth

braiding

Particleantiparticle pairs are created out of the vacuum.

death

afterlife?

6

But before we can implement this model in the real world, we must design and build a suitable 2dimensional structure.

• The design would be much easier if we already had a quantum computer!?!

• So we use instead powerful mathematical ideas coming from algebras and the theory of Vaughan Jones.

* -£

7

We will define a Hamiltonian with both large and small terms. The large terms will define “multi loops” on a surface and the small terms will be studied perturbatively. The small terms create an effective action which willbe a sum of projectors.

The projectors in define “disotopy” of curves: This is the (previously mentioned) rich mathematical theoryderived from algebras.

H%

H%

H

8

S set of curves on a surface [S] set of isotopy class of curves on

is a surface:

An example of a “multi loop” on

, , etc…

9

For 2 strand relation

2 1, 1.a a aÈ

= Þ = =±Ç

so a = d. In both cases:

( ) ( )1 2,H ZV å= å @ @C functions on Z homology.2

a ad 1

10

It turns out that the only possible relation on 3strands is:

This gives something much more interesting than homology.

The 4 strand relation is even more interesting: it yields a computationally universal theory.

È È È ÈÇ Ç Ç Ç

+ d d 0

for 2.d =±

11

( ) ( ) isotopyY =Y

Consider:

Vd is the associated TQFT Vd with a rich and known structure.1

( )

1 5If 2cos : 1, 2, , a unique (positive) local 2 2subspace . For other no local condition exists capable of

removing the log extensive degeneracy of .d d

d

d k

V V d

V

p +=± ± ± ± $+

Ì

K

( ) ( ) isotopyd dY =Y -

[ ] s ÌCsC

dV ÌdV Ì

1. In A magnetic model with a possible Chern-Simons phase. With an appendix by F. Goodman and H. Wenzl. Comm. Math. Phys. 234 (2003), no. 1, 129—183 and A Class of P, T-Invariant Topological Phases of Interacting Electrons, ArXiv:cond-mat/0307511, it is argued that Vd as likely to collopse to Vd.

12

Locating Topological Phases Inside Hubbard Type Models.

Kirill Shtengel Michael

Freedman

Chetan Nayak

13

A two dimensional lattice of atoms, partially filled with a

population of donated electrons can have it’s parameters tuned

to become a (universal) quantum computer.

14

In our model the sites (atoms) are arrayed on the Kagome lattice

The colors encode differing chemical potentials .

Tunneling amplitudes tab also vary with colors. c

, , ca abU vm

Hubbard Model

15

We work with an equivalent triangular representation.

• In this representation particles (e.g. electrons) live on edges. The important feature for us is that the triangular lattice is not bipartite.

16

We discuss an “occupation model” at 1/6 fill. For example, imagine that each green atom has donated one electron which is now free to localize near any atom = site of Kagome (K).Let’s look at a “game”.

17

Hamiltonian Ground State Manifold

H 1/6 all particle positions}

U0 large)one particle per bond

dimer cover TNow small terms:

Ì 16

H

, 1i j

i jU n n

+

=å

2 2 ij iVt

U U U

me e e» » »

( )0

† †

, 60ij i j j i i i ij i j

i ii jt c c c c n V n nm+

å + +å +å

20 i

iU n

+

å

j

18

( )

( )( )'

0

0

0

2

' '

2

,'

| |

| |

write | | Q |

| |1Q

| | , so

| | |

| | | |

| |

n

n

kn n k

k n k k nn k n k n k

H H V

H E

E H V

P

k kR Q

E H E

Q RV

RV P

P RV RVRV

kVk k VnkVnn

E E E

l

l

e

l

l

l l

l le e e¹ ¹

= +

Y = Y

- Y = Y

Y = Y + Y

= =S- -

Y = Y

Y = Y + Y

Y = Y + Y + Y +

´Y = + S + S

- - -

L

1444444444444444 2

'

0

22 3

,

| | | |

n n n

k nn

nk

nk k n

n E

nVk k

H E n V

E nV

EnV

U

n

e l

lle

e l¹¹

+

- Y = - = Y

= + + + S +S-

L444444444444444444444444444434

L

don’t like: perturbed, but can recurse

R´

|n

diagonal terms of projectors

dynamic, off diag. terms of projectors

balanced to keep

i jm m=% %

ReviewPerturbation Theory

function of

19

To each “small” process there will be a contribution to an “effective Hamiltonian”:

: , span {dimer covering}H D D D® =%

We work in powers of rbbt

Ue=

( )0 0

set 1

set

, 0

r bbb gb

brb

gbb

U

t t

t c c

t o

e

e

e

=

= =

= >

=

g

g

20

These matrix equations control all small processes:

( )

( )

20

120

'

20

20

Type 1 rhombus:

2 12

2 12

Type 1 ,

2 12

2 2

b b b bgb rb gb gbb b b brb gb rb rb

b b b bbb rb gb bbb b b brb gb rg rb

v t t av c

t t v ac v

v t t av c

t t v c v

e

e

e

e

-

æ ö æ ö æ ö- --÷ ÷ç ç ÷ç÷ ÷ç ÷=ç ç÷ ÷ ÷ç ç ç÷ ÷ç÷÷ç÷- -çç - è øè øè ø

æ ö æ ö- --÷ ÷ç ç÷ ÷ç =ç÷ ÷ç ç÷ ÷÷ç÷- -çç -è øè ø

:

:

( )

( )

( )( )

( )

( )

1

22

22

2

2

1

Type 2 ,

2 1 12

1 122

Finally, Type 3 rhombus:

2 0

02

r r rbb bb bb

rr r

bbbb bb

g g gbb bb bb

gg g

bbbb bb

a

v t v

vt v

v t v

vt v

e

e

-

æ ö÷ç ÷ç ÷ç ÷çè ø

æ ö÷-ç æ ö æ ö--÷ç ÷ç ÷÷ çç ÷ ÷=ç÷ ç÷ç ÷÷ ç ç ÷ç÷÷ç ç -÷ - è øè øç ÷- ÷çè ø

æ ö÷-ç æ ö÷ç ÷ç÷ç ÷=ç÷ ÷ç ÷ ç ÷ç ç÷ è øç ÷- ÷çè ø

:

0 0, killing this process completely.

0 0

æ ö÷ç ÷=ç ÷ç ÷ç÷ è ø

21

( ) ( )

( )

( )

20

1 20

2

Types 1 & 1' : 2

and 2

Type 2 : 2

Type 3 : 0

b bgb bb

b brb rg

rbb

gbb

v v ac

v v a c

v

v

e

e

e

-

= =

= =

=

=

To make all processes projections, and thus obtain an exactly soluble point, we must impose:

( )11 1

1 1 0 1Note: and

1 0 1 1 1

a a a aa a

a a a-

- -

æö æö æ ö æ ö- - - -÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= = +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç- -è ø è ø è ø è ø

1

1 sum of "projectors" id

1

aH

a--

= ~ Ä-

%

22

And if there is a Ring term:

2 21 1 1

2 1 22 1 1

11 2

4

so and

and

v r bc cR

r v c b c

v br v b r

ad

b

e e

e e-

-

æ öæ ö- - ÷÷ çç ÷÷= =çç ÷÷ çç ÷ç ÷÷ç- -è ø è ø

=- =-

=

23

Some choice about how to treat : R JÇ

e.g. democracy: all loops dad bd 3

aristocracy: a1 bd1

However is most general.4a

db=

mob rule: ad1/4 b1

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dimerization multiloops

B R B

ì ®ïïíï ® Èïî

dimerizations multiloops

+:

1

for type 1 process - isotopy is enforced by:

1

1

d

d

d -

ìïïïï -íïï -ïïî 1

for type 3 need

0 0

0 0

to prevent process

3

+:

3

3

if there is a ring exchange term

1

1

d

d -

-

-+:

for type 2 processes "isotopy" is enforced by:

1 1

1 1

ìïïïï -íïï -ïïî2

5

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CONCLUDING REMARKS

• Ring terms vs. Rsublattice defects• Fermionic vs. Bosonic models