Transcript The Topological approach to building a quantum computer

The Topological approach to
building a quantum computer.
Michael H. Freedman
Theory Group
Microsoft Research
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 Classical computers work with
bits: {0,1}.
 Quantum computers will store
information in a superposition of
| 0ñ and |1ñ, i.e. a vector in C2, a
“qubit”.
 The standard model for quantum
computing:
• Local gates on Ä C2, followed
my measurement of the qubits.
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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
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The Topological Model
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.
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Particle-antiparticle pairs are
created out of the vacuum.
time
birth
braiding
death
afterlife?
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But before we can implement this model in
the real world, we must design and build a
suitable 2-dimensional 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.
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We will define a Hamiltonian H 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
% action which will
create an effective H
be a sum of projectors.
%
The projectors in H
define “d-isotopy” of
cu r v es : Th i s i s t h e ( p r ev i o u s l y
mentioned) rich mathematical theory
derived from C*-algebras.
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S is a surface:
,
, etc…
An example of a “multi loop” d on S
S = set of curves on a surface S. [S] = set of
isotopy class of curves on S
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For 2 strand relation
È
2
= a
Þ a = 1, a = ± 1.
Ç
=
=
a
-1
ad
=
so a = d. In both cases:
H1(å , Z2 )
V = (å )@C
@
functions on Z2 -homology.
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It turns out that the only possible relation on 3- strands is:
+
È
Ç
for d = ±
+
È -d È
Ç
Ç
È
=0
-d
Ç
2.
This gives something much more interesting than homology.
The 4- strand relation is even more interesting: it yields a
computationally universal theory.
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Consider:
Vd Ì Vd Ì
C
[s ]
s
Ì C
Y( )= Y(
)
d Y( )= Y(
)
isotopy
d - isotopy
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Vd is the associated TQFT Vd (S) with a rich and known structure.
If d = ± 2 cos p
k+ 2
: ± 1, ± 2, ±
1+
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, K $ a unique (positive) local
2
subspace Vd Ì Vd . For other d no local condition exists capable of
removing the (log) extensive degeneracy of Vd .
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 V-d as likely to collopse to Vd.
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Locating Topological Phases Inside
Hubbard Type Models.
Kirill
Shtengel
Michael
Freedman
Chetan
Nayak
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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.
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Hubbard Model
In our model the sites (atoms) are arrayed on the
Kagome lattice
c
ab .
The colors encode differing chemical potentials ma ,U , v
c
Tunneling amplitudes tab also vary with colors.
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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.
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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 ”.
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Hamiltonian
H=
Ground State Manifold Ì H 1
H1/6 = {all particle positions}
+
U 0 å ni2 (U0 large)
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{one particle per bond}
i
+
U å ni n j
i, j = 1
D ={dimer cover T}
Now small terms:
t
» e
U
å
Vij
U
» e
2
mi
» e2
U
+
†
†
t
c
c
+
c
c
+
å
m
n
+
å
V
n
n
(
)
ij
i
j
j
i
i
i
ij
i
j
0
i , j 60
i
i
j
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Review - Perturbation Theory
H = H0 + l V
function of l
H | Y = En | Y
(En - H 0 )| Y = l V | Y
write | Y = P | Y + Q | Y
´ R
R= Q
|k k|
1
Q= S
k E - e
En - H 0
n
k
Q | Y = l RV | Y , so
don’t like: perturbed, but can recurse
| Y = l RV | Y + P | Y
n|
| Y = P | Y + l RV | Y + l 2 RVRV | Y + L
k Vk ' ´ k 'Vn
k Vn
|Y = | n + l S
+ l 2 S'
+L
k¹ n E - e
k , k ¹ n (E - e )(E - e )
n
k
n
k
n
k'
1444444444444444
444444444444444444444444444
2
434
n | En - H 0 | Y = En - en = l
diagonal terms
of projectors E = e + l
n
n
nVn + l
2
S
k¹ n
dynamic, off diag.
terms of projectors
n |V |Y
nVk
k Vn
En - ek
. .
+l
3
S
k ,k ' ¹ n
+L
U 2 balanced to keep
%i = m
%j
m
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To each “small” process there will be a contribution to an
“effective Hamiltonian”:
%: D ® D, D = span {dimer covering}
H
We work in powers of e =
g set U = 1
r
tbb
U
r
b
g set tbb
= t gb
= e
b
trb
= c0 e, c0 > 0
g
tbb
= o (e )
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These matrix equations control all small processes:
Type (1) rhombus:
b
b ö
æ vgb
- 2trbb t gb
÷
çç
÷
=
÷
b
çç- 2t b t b
÷
vrb ø
è rb gb
b
2ö
æ vgb
2
c
e
÷
0
çç
÷
:
÷
b
ççè- 2c e 2
÷
vrb ø
0
æ a - 1ö÷
çç
÷
÷
çè- 1 a- 1 ø÷
b
æ vbb
- 2c0 e 2 ö÷
çç
÷
:
÷
b
ççè- 2c e 2
÷
vrb ø
0
æ a - 1ö÷
çç
÷
çè- 1 a- 1 ø÷
÷
Type (1' ),
b
b ö
æ vbb
- 2trbb t gb
÷
çç
÷
÷=
b
b
b
çç- 2t t
v
è rb gb
ø÷
rg
Type (2),
æ r
r 2ö
v
2
t
çç bb
( bb ) ÷÷÷
çç
=
÷
÷
çç- 2 t r 2
r
÷
vbb ÷
è ( bb )
ø
æ vbbr
ö
- 2e 2 ÷
çç
÷:
r ÷
ççè- 2e 2
vbb ÷
ø
æ 1 - 1ö÷
çç
÷
çè- 1 1 ÷
÷
ø
Finally, Type (3) rhombus:
æ g
g 2ö
- 2 (tbb ) ÷
çç vbb
÷
÷
çç
=
÷
÷
2
çç- 2 t g
vbbg ÷
÷
è ( bb )
ø
ævbbg
çç
ççè 0
ö
0÷
÷=
g ÷
vbb ÷
ø
æ0 0÷
ö
çç
÷, killing this process completely.
÷
çè0 0÷
ø
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a - 1
%
H = sum of "projectors" ~
Ä id
- 1
- 1 a
æ ö
a - 1æ
1ö÷ æ
0ö÷
a - 1æ
- aö÷
- 1 ç- a÷
ç
ç
ç
÷= ç ÷
÷
Note:
and
= (a + a )ç ÷
- 1 ç ÷
- 1 ç
÷
÷
÷
÷
÷
÷
ç
ç
ç
- 1 a èaø è0ø
- 1 a è1ø
èç 1 ø÷
To make all processes projections, and thus obtain an
exactly soluble point, we must impose:
b
b
Types (1)& (1'): vgb
= vbb
= 2ac0 e 2
b
b
and vrb
= vrg
= 2a- 1c0 e 2
r
Type (2): vbb
= 2e 2
g
Type (3): vbb
= 0
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And if there is a Ring term:
æbc1e 2
æv1 - r ö
÷
çç
÷
R = çç
=
÷
çè- r v2 ø
÷ ççè- c1e 2
ö
- c1e ÷
÷
- 1
2÷
b c1e ÷
ø
2
so v1 = - br and v2 = - b- 1r
4
a
and
= d
b
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Some choice about how to treat :
RÇJ
e.g. democracy: all loops = d
a=d b=d 3
aristocracy:
a=1
b=d-1
mob rule: a=d1/4 b=1
a4
= d is most general.
However
b
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ïìï dimerization ® multiloops
í
ïïî
B ® RÈ B
Cdimerizations
Cmultiloops
ìï for type 2 processes "isotopy" is enforced by:
ïï
ïí
1 - 1
2
+
ïï
:
- 1 1
ïïî
ìï for type 1 process
ïï
ïí
ïï
ïïî
1
for type 3
3
d - isotopy is enforced by:
+
:
d
- 1
- 1 d- 1
need
0
0
0
0
to prevent process
if there is a ring exchange term
5
+
:
d3
- 1
- 1
d-
3
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CONCLUDING REMARKS
• Ring terms vs. R-sublattice defects
• Fermionic vs. Bosonic models
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