The Topological approach to building a quantum computer.

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
  C2, a qubit.
• The standard model for quantum computing
• Local gates on C2, followed my measurement of
  the qubits.

3

• Successes
• Shor's factoring algorithm
• Grovers 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.
4
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

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

5
Particle-antiparticle pairs are
created out of the vacuum.
birth braiding
time
death
afterlife?
6
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.

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 will be a sum
of projectors. The projectors in define
d-isotopy of curves This is the (previously
mentioned) rich mathematical theory derived from
C-algebras.
8
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
9
For 2 strand relation
ad
a
-1



so a d. In both cases
functions on Z -homology.
2
10
It turns out that the only possible relation on
3- strands is
- d
- d
0


This gives something much more interesting than
homology. The 4- strand relation is even more
interesting it yields a computationally
universal theory.
11
Consider
Vd is the associated TQFT Vd (S) with a rich and
known structure.1
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, 129183 and A Class of P,
T-Invariant Topological Phases of Interacting
Electrons, ArXivcond-mat/0307511, it is argued
that Vd as likely to collopse to Vd.
-
12
Locating Topological Phases Inside Hubbard Type
Models.
Kirill Shtengel
Chetan Nayak
MichaelFreedman
13
A two dimensional lattice of atoms, partially
filled with a population of donated electrons can
have its parameters tuned to become a
(universal) quantum computer.
14
In our model the sites (atoms) are arrayed on
the Kagome lattice
Hubbard Model
The colors encode differing chemical potentials
. Tunneling amplitudes tab
also vary with colors.
c
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).Lets look at a game.
17

• Hamiltonian Ground State Manifold
• H H1/6 all particle positions
• (U0 large) one particle per bond
• D dimer cover T
• Now small terms

j
18
Review - Perturbation Theory
function of l
dont like perturbed, but can recurse
dynamic, off diag. terms of projectors
.
.
diagonal terms of projectors
balanced to keep
19
To each small process there will be a
contribution to an effective Hamiltonian
20
These matrix equations control all small
processes
21
To make all processes projections, and thus
obtain an exactly soluble point, we must impose
22
And if there is a Ring term
23
Some choice about how to treat
e.g. democracy all loops d
ad bd 3
aristocracy
a1 bd-1
mob rule ad1/4 b1
However is most general.
24
Cdimerizations Cmultiloops
to prevent process
25
CONCLUDING REMARKS

• Ring terms vs. R-sublattice defects
• Fermionic vs. Bosonic models