Download Topological quantum computation

Topological
quantum
computation
JKP
Pavia, January 2013
Why?
Topology promises to solve the
problem of errors that inhibit
the experimental realisation of
quantum computers…
…and it is lots of fun :-)
Geometry – Topology
•  Geometry
–  Local properties of object
•  Topology
–  Global properties of object
geom.
⇔
⇔
topo.
Quantum Physics
Quantum mechanics
Particle state -> wave function Ψ(x, t)
d Ψ(t)
i
= H Ψ(t)
dt
Ψ(t) = U(t) Ψ(0)
Ψ(x)
x
P(x) = Ψ(x)
Uncertainty:
€
•  Intrinsic
ΔxΔp ≥
2
•  Control errors
H → H ' ⇒ U → U€
'€
⇒ Ψ(t) → Ψ '(t)
give measurement errors P(x) ≠ P'(x)
2
(Classical) computers
•  Encoding: binary numbers 0 and 1: 10101110
•  Binary gates:
10101110 -> 01101000
•  Error correction
error
1011 ! !!
→1001
–  Error:
–  Avoid errors by redundant encoding (copy)
1011 !enciding
!!→(111)(000)(111)(111)
!error
!!
→(011)(010)(110)(011)
!correction
!!!
→(111)(000)(111)(111)
!decoding
!!→1011
Quantum computers
•  Quantum binary states 0 , 1 (spin,...)
Ψ = a 0 +b1 ,
a 0 ⊗ 1 ⊗ 0 ⊗ ...+ ...
•  Quantum binary€gates-> Unitary matrices
€
U 0 ⊗ 0 = a 0 ⊗ 0 + b1 ⊗ 1
•  Use them for computation (can find faster paths
to get to the answer)
€
|input>
|answer>
Quantum computers: Why?
•  Factoring
quantum hackers exponentially better than
classical hackers!
•  Searching objects: where is ❥?
¢®¶♬ΙΠÃ≥⅙⏎✜Ψ✪❥⅖ű
Quantum error correction
•  Error correction
1011 !error
!!
→ 1001
–  Error
–  Avoid errors by redundant encoding
Copying or “cloning” quantum states is not
allowed!
UΨ ⊗ 0 ≠ Ψ ⊗ Ψ
Quantum error correction
(Shor ‘96, Steane ’96)
€
Topological
quantum computers...
Quantum error correction
•  Quantum redundancy:
1 = 0
L
1
O1
€2
€
€
6
O2
€
3
€
€4
€
€ 4 =1L
5
Too costly…
Topological quantum effects
Aharonov-Bohm effect
Magnetic flux Φ and charge
Ψ(x) → e
€
ineΦ
Ψ(x)
Φ
e
e−
The phase is a function
of winding number
€n
€
Topological effect:
n is€the integer number
of rotations
€
Particle statistics
Exchange two identical particles:
x1
x2
Statistical symmetry:
Physics stays the same, but Ψ could change!
€
2×
€
€
Ψ(x1, x 2 ) = ???Ψ(x 2 , x1 )
€
=
Anyons and statistics
3D
2D
Bosons
Ψ → Ψ
Fermions
Ψ → ei 2π Ψ
Ψ → ei 2ϕ Ψ
Ψ →U Ψ
Anyons
Anyons ~ particles with mag. flux and el. charge
Statistical phase ~ Aharonov-Bohm effect
Anyons and physical systems
2D
Ψ → ei 2ϕ Ψ
Ψ →U Ψ
Anyons
Anyonic properties can be found in 2-dimensional
topological physical systems:
(c)!
(a)!
(b)!
• Fractional quantum Hall effect
• Topological insulators
• Bose-Einstein condensates
[Giandomenico Palumbo
& JKP,
“CSM theory
from lattice fermions”,
arXiv:1301.2625]
Anyonic evolutions are topological
Anyonic evolutions have
topological properties like
the Aharonov-Bohm effect.
Φ
(b)!
e−
€
€
Many different states of
physical system give the
Redundant encoding
same anyonic states
Topological anyonic states
Quantum states of
anyons similar to spin,
but cannot be locally
determined.
σ σ
σ
σ
€
time
These states change
when we exchange
anyons.
1L
0L
Measure:
σ
×
σ
=
1+
ψ
fuse
#1 1
&
% €⊗ = 0 +1(
€
$2 2
'
1 vacuum 1
ψ
ψ
Topological quantum computers
67
- Quantumtinformation
encoded in state of anyons.
- Their statistics is used
for quantum gates.
€
!
4.2 Anyonic quantum computation
!
€ €
σa σa
a σ
a
σ
σa
!
!
aσ
!
!
€ €
€
1
3
-  avoid errors:
-- Redundancy
equivalence between states
-- Hamiltonian
e
e
e
e
e
e
1
tFig. 4.10 A possible configurationMeasure
protects information
of topological quantum computation. Initially, pairs of
information
2
4
5
6
created from the vacuum. Braiding operations between them unitarily evolve
Finally, fusing the
gives
set of outcomes
! anyons
! a!
!
! ei , i = 1, ..., which
! together
!
of the computation.
by fusing.
Conclusions
The problem of error
correction in QC can be
resolved if we find a way to
engineer topological systems
that can support anyons.
http://quantum.leeds.ac.uk/~jiannis