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Geometric Quantum Computation and Errors
Vlatko VedralVlatko
[email protected]@imperial.ac.uk
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Team
w Jiannis Pachos (post-doc)w Angelo Carollo (finishing
PhD)wMarcelo Santos (post-doc)w Ivette Fuentes-Guridi (ex student,
Perimeter,
Oxford).w Collaboration: A. Ekert, J. A. Jones, J. Anandan, E.
Sjoqvist,
M. Ericsson, M. Palma, R. Fazio, G. Falci, J. Siewert…
Support: EU, EPSRC, ESF, Hewlett-Packard, Elsag Spa,
QUIPROCONE.
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wBerry phase of a spin-1/2 particle in a magnetic field.
wAny quantum computation can be executed with geometric phases
only.
wWe analyze the effect of simple errors and show that there is
some natural inbuilt resistance.
wHow far can this be generalized?
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Adiabatic theorem
))(())(())(( tREtRtRH Ψ=Ψ
If H changes slowly through some parameter, the adiabatic
theorem assures that the system remains in the eigenstate of the
Hamiltonian.
And if we change the Hamiltonian so that in a time τ it returns
to its initial form…
)()( 0== tHH τCyclic evolution
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Berry phaseM. Berry, Proc. Roy. Soc. A 392, 45 (1984)
)0()( )()( Ψ=Ψ titi nn eet γβ
Then the state returns to its initial form but since eigenstates
are defined up to a phase factor, the state could acquire a phase
due to the adiabatic and cyclic evolution that took place.
dttETT
nn )()(0∫−=β
Usual dynamical phase
dttdtd
tiTn )()()( ΨΨ= ∫γGeometrical phase: It depends only on the path
taken in the configuration space of parameter R.
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A typical example is the phase acquired by a spin-1/2 particle
interacting with a slowly varying magnetic field:
Spin 1/2 particle:Canonical example of Berry phase
The eigenstates acquire a geometric phase proportional to the
area γ enclosed in the path traversed by the field B.
→→
⋅−= σµ )()(ˆ tBtH
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Classical Berry Phase
1. Cats and Astronauts;2. The Earth is a sphere - Foucault’s
pendulum;
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Spin-1/2 interacting with an e-m fieldSemi-classical
description
A 2-level system with Bohr frequency ω, interacting with a
classical oscillating field with frequency ν and amplitude α, in
the rotating frame is described:
( )ϕϕ ασασλσ iiz eeH −−−+ ++∆= 2ˆ νω −=∆
By rotating (adiabatically) the vector R as shown in the picture
(the phase ϕ is rotated from 0 to 2π), the eigenstates acquire the
geometric phase:
)cos1(2 θπγχ −±=±=±22 )(4cos αλθ +∆∆=where
)2/,sin,cos( ∆= ϕλϕλRWhere:
→
⋅= σRĤWe can rewrite the Hamiltonian as
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Spin-1/2 interacting with an e-m fieldFully quantised
description
In this case the interaction is described by the Jaynes Cumming
Hamiltonian:
In analogy with the semi-classical case, we apply an adiabatic
transformation,by varying ϕ from 0 to 2π. The eigenstates of the
system acquire the geometric phases:
( )††2
ˆ aaaaH z −+ +++= σσλσω
ν
The change in the Hamiltonian is implemented through:
aaieU†
)( φϕ −=)(ˆ)()(ˆ † ϕϕϕ UHUH =
)1(4cos 22 ++∆∆= nn λθ
Carollo, Fuentes-Guridi, Bose and Vedral, PRL (2002).Carollo,
Santos, Vedral, PRA (2003).
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Controlled Not = Phase shift
=
φie000010000100001
φ
→ 1111 −π
φ operates symmetrically on the two qubits: it does not
distinguish between control and target bits
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Conditional Geometry
Jones, Vedral, Ekert and Castagnoli, Nature (2000).
1H13CCl3
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Fault-tolerance
Random motion Systematic displacementGeometric phase is
invariant under the above.
G. De Chiara and G. M. Palma, quant-ph (2003).
(but see also Blais and Tremblay, PRA (2003))
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Master Equation
wMaster Equation
[ ] { }† † †1
1 1, 2
2
n
k k kk k kk
Hi
ρ ρ ρ ρ ρ=
= − Γ Γ + Γ Γ − Γ Γ∑&
0 1̂W iH t= − ∆%
( ) †0
( )n
k kk
t t W t Wρ ρ=
+ ∆ ≈ ∑
k kW t= Γ ∆
†
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n
k kk
iH H
=
= − Γ Γ∑%
w For small time intervals,
where
Carollo, Fuentes-Guridi, Santos and Vedral, PRL (2003).
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Quantum Jumps
w For a given jump “k”,with probability
w Different trajectories = different set of W’s
= different set of pure states
i(l) 01
W m
im
l=
Ψ = Ψ∏
†1( ) ( )m k m kt W t Wρ ρ+ ≈
{ }†Tr ( )k k m kp W t Wρ=
{ }0 0, ,...,i iNΨ Ψ Ψ
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Geometric phase
w Pantcharatnam formula
Carollo, Santos, Fuentes-Guridi, Vedral, Phys. Rev. Lett.
(2003).
{ }0 1 1 2 1 0arg ...g N N Nγ −= − Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ
w Continuous limit
( ) ( )( ) ( )
( ) ( )0
ddtIm dt arg 0
T
g
t tT
t tγ
Ψ Ψ= − − Ψ Ψ
Ψ Ψ∫
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No Jump
( ) ( )0 0ddt
i t H tΨ = Ψ%
( ) ( )( ) ( )
( ) ( )0 0
0 0 00 00
arg 0T
gt H t
Tt t
γΨ Ψ
= − − Ψ ΨΨ Ψ∫
w In the limit N >> 1,
( )0 0 0 0ˆ W 1N
tTm
mT
i HN
Ψ = Ψ = − Ψ
%
0 1̂W iH t= − ∆%†
12
n
k kk
iH H
=
= − Γ Γ∑%
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w In particular,
( )† 01
ˆ ˆ ˆIf 1 then 1 1n
k kk
W iH tα=
Γ Γ ∝ = − + ∆∑
and the geometric phase for the no-jump trajectory coincides
with the one for thedecoherence free evolution.
No Jump
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1 jump
w For 1 jump at time t1( ) ( )
( ) ( )( ) ( ){ }
( ) ( )
( ) ( )( ) ( ){ }
1
1
' '
1 ' '1 1' '0
'' ''
'' ''' ''
ddt dt arg
ddt dt arg 0
tj j
T
t
t tt t
t t
t tT
t t
γΨ Ψ
= − Ψ Γ Ψ +Ψ Ψ
Ψ Ψ− Ψ Ψ
Ψ Ψ
∫
∫
( ) ( ) ( )' '' '0 1 10 , jt W tΨ = Ψ Ψ = Ψ
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Example: Spin 1/2
zˆH 2ω
σ=
†z
ˆˆ , 1λσΓ = Γ Γ ∝
( ) ( )0 0ˆ1 1 coszγ π σ π θ= − Ψ Ψ = −
w Spin 1/2 coupled to a magnetic field.
w Dephasing, no-jump - same phase!
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Dephasing: 1 jump - same phase!!
( ) ( ) ( ) ( ){ }( ) ( ){ }( ) ( )( ) ( )
2
ˆ 0
ˆ
ˆ ˆ0 0 dt arg 0 02
ˆarg 0 0
ˆ1 0 0 1 cos
z
z
kkz z
kiz
z
e
πω
σ
πσ
ωγ σ σ
σ
π σ π θ
= − Ψ Ψ − Ψ Ψ
− Ψ Ψ
= − Ψ Ψ = −
∫
( ) ( ) ( ) ( ){ }
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( )
1
1 1
1
1ˆ
0
ˆ ˆ2 22 2
ˆ ˆ0 0 dt arg 0 02
ˆ ˆ0 0 dt arg 0 02
ˆ1 0 0 1 cos
z
z z
t
z z
i t i t
z zt
z
e e
σ
σ σπ π ω ωω
ωγ σ σ
ωσ σ
π σ π θ
−
= − Ψ Ψ − Ψ Ψ
− Ψ Ψ − Ψ Ψ
= − Ψ Ψ = −
∫
∫
w Dephasing, k jumps - same phase!!!
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Spontaneous emission
( ) ( )ˆ20
ˆ ln 0 02
ze
ωπ σ
ασω
γ πα−
− = + Ψ Ψ
( ) ( )0 2 2ˆ 1 cos 2 sin ( ) , for Oσ α αγ π θ π θ ω αω ω−
≈ − + + >>
ˆασ −Γ =
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Implementations
1. NMR - confirmed experimentally - Jones et al, Nature
(2000)
2. Josephson Junctions - Fazio et al, Nature (2000).
0 0 Cooper pairs
1 1 Cooper pair
Y.Nakamura, Y.A.Pashkin, J.S.Tsai, Nature 398, 786 (1999)
(for error analysis see, Whitney and Gefen, PRL (2003))
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Geometry in Josephson
2( ) ( )cos( )ch x JH E n n E θ α= − − Φ −
1 2 1 2
2 2
0
( ) ( ) 4 cosJ J J J JE E E E E π Φ
Φ = − + Φ
1 2
1 2 0
( )tan( ) tan
( )J J
J J
E E
E Eα π
− Φ= + Φ
0 / 2h eΦ =
Regime:
1 2,J J chE E E
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Josephson = Spin 1/2
chT E
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Aharonov-Bohm Effect
1 2( ) /c
d d Adsφ π λ= − + ∫Ñ
solenoid
Electrons in Phase
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Degeneracy
0 0
01 1
exp ( )t
adiab P A dψ ψ
τ τψ ψ
→
∫
0 0 0 1
1 0 1 1
d dd dAd dd d
ψ ψ ψ ψτ τ
ψ ψ ψ ψτ τ
=
where
Wilczek and Zee, PRL 1984.
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Summary and Future
w Geometry offers some protection.w Topology – how far?w
Implementations?w Combining other mechanisms of protection.
V. Vedral, Int. J. Q. Info. (2003).J. Pachos and V. Vedral,
quant-ph (2003)
