Download - Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”
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Quantum computation with solid state devices
-“Theoretical aspects of superconducting qubits”
Quantum Computers, Algorithms and Chaos, Varenna 5-15 July 2005
Rosario Fazio
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OutlineLecture 1
- Quantum effects in Josephson junctions- Josephson qubits (charge, flux and phase)- qubit-qubit coupling- mechanisms of decoherence- Leakage
Lecture 2
- Geometric phases- Geometric quantum computation with Josephson qubits- Errors and decoherence
Lecture 3
- Few qubits applications- Quantum state transfer- Quantum cloning
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Adiabatic cyclic evolution
The Hamiltonian of a quantum systemdepends on a set of external parameters r
The external parameters are changed in time r(t)
Adiabatic approximation holds
e.g. an external magnetic field B
e.g. the direction of B
If the system is in an eigenstateit will adjust to the instantaneousfield
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What happens to the quantum state if
r(0) = r(T)????
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Parallel Transport e(0) After a cyclic change
of r(t) the vector e(t)does NOT come backto the original direction
The angle depends onThe circuit C on the sphere
r(T)=r(0)
e(T) ≠
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Quantum Parallel TransportSchroedinger’s equation implements phase parallel transport
|)]([| tHdtdi r
)]([)()]([)]([ tntEtntH n rrr
)]([)( )()(tneet tidttEi
nn r
Schroedinger’s equation:
Adiabatic approx:Instantaneous eigenstates
Look for a solution:
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Berry phase
)()( tndtdtni
The geometrical phase change of |> along a closedcircuit r(T)=r(0) is given by
C r dnni rrr )()(
- M.V.Berry 1984
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Spin ½ in an external field
B
C
The Berry phase is relatedto the solid angle that Csubtends at the degeneracy
)(21
/ C
Bσ HAdiabatic condition B << 1
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Aharonov-Anandan phase
Geometric phases are associated to the cyclic evolution of the quantum state (not of the Hamiltonian)
Generalization to non-adiabatic evolutions
Consider a state which evolves according to the Schrödiger equation such that
)0()( ieT Cyclic state
- Y. Aharonov and J. Anandan 1987
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Aharonov-Anandan phase
dttdtdtidttHt
TT
00
)(~)(~)(~)(~
Introducing such that ~ )0(~)(~ T
Dynamical phase Geometrical phase
Evolution does not need to be adiabaticAdiabatic changes of the external parameters are a way to have a cyclic stateIn the adiabatic limit )()(~ tnt
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Aharonov-Anandan phase (Example)
zBH The Hamiltonian
|)2/sin(|)2/cos()0(| Initial state
|)2/sin(|)2/cos()(| iBtiBt eet evolves as
The state is cyclic after T=/B
)]cos(1[)(~)(~0
dttdtdti
T
AA
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Experimental observations
Geometrical phases have been observed ina variety of systems
Aharonov-Bohm effect
Quantum transport
Nuclear Magnetic Resonance
Molecular spectra
…
see “Geometric Phases in Physics”, A. Shapere and F. Wilczek Eds
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Is it possible to observegeometric phases
in amacroscopic system?
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Geometric phases in superconducting nanocircuits
Possible exp systems: Superconducting nanocircuits
Implications:
• “Macroscopic” geometric interference
•Solid state quantum computation
•Quantum pumping
-G. Falci, R. Fazio, G.M. Palma, J. Siewert and V. Vedral 2000-F. Wilhelm and J.E. Mooij 2001-X. Wang and K. Matsumoto 2002-L. Faoro, J. Siewert and R. Fazio 2003-M.S. Choi 2003-A. Blais and A.-M. S. Tremblay 2003-M. Cholascinski 2004
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Cooper pair box
CHARGE BASIS
Charging Josephson tunneling
nN x nnnn
2JE
nnnnCE 112
IJ
Cj
V
Cx
n
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From the CPB to a spin-1/2
Hamiltonian of a spin In a magnetic field
In the |0>, |1>subspace
H =Magnetic field in the xz plane
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Asymmetric SQUID
H = Ech (n -nx)2 -EJ ( cos (
0
221
221 cos4)()( JJJJJ EEEEE
021
21 tantan JJ
JJ
EEEE
EJ2 C
Cx
EJ1 C
Vx
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From the SQUID-loop to a spin-1/2
In the {|0>, |1>} subspace
HB = - (1/2) B .
Bx = EJ cos By = EJ sin Bz = Ech (1-2nx)
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“Geometric” interference in nanocircuits
HB = - (1/2) B . Bx = EJ cos
By = EJ sin
Bz = Ech (1-2nx)B
C“ ”
In order to make non-trivial loops in the parameterspace need to have both nx and
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Berry phase in superconducting nanocircuits
1/2
nx
M
Role of the asymmetry
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Berry phase - How to measure
•Initial state
•Sudden switch to nx=1/2
•Adiabatic loop
|0>
(1/2½)[|+> + |->]
(1/2½)[ei+i|+> +e-i-i|->]
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Berry phase - How to measure
•Swap the states
•Adiabatic loop with opposite orientation
•Measure the charge
(1/2½)[ei+i|-> +e-i-i|+>]
(1/2½)[e2i|-> +e-2i|+>]
P(2e)=sin22
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Quantum computation
• Two-state system• Preparation of the state• Controlled time evolution• Low decoherence• Read-out
Phase shifts of geometric origin
Intrinsic fault-taulerant for area-preserving errors
- J. Jones et al 2000- P. Zanardi and M. Rasetti 1999
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Geometric phase shift
1111
1010
0101
0000
2/)1(
2/)1(
2/)0(
2/)0(
i
i
i
i
e
e
e
e
z- interaction
Spin 1 Spin 2
Controlled phase gate
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Geometric phase shift
Two Cooper pair boxes coupled via a capacitance
Hcoupling = - EK z1z2
-G. Falci et al 2000
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Non-abelian case
λH
,...t,...,λt,λtλλ μ21
iii ηEηH
0exp
0
ψUE(t)dthi(T)ψ
T
NdλAPU
C
μ
μ
μ ,...1, exp
η
ληA
μμ
When the state of the system is degenerate over the full course of its evolution, the system need not to return to the original eigenstate, but only to one of the degenerate states.
Control parameters:N degenerate
tηtUt
00 0 nt
Adiabatic assumption:
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Holonomic quantum computation
System S, with state space H ,
perform universal QC
Dynamical approach
Geometric approach
able to control a set of parameters on which depend a iso-degenerate
family of Hamiltonian
information is encoded in an N degenerate eigenspace C of a distinguished
Hamiltonian
Universal QC over C obtained by adiabatically driving the control parameters
along suitable loops rooted at
Mλ
λH
0λH
γ0λ
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Josephson network for HQC
L. Faoro, J. Siewert and R. Fazio, PRL 2003
L.M.Duan et al, Science 296,886 (2001)
There are four charges states |j> corresponding to the position of the excess Cooper pair on island j
One excess Cooper pair in the four-island set up
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Josephson network for HQC
0,0,1,0J0,1,0,0JD LM1
0,0,0,1JJ0,0,1,0J0,1,0,0JJD 2M
2L
*M
*LR2
DEGENERATE EIGENSTATES WITH 0-ENERGY EIGENVALUE
1,0,0
21
DDJJ RL
control parameters
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One-bit operations
1111
iΣeU Rotation around the z-axis
yσiΣeU 1
2 Rotation around the y-axis
Intially we set 0 RL JJ so the eigenstates
correspond to the logical states
21 D,D
10 ,
In order to obtain all single qubit operationsexplicit realizations of :
0110 iiyσ
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Adiabatic pumping
LR
V1(t)
V2(t) Open systems – modulation of the phase of the scattering matrix
Closed systems – periodic lifting of the Coulomb Blockade
Charge transport, in absence of an external bias,by changing system parameters
Charge is transferred coherently
Quantization of transferred charge
-P.W. Brouwer 1998-….
-H. Pothier et al 1992
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Cooper pair pumping vs geometric phases
Relation between the geometric phase and
Cooper pair pumping
- J.E. Avron et al 2000-A. Bender, Y. Gefen, F. Hekking and G. Schoen 2004
- M. Aunola and J. Toppari 2003-R. Fazio and F. Hekking 2004
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Cooper pair pumping vs geometric phases
In order to relate the second term to the AA phase
~||~tAAt i Take the derivative with respect
to the external phase
][2/ AADT eQ
dttdtdti
T
AA 0
)(~)(~ dttHtT
D 0
)()(
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Cooper pair sluice
t
Iright coil
Vg
tt
Ipumped
Can be generalized to pump 2Ne per cycle. (N = 1,2,…?)
A. O. Niskanen, J. P. Pekola, and H. Seppä, (2003).
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Cooper pair sluice - exp
The measured device
Input coilsSQUIDloops
Gate lineJunctions