Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”
40
Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits” Quantum Computers, Algorithms and Chaos, Varenna 5-15 July 2005 Rosario Fazio
description
Quantum Computers, Algorithms and Chaos , Varenna 5-15 July 2005. Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”. Rosario Fazio. Scuola Normale Superiore - Pisa. Outline. Lecture 1 - Quantum effects in Josephson junctions - PowerPoint PPT Presentation
Transcript of Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”
Quantum computation with solid state devices
-“Theoretical aspects of superconducting qubits”
Quantum Computers, Algorithms and Chaos, Varenna 5-15 July 2005
Rosario Fazio
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
Quantum information protocols without external control
Choose a given model and use just the time evolution (less flexible but more stable)(less flexible but more stable)
Easier to implement in solid state systems
Implementation of Quantum communication schemes in solid state devices
Josephson arrays in quantum communication
Motivations
Alice
Bob
Quantum channel
Protocols
Cloning
Quantum state transfer
Alice
Bob~ |>
Quantum communications with spin chains
|>=a|0>+b|1>
Quantum channel
Alice
Bob
. . .|> ||0> |> |0>
J J J
Initial state
i
zi
iiii hJH 1
S. Bose (2002), M. Christandl et al (2003), F. Verstraete, M. Martin-Delgado and J.I. Cirac (2003), D. Burgarth and S. Bose (2004), D. Burgarth, V. Giovanetti and Bose (2005), V. Giovannetti and R. Fazio (2005), A. Romito, C. Bruder and R. Fazio (2005), G. De Chiara D. Rossini, S. Montangeroand R. Fazio (2005), …
Quantum communications with spin chains
Alice
Bob
. . . J J J
000...0000||
i
zi
iiii hJH 1
Quantum communications with spin chains
000...0000||
• Initial state
• Time evolution
000...0000||
)(||)()( }1,...1{ tUtUTrt LL
1|sin0|cos|
ie
•Sender at site 1•Receiver at site L
Quantum communications with spin chains
000...0000|
i
zi
iiii hJH 1
000...0000|
Quantum communications with spin chains
Total magnetizationIs a constant of motion
ja j |000...1000|
where
thj
j
000...1...000|| Le iHt ||1
dtF
|)(|
Jt
Quality of the channel – average fidelity
Fidelity ~ L-1/3
L
Quantum communications with spin chains
Perfect communications with spin chains
xiJSetU )(
No cloning theorem
U|a>|0> |a>|a>
U(|a>+|b>) |0> |aa>+|bb> ≠ (|a>+|b>) (|a>+|b>)
Quantum cloning
W. Wootters and W. Zurek (1982)
A quantum copying machine does not exist!
Although perfect cloning is not possible …..
… Imperfect cloning has been considered
Quantum cloning
V. Buzek and M. Hillery (1996), D. Bruβ et al (1998), D. Bruβ, A. Ekert and C. Macchiavello (1998), R. Werner (2000), …
A. Lama-Linares et al (2002), De Martini et al (2004),J. Du et al (2004), …
Central quantity
Fidelity for n m cloning
n states to be copied belonging to a portion of the Hilbert space
the m cloned statesare in in the mixedstate
||, mnF
IndependentIndependent on on
83.021 UCF
Universal ClonerUniversal Cloner
Phase Covariant ClonerPhase Covariant Cloner
Examples
854.021 PCCF Fidelity at
the equator
Ry(/2)
21
31
Quantum circuits
• XY Model
• Heisenberg Model
• Ising Model
•Start from the state to clone
•Wait for a time (independent on the state to be cloned)
•Cloning 1 m
| > =|0>
| > = cos|0>+sinei |1>
G. De Chiara et al. (2004,2005)
Spin network cloning
Fid
elity
Heisenberg
Ideal cloner coincides with the XY model
21
Phase covariant cloner
Best cloner
Heisenberg
XY
m1
m
Fid
elity
n m F Jt
2 3 0.94 81
2 5 0.87 73
2 7 0.81 69
3 5 0.97 584
Cloning from n to m
F
Cloning in the presence of noise
/J
Vg
Vg
Josephson coupling realizes the XY model
Quantum cloning with Josephson qubits
..2 2112 ch
EH J
..2 3113 ch
EH J
Vg
Vg
Josephson couplingrealizes the XY coupling
F
U1U2
Quantum cloning with Josephson qubits
Josephson arrays as
artificial 1D magnets
Charge regime - C. Bruder R, Fazio, G. Schön, PRB 47, 342 (1993) Flux regime - L. Levitov, T.P. Orlando, J.B. Mayer, J.E. Mooij cond-mat 0108266
Quantum communication with JJA
Bose-Hubbard = Quantum Phase Model = XXZ model
prep
arat
ion
measurem
entstate propagation
).(2 1 ch
EhUH
iii
J
i
zii
ij
zjij
zi
C. Bruder et al (1993)
Quantum communication with JJAs
Example - N=3 tF
idel
ity
EJ /EC=0.1
C/C0=10
Fidelity ~ 0.999
|)(| tF
| N0|
Averaged over the initial state
Fmax
N
Fidelity vs N
Fmax
C0/C
Dependence on the electrostatic interaction
V
t = tmax|>~
The current is proportional to the fidelity
Vg
VgVg
Vg
The charge state of the N island as a function of tp
Current correlation <I1(t)IN(t+tp)>
Charge correlation <n1(t)nN(t+tp)>
Fidelity ~ 0.95
EJ/EJ=10%
nx/nx=10%
Imperfections – N=3
t
L. Levitov, T.P. Orlando, J.B. Mayer, J.E. Mooij cond-mat 0108266
x x x x x x
State transfer with flux qubits
Alice
Bob
Quantum channel
Entangled
Entanglement sharing
{ {………………………………………..
singletsinglet
● ● ● ● ● ●
● ● ● ● ● ● time
-1 0 1 2 3 4site #
● ● ● ● ● ●
Singlet propagation
Entanglement
Singlet initial state
Entropy for symmetric sites
Entropy for sites (-6,7)
Fidelity to the initial singlet
JJ arrays can be used in quantum communication
Entanglement sharing Quantum Cloning State transfer
Experiments seems to be possible at present
Conclusions
