Post on 20-Mar-2016
description
Quantum computation with solid state devices
-“Theoretical aspects of superconducting qubits”
Quantum Computers, Algorithms and Chaos, Varenna 5-15 July 2005
Rosario Fazio
• Two-state system• Preparation of the state• Controlled time evolution• Low decoherence• Read-out
“DiVincenzo list”
(Esteve)
(Averin)
Geometric quantum computation
Applications
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
Solid state qubits
Advantages- Scalability- Flexibility in the design
Disadvantages- Static errors- Environment
Qubit = two state system
How to go from N-dimensional Hilbert space (N >> 1)
to atwo-dimensional one?
All Cooper pairs are ``locked'' into the same quantum state
)()()( ris ernr
There is a gap in the excitation spectrum
T/Tc
Quasi-particle spectrum
•Cooper pairs also tunnel through a tunnel barrier
•a dc current can flow when no voltage is applied
•A small applied voltage results in an alternating current
Josephson junction
t)eV(II J 2sin
sinJII
21
Vet
2 relation Josephson
Energy of the ground state ~ -EJcos
SQUID Loop
ehc20
LR
nRL 220
RLI sinsin
0
2cos
JeffJ EI
Dynamics of a Josephson junction
+ + + + + + +_ _ _ _ _ _ _
HC
Q2
2
cosJE
X
=
0sin2
2
mg
ttm
Mechanical analogy
IIRV
tVC J sin
IItRt
C J
sin1
2
2
JIIU cos)(
U
Washboard potential
Quantum mechanical behaviour
HC
Q2
2
cosJE
The charge and the phase are Canonically conjugated variable
ie
Q
,2
From a many-body wavefunctionto a one (continous) quantum mechanical degree of freedom
Two state system
Josephson qubits
Josephson qubits are realized by a proper embedding of the Josephson junction in a superconducting nanocircuit
Charge qubit
Charge-Phase qubit
Flux qubit
Phase qubit
C
J
EE
1
104
Major difference is in the form of the non-linearity
Phase qubit
Current-biased Josephson junction
)sin()cos()( 0101 tItItIII wswcdcdc
U
2 4 6 8 10 12 14
-8
-6
-4
-2
zH 2
01
z x y
The qubit is manipulated by varying the current
Flux qubit
The qubit is manipulated by varying the flux through the loop f and the potential landscape (by changing EJ)
X
(t)
1 2
Cooper pair box
tunable:tunable: - external (continuous) gate charge nx
- EJ by means of a SQUID loop
JC
x
exVCn xx 2/
Cooper pair box
phase difference
Cooper pair number,
voltage across junction
current through junction
ˆsinˆ
2ˆ
)(2ˆ2
ˆ
1||
||ˆˆ
J
x
i
IdtndeI
nnC
edtd
eV
nne
nnnn
Cooper pair box
Cooper pair box
CHARGE BASIS
Charging Josephson tunneling
nN x nnnn
2JE
nnnnCE 112
IJ
Cj
V
Cx
n
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
Coherent dynamics - experiments
Nakamura et al 1999
0 50 1000
20
40
60
Rab
i fre
quen
cy (M
Hz)
nominal URF
(µV)0.0 0.5 1.0
30
35
40
45
50
55
switc
hing
pro
babi
lity
(%)
RF pulse duration t (s)Vion et al 2002
Chiorescu et al 2003
•Chalmers group•NTT group•…
See also exps bySchoelkopf et al, Yale
NIST
Inductance
EJ2 Cnx Cx
EJ1 C
EJ2 Cnx Cx
EJ1 C
L
Vx
Vx
Charge qubit coupling - 1
yyLEH 2112
Capacitance
EJ2 Cnx Cx
EJ1 C
EJ2 Cnx Cx
EJ1 C
Charge qubit coupling - 2
zzCCEH 2112
Josephson Junction
EJ Cnx Cx
EJ1 C
EJ2 C
Charge qubit coupling - 3
..2 2112 chEH J
Tunable coupling
Untunable couplings = more complicated gating
)2cos(~ qEcc
Variable electrostatic transformer
The coupling can be switched off even in the presence of parasiticcapacitances
The effective coupling is due to the (non-linear) Josephson element
zzvH 2112
Averin & Bruder 03
Leakage
The Hilbert space is larger than the computational spaceConsequences:a) gate operations differ from ideal ones (fidelity)b) the system can leak out from the computational space (leakage)
Leakage One qubit gate Fidelity
Two qubit gate Fidelity
|m+1>|m>
~EcEj
|1>|0>qubit
Sources of decoherence in charge qubits
Z
electromagnetic fluctuations of the circuit (gaussian) discrete noise due to fluctuating
background charges (BC)trapped in the substrate or in the junction
Quasi-particle tunneling
Full density matrix
TRACE OUT the environment
RDM for the qubit: populations and coherences
Reduced dynamics – weak coupling
”Charge degeneracy”
(= 0 , = EJ) no adiabatic term optimal point ”Pure dephasing”
(EJ =0 , = ) no relaxation
Reduced dynamics – weak coupling
Background charges in charge qubits
HQz
x
Fluctuations due to the
environmentE
E is a stray voltage or current or charge polarizing the qubit
electrostatic coupling
Charged switching impurities close to a solid state qubit charged impurities
Electronic bandEdi
+di
1g““Weakly coupled” chargeWeakly coupled” charge
Decoherence only depends on
= oscillator environment
““Strongly coupled” chargeStrongly coupled” charge• large correlation times of environment• discrete nature• keeps memory of initial conditions• saturation effects for g >>1• information beyond needed
/vg
1g
g=v/ weak vs strongly coupled charges
EJ=0 – exact solution
envzz HEH
)()(1010
1111
0000
)0()(
)0()(
)0()(
tiEtet
t
t
Constant of motion
Equbit
E
tiHtiH
HXHH
eett
ln)0()(ln)(
12
12
In the long time behavior for a single Background Charge
1g tv
2
The contribution to dephasing due to “strongly coupled” charges (slow charges) saturates in favour of an almost static energy shift
t
~
1gt 2/vttE t
~
EJ=0 – exact solution
Standard model: BCs distributed according to
with
yield the 1/f power spectrum
from experiments 68 1010 A
Experiments: BCs are responsibe for 1/f noise in SET devices.
Warning: an environment with strong memory effects due to the presence of MANY slow BCs
Background charges and 1/f noise
“Fast” noisein general quantum noise• fast gaussian noise• fast or resonant impurities
)|()]([)()( sltf
tslPtsl
t D
EfH
fzt
slz ˆ
2
1)(
2
1Split
Two-stageelimination
Slow noise ≈ classical noise• slow 1/f noise
Slow vs fast noise
222
• expand to second order in → quadratic noise
• Large Nfl central limit theorem → gaussian distributed 0P
• Slow noise: (t) random adiabatic drive M →adiabatic approximation
• Retain fluctuations of the length of the Hamiltonian → longitudinal noise
Paladino et al. 04
see also Shnirman Makhlin, 04 Rabenstein et al 04
)(/1
ln1641 222
SN M
meas
MC t
dAEv fl
• Static Path Approximation (SPA)
variance
HQ
z
xOptimal point
Initial defocusing due to 1/f noise
Standard measurementsno recalibration
with recalibration
Falci, D’Arrigo, Mastellone, Paladino, PRL 2005, cond-mat/0409522
SPA
HQ
z
xOptimal point
Initial suppressionof the signal due
essentially to inhomogeneuos
broadening(no recalibration)
Initial defocusing due to 1/f noise