Quantum Computers, Algorithms and Chaos , Varenna 515 July 2005. Quantum computation with solid state devices  “Theoretical aspects of superconducting qubits”. Rosario Fazio. Scuola Normale Superiore  Pisa. “DiVincenzo list”. Twostate system Preparation of the state
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Quantum Computers, Algorithms and Chaos, Varenna 515 July 2005
Quantum computation with solid state devices“Theoretical aspects of superconducting qubits”
Rosario Fazio
Scuola Normale Superiore  Pisa
“DiVincenzo list”
(Esteve)
(Averin)
Geometric quantum computation
Applications
Outline
Lecture 1
 Quantum effects in Josephson junctions
 Josephson qubits (charge, flux and phase)
 qubitqubit 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
Ndimensional Hilbert space (N >> 1)
to a
twodimensional one?
All Cooper pairs are ``locked'' into the
same quantum state
Quasiparticle spectrum
There is a gap in the
excitation spectrum
D
D
T/Tc
j1 I j2
Josephson junction
Energy of the ground state
~ EJcosj
SQUID Loop
F
jR
jL
X
Dynamics of a Josephson junction
+ + + + + + +
_ _ _ _ _ _ _
j1 j2
=
Mechanical analogy
Washboard potential
U(f)
Quantum mechanical behaviour
The charge and the phase are
Canonically conjugated variable
From a manybody wavefunction
to 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
ChargePhase qubit
Flux qubit
Phase qubit
1
104
Major difference is
in the form of the
nonlinearity
U(f)
Phase qubit
Currentbiased Josephson junction
The qubit is manipulated
by varying the current
X
Flux qubit
(t)
j2
j1
The qubit is manipulated
by varying the flux through
the loop f and the potential
landscape (by changing EJ)
Cooper pair box
tunable:  external (continuous) gate charge nx
 EJ by means of a SQUID loop
Cooper pair box
Cooper pair number,
phase difference
voltage
across junction
current
through junction
Cooper pair box
Cooper pair box
V
IJ
Cx
Cj
E
E
C
J
2
CHARGE BASIS
n
(
)
(
)
å
å
2


+
+
+
n
n
n
n
n
n
1
n
1
n
x
n
N
Charging
Josephson tunneling
From the CPB to a spin1/2
H =
In the 0>, 1>
subspace
Hamiltonian of a spin
In a magnetic field
Magnetic field in the xz plane
Coherent dynamics  experiments
Schoelkopf et al, Yale
NIST
Chiorescu et al 2003
Nakamura et al 1999
See also exps by
Vion et al 2002
Charge qubit coupling  1
EJ1 C
F
nx Cx
EJ2 C
F
EJ2 C
Vx
EJ1 C
Vx
Inductance
nx Cx
L
Charge qubit coupling  2
EJ1 C
F
nx Cx
EJ2 C
F
EJ2 C
Capacitance
EJ1 C
nx Cx
Charge qubit coupling  3
EJ1 C
F
nx Cx
EJ C
F
EJ2 C
Josephson Junction
F
Tunable coupling
Variable electrostatic transformer
Untunable couplings = more complicated gating
The effective coupling is due to
the (nonlinear) Josephson element
The coupling can be switched off
even in the presence of parasitic
capacitances
Averin & Bruder 03
m>
m+1>
~Ec
qubit
Ej
0>
1>
Leakage
The Hilbert space is larger than the computational space
Consequences:
a) gate operations differ from ideal ones (fidelity)
b) the system can leak out from the computational
space (leakage)
Leakage
Two qubit gate Fidelity
One qubit gate Fidelity
Sources of decoherence in charge qubits
electromagnetic fluctuations
of the circuit (gaussian)
discrete noise due to fluctuating
background charges (BC)
trapped in the substrate or in the junction
Z
Quasiparticle tunneling
Reduced dynamics – weak coupling
Full density matrix
TRACE OUT the environment
RDM for the qubit: populations and coherences
Reduced dynamics – weak coupling
(e = 0 , W = EJ)
no adiabatic term
optimal point
(EJ =0 , W = e)
no relaxation
E
charged impurities
Electronic band
Fluctuations due to the environment
HQ
E
z
di+di
x
Background charges in charge qubits
E is a stray voltage or current or charge polarizing the qubit
Charged switching impurities
close to a solid state qubit
electrostatic coupling
g=v/g weak vs strongly coupled charges
“Weakly coupled” charge
Decoherenceonlydepends on
= oscillator environment
EJ=0 – exact solution
Constant of motion
~
EJ=0 – exact solution
In the long time behavior for a single Background Charge
~
The contribution to dephasing due to “strongly coupled” charges
(slow charges) saturates in favour of an almost static energy shift
Background charges and 1/f noise
Experiments: BCs are responsibe for 1/f noise in SET
devices.
Standard model: BCs distributed according to
with
yield the 1/fpower spectrum
from experiments
Warning:an environment with strong memory effects due
to the presence of MANY slow BCs
Split
Slow vs fast noise
Twostage
elimination
Initial defocusing due to 1/f noise
z
HQ
x
Optimal point
s
2
Paladino et al. 04
variance
see also Shnirman Makhlin, 04
Rabenstein et al 04
Initial defocusing due to 1/f noise
z
HQ
Initial suppression
of the signal due essentially to inhomogeneuos
broadening
(no recalibration)
x
Optimal point
Falci, D’Arrigo, Mastellone, Paladino, PRL 2005, condmat/0409522
with recalibration
Standard measurements
no recalibration
SPA