Diapositiva%201

1
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
2
Outline
Lecture 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
3
Adiabatic cyclic evolution

• The Hamiltonian of a quantum system
• depends 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 eigenstate it will adjust
to the instantaneous field
4
What happens to the quantum state if r(0)
r(T) ????
5
Parallel Transport
e(0)
e(T) ?
After a cyclic change of r(t) the vector
e(t) does NOT come back to the original direction
a
r(T)r(0)
The angle a depends on The circuit C on the
sphere
6
Quantum Parallel Transport
Schroedingers equation implements phase parallel
transport
Schroedingers equation
Adiabatic approx Instantaneous eigenstates
Look for a solution
7
Berry phase
The geometrical phase change of ygt along a
closed circuit r(T)r(0) is given by
- M.V.Berry 1984
8
Spin ½ in an external field
Adiabatic condition wB ltlt 1
The Berry phase is related to the solid angle
that C subtends at the degeneracy
9
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
Cyclic state
- Y. Aharonov and J. Anandan 1987
10
Aharonov-Anandan phase
Introducing such that
Dynamical phase
Geometrical phase

• Evolution does not need to be adiabatic
• Adiabatic changes of the external parameters are
  a way to have a cyclic state
• In the adiabatic limit

11
Aharonov-Anandan phase (Example)
The Hamiltonian
Initial state
evolves as
The state is cyclic after Tp/B
12
Experimental observations
Geometrical phases have been observed in a
variety of systems

• Aharonov-Bohm effect
• Quantum transport
• Nuclear Magnetic Resonance
• Molecular spectra

see Geometric Phases in Physics, A. Shapere and
F. Wilczek Eds
13
Is it possible to observe geometric phases in
a macroscopic system?
14
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

15
Cooper pair box
CHARGE BASIS
n
(
)
(
)
å
å
2



-
-
n
n
n
n
n
n
n
n
1
1
x
n
N
Charging
Josephson tunneling
16
From the CPB to a spin-1/2
In the 0gt, 1gt subspace
Hamiltonian of a spin In a magnetic field
Magnetic field in the xz plane
17
Asymmetric SQUID
H Ech (n -nx)2 -EJ (F) cos (f - a)
18
From the SQUID-loop to a spin-1/2
In the 0gt, 1gt subspace
HB - (1/2) B . s
19
Geometric interference in nanocircuits
HB - (1/2) B . s

20
Berry phase in superconducting nanocircuits
1/2
nx
Role of the asymmetry
21
Berry phase - How to measure

• Initial state
• Sudden switch to nx1/2
• Adiabatic loop

0gt (1/2½)gt -gt (1/2½)eigibgt
e-ig-ib-gt
22
Berry phase - How to measure

• Swap the states
• Adiabatic loop with opposite orientation
• Measure the charge

(1/2½)eigib-gt e-ig-ibgt (1/2½)e2ig-gt
e-2iggt P(2e)sin22g
23
Quantum computation

• Two-state system
• Preparation of the state
• Controlled time evolution
• Low decoherence
• Read-out

Intrinsic fault-taulerant for area-preserving
errors
Phase shifts of geometric origin
- J. Jones et al 2000 - P. Zanardi and M. Rasetti
1999
24
Geometric phase shift
sz- interaction
Controlled phase gate
25
Geometric phase shift
Two Cooper pair boxes coupled via a capacitance
Hcoupling - EK sz1sz2

• G. Falci et al 2000

26
Aharonov-Anandan phase is sup. nanocircuits

• AA phase in symmetric SQUID
• No dynamical phase
• Fast manipulation for qubits

H

• A. Blais and A.-M. S. Tremblay 2003

27
Non-abelian case
N degenerate
Control parameters
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.
Adiabatic assumption
28
Holonomic quantum computation
Dynamical approach
System S, with state space H , perform universal
QC
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

29
Josephson network for HQC
L. Faoro, J. Siewert and R. Fazio, PRL 2003
One excess Cooper pair in the four-island set up
There are four charges states jgt corresponding
to the position of the excess Cooper pair on
island j
L.M.Duan et al, Science 296,886 (2001)

30
Josephson network for HQC
31
One-bit operations
In order to obtain all single qubit
operations explicit realizations of
Rotation around the z-axis
Rotation around the y-axis
Intially we set
so the eigenstates
correspond to the logical states
32
One-bit operations
33
Two-bit operations

• The qubits are coupled by symmetric SQUIDS (can
  be switched off)
• Capacitive coupling can be neglected if the
  capacitances of the junctions are sufficiently
  small.

No 1st-order contributions only one Cooper pair
is allowed in each qubit set-up
34
Two-bit operations
No 1st-order contributions only one Cooper pair
is allowed in each qubit set-up
The non-vanishing 2nd-order contributions
unwanted diagonal 2nd-order contributions
35
Conditional phase shift
2nd-order energy shifts can be compensated by
adjusting gate voltages
DEGENERATE EIGENSTATES WITH 0 ENERGY EIGENVALUE
36
Adiabatic pumping
Charge transport, in absence of an external
bias, by changing system parameters

• Open systems modulation of the phase of the
  scattering matrix
• Closed systems periodic lifting of the Coulomb
  Blockade

Charge is transferred coherently Quantization
of transferred charge

• P.W. Brouwer 1998
• .
• H. Pothier et al 1992

37
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

38
Cooper pair pumping vs geometric phases
EXAMPLE
0gt 1gt eif1gt
39
Cooper pair pumping vs geometric phases
In order to relate the second term to the AA phase
Take the derivative with respect to the external
phase d
40
Cooper pair sluice
A. O. Niskanen, J. P. Pekola, and H. Seppä,
(2003).
t
Can be generalized to pump 2Ne per cycle. (N
1,2,?)
41
Cooper pair sluice - exp
The measured device
Gate line
Input coils
Junctions
SQUID loops