pptx - EPIQ

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Nov 15, 2013 (3 years and 11 months ago)

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Josephson
qubits

P.
Bertet

SPEC, CEA Saclay (France
),

Quantronics

group

0
100
200
300
400
0.0
0.2
0.4
0.6
0.8
1.0
11
00
01


switching probability
swap duration (ns)
10
Introduction : Josephson circuits for quantum
physics

M.H. Devoret, J.M. Martinis and J. Clarke,
PRL

85
, 1908 (1985)

M.H. Devoret, J.M. Martinis and J. Clarke,
PRL

85
, 1543 (1985)

YES THEY CAN

Discrete

energy

levels

… to
genuine

quantum information and quantum
optics

on a chip

CAN MACROSCOPIC «

MAN
-
MADE

» ELECTRICAL


CIRCUITS BEHAVE QUANTUM
-
MECHANICALLY ????

From

a
fundamental

question
(25
years

ago
) ….

M.
Hofheinz

et al., Nature (2009)

Q. state engineering and

Tomography

M.
Neeley

et al., Nature (2011)

3
-
qubit

entanglement

… AND

MUCH MORE

TO COME !!

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2: Qubit readout and circuit quantum electrodynamics

Lecture 3: 2
-
qubit gates and quantum processor architectures

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2:
Qubit

readout

and circuit quantum
electrodynamics

Lecture 3: 2
-
qubit

gates

and quantum processor architectures

1) Introduction:
Hamiltonian

of an
electrical

circuit

2) The Cooper
-
pair box

3) Decoherence of superconducting qubits

Energy (a.u)
x (a.u.)
Real atoms

Hydrogen atom

2
0
( )
4
e
V r
r

 
2
ˆ
ˆ
ˆ
( )
2
p
H V r
m
 
quantization



ˆ ˆ
,
r p i

|0>

|1>

|2>

|3>

«

two
-
level atom

»

E
01
=E
1
-
E
0
=h
n
01

I.1) Introduction

Real atoms

Hydrogen atom

2
0
( )
4
e
V r
r

 
2
ˆ
ˆ
ˆ
( )
2
p
H V r
m
 


ˆ ˆ
,
r p i

Frequency
n

(a.u)

n
01

FWHM

G/2

P(1) (a.u)

Spectroscopy


(weak field)

laser

( ) ( )
I
H t d E t
 
0
( ) cos 2
E t E t
n

+ spontaneous emission
G

Optical

Bloch

equations

P(1)

0

1

Time (a.u)

Rabi oscillations

(short

pulses, strong field at
n
=
n
01
)

0
R
d E
f
h

|0>

|0>

|1>



1
0 1
2

I.1) Introduction

quantization

X
,
f

E

0

|0>

|1>

|2>

|3>

|4>

Quantum regime ??



ˆ ˆ
,
x p i

ˆ
ˆ
,
i
Q
 

 

Electrical harmonic oscillator

k

m

x

2 2
(,)
2 2
kx p
H x p
m
 
+
q

f

-
q

2 2
(,)
2 2
H
Q
Q
L C

  
0
k
m


0
1
LC


( ) (')'
t
t V t dt

 

( ) (')'
t
Q t i t dt



I.1) Introduction

LC oscillator in the quantum regime ?

2 conditions :

X
,
f

E

|0>

|1>

|2>

|3>

|4>

0
kT


Typic :

L nH

C pF

0
5
GHz
n

At
T=30mK

:

0
8
h
kT
n

X
,
f

E

|0>

|1>

|2>

|3>

|4>

1
Q

OK if dissipation negligible

Superconductors

at T<<Tc

I.1) Introduction

T<<Tc

: dissipation negligble at GHz frequencies

Q=4 10
10

at 51GHz

and at 1K

Microwave superconducting resonators

T
c
(Nb)=9.2K

S. Kuhr et al.,
APL

90
, 164101 (2006)

Quantum

regime

I.1) Introduction

Necessity of anharmonicity

How to prepare |1> ?

X
,
f

E

0

|0>

|1>

|2>

|3>

|4>

+
q

f

-
q

Need non
-
linear
and
non dissipative
element : Josephson junction

I.1) Introduction

Basics of the Josephson junction

The building block of

superconducting qubits

Josephson AC relation :

2
d Q
V
e dt C

 
Josephson DC relation :

sin
C
I I




R L
= mod(2 )=
θ -θ 0,
/2e
θ
2
 


N
=
Q
/2e

R
θ
L
θ
( ) (')'
t
t V t dt

 

( ) (')'
t
Q t i t dt



B. Josephson,
Phys. Lett.

1
, 251 (1962)

P.W. Anderson & J.M. Rowell,
Phys. Rev.

10
, 230 (1963)

S. Shapiro,
Phys. Rev.

11
, 80 (1963)

I.1) Introduction

Basics of the Josephson junction

The building block of

superconducting qubits

Classical variables ??

NON
-
LINEAR INDUCTANCE



2
( )
2 1/
J
C C
L I
eI I I


POTENTIAL ENERGY

( ) cos cos
2
C
J J
I
E E
e
  
   
Josephson AC relation :

2
d Q
V
e dt C

 
Josephson DC relation :

sin
C
I I




R L
= mod(2 )=
θ -θ 0,
/2e
θ
2
 


N
=
Q
/2e

R
θ
L
θ
( ) (')'
t
t V t dt

 

( ) (')'
t
Q t i t dt



I.1) Introduction

Hamiltonian

of an
arbitrary

circuit

=

HAMILTONIAN ???

M. H. Devoret, p. 351 in
Quantum fluctuations
(Les Houches 1995)

G.
Burkard

et al.,
Phys. Rev. B
69
, 064503 (2004)


Correct procedure described in :

G.
Wendin

and V.
Shumeiko
,
cond
-
mat/0508729

M.H. Devoret, lectures
at

Collège de France (2008) accessible online

=

Hamiltonian

of an
arbitrary

circuit

=

HAMILTONIAN ???

=

1)
Identify

the relevant
independent

circuit variables

2)
Write

the circuit
Lagrangian

3)
Determine

the canonical
conjugate

variables and the
Hamiltonian

Hamiltonian

of an
arbitrary

circuit

=

=

1)
Choose

reference

node

(
ground
)

branch

node

Identifying

the
relevant
independent

circuit variables

Hamiltonian

of an
arbitrary

circuit

=

=

1)
Choose

reference

node

(
ground
)

2)
Choose

«

spanning

tree

» (no
loop
)

Identifying

the
relevant
independent

circuit variables

Hamiltonian

of an
arbitrary

circuit

=

=

1)
Choose

reference

node

(
ground
)

2)
Choose

«

spanning

tree

» (no
loop
)

3)
Define

«

tree

branch

fluxes

»


a


c


d


e


b

( ) (')'
t
i
t V t dt

 

Identifying

the
relevant
independent

circuit variables

Hamiltonian

of an
arbitrary

circuit

=

=

1)
Choose

reference

node

(
ground
)

2)
Choose

«

spanning

tree

» (no
loop
)

3)
Define

«

tree

branch

fluxes

»


a


c


d


e


b

4)
Define

node

fluxes
=
sum

of
branch

fluxes
from

ground


4
=


b
+


d


5


1


3


2

( ) (')'
t
i
t V t dt

 

Identifying

the
relevant
independent

circuit variables

Hamiltonian

of an
arbitrary

circuit

=

=

Write

Lagrangian

(,) ( ) ( )
i i el i pot i
L L L
     
taking

into

account

constraints

imposed

by
external

biases

(fluxes or charges)


a


c


d


e


b


4
=


b
+


d


5


1


3


2


ext


1


2





2
1 2
2 1
cos
2
ext
pot J
L E
L
  
   
Hamiltonian

of an
arbitrary

circuit

=

=

Conjugate

variables :

i
i
L
Q



Classical

Hamiltonian


(,)
i i i i
H Q Q L
   

Quantum
Hamiltonian


ˆ
ˆ
(,)
i i
H Q

With


ˆ
ˆ
,
i i
Q i
 
 
 
ˆ
ˆ
(,)
i i
H n

ˆ
ˆ
,
i i
n i

 

 
or

with

ˆ
ˆ
/2
i i
n Q e

ˆ
ˆ
(2/)
i i
e



a


c


d


e


b


4
=


b
+


d


5


1


3


2

Different types of qubits

Cooper
-
pair boxes

Phase qubits

C
J
E
E
/
0.1 1 50
 
Flux qubits

50

4
10
Junctions

sizes

.01 to 0.04
m
m
2

.04
m
m
2

100
m
m
2

NIST

Santa Barbara

TU Delft

MIT

Berkeley

NEC

Saclay

Chalmers

NEC

Yale

ETH Zurich

Shape

Of the

Potential

Energy

-1
0
1
E
p
(a.u)
/
(rad)
-2
0
2

Ep (a.u)


m
/

-2
0
2
4
Ep (a.u)

(rad)
Charge qubit/Quantronium/Transmon

I.2) Cooper
-
Pair Box

Different types of qubits

Cooper
-
pair boxes

Phase qubits

C
J
E
E
/
0.1 1 50
 
Flux qubits

50

4
10
Junctions

sizes

.01 to 0.04
m
m
2

.04
m
m
2

100
m
m
2

NIST

Santa Barbara

TU Delft

MIT

Berkeley

NEC

Saclay

Chalmers

NEC

Yale

ETH Zurich

Shape

Of the

Potential

Energy

-1
0
1
E
p
(a.u)
/
(rad)
-2
0
2

Ep (a.u)


m
/

-2
0
2
4
Ep (a.u)

(rad)
Charge qubit/Quantronium/Transmon

I.2) Cooper
-
Pair Box

V
g
The
Cooper
-
Pair Box

1 degree of freedom

1 knob

ˆ
θ
,
ˆ
N
 

 
i
ˆ
ˆ
ˆ
cos
2
g
C
J
E
H = ( -
E N
N ) -

J
C
E
,
E
2
(2 )
2
c
e
E
C


«

charging

energy

»

V
g


2
0
ˆ
cos
ˆ
cos
ˆ
ˆ
ˆ
J
2
g
C
H = ( -N ) -
2
E
E
2L
N






1
ˆ
θ
,
ˆ
1
N
 

 
i
2 d
°

of freedom

J
C
E
,
E
1
N
2
N
2
ˆ
θ
,
ˆ
2
N
 

 
i
θ
1
θ
2
L
inductance

2 knobs

small

0
2
J
L
E


2 1
1 2
ˆ ˆ
θ -θ
ˆ ˆ ˆ
2
,
ˆ
θ= =
N N -N
 

 
 
i
or

1 2
1 2
ˆ ˆ
ˆ
ˆ ˆ ˆ
=
θ +θ =
2
,
N +N
K

 

 
 
 
i

θ
The split CPB

V
g
ˆ
cos cos
ˆ
ˆ
2
J
g
C
H = (
E N
E
-N ) -
2


ˆ
θ
,
ˆ
N
 

 
i
1 d
°

of freedom

J
C
E
,
E
N
2 knobs

0
δ=


tunable

J
E
The split CPB

Energy levels of the CPB

ˆ
(,) ( )cos
ˆ
ˆ
 

C
J
2
g g
H N = ( -
E
N ) -
E N
Solve
either

in
charge

basis |N>



(,),(,)
k g k g
E N N

 
N

(

)

,
k k N
N
c N





ˆ
g
N
J
2
C
E
E N N N N+1 N
H = ( -N ) -
2
N N+1



......
...
...
...
...
...
2
C
J
g
2
C g
2
g
J J
C
J
N=-1 N=0 N=1
.........
0...
......
...0
........
...
E (-1- N )
E (0 - N )
-E/2
-E/2 -
E (1- N
E/2
-E
.
/2
)
 
 
 
 
 
 
 
 
Diagonalize

I.2) Cooper
-
Pair Box

Energy levels of the CPB

ˆ
(,) ( )cos
ˆ
ˆ
 

C
J
2
g g
H N = ( -
E
N ) -
E N

or

in
phase

basis |

>



(,),(,)
k g k g
E N N

 


0,2
 

(

)

2
0
( )
k k
d

 


ˆ
(,) ( )cos




 
2
g g
C
J
H N = ( -N ) -
E
1
E
i
Solve

Mathieu
equation

( )cos

 





J
k k
g
C
k
k
2
( ) (
( -N ) - =E
)
1
E
i
(
E
)
I.2) Cooper
-
Pair Box

Two simple limits : (1)

( )
J C
E E

0
0.5
1
1.5
2
2.5
3
0
1
2
N
g

Energy

0
0.5
1
1.5
2
2.5
3
0
1
2
Ej
Ec
0
E
J
/
E
c
=0

N

0
( )
c N
N

1
( )
c N
3
2
1
0
1
2
3
1
3
2
1
0
1
2
3
1
0
0
0.5
0
0
0.5




2
0
( )

2
1
( )

(charge
regime
)

Two simple limits : (1)

( )
J C
E E

0
0.5
1
1.5
2
2.5
3
0
1
2
N
g

Energy

E
J
/
E
c
=0.1

E
0
(
N
g
)

E
1
(
N
g
)

E
2
(
N
g
)

3
2
1
0
1
2
3
1
3
2
1
0
1
2
3
1
0
( )
c N
1
( )
c N
0
0
0.5
2
0
( )

2
1
( )

0
0
0.5
N
g
=0.01

(charge
regime
)

QUBIT

0
0
0.5
Two simple limits : (1)

( )
J C
E E

0
0.5
1
1.5
2
2.5
3
0
1
2
N
g

Energy

E
J
/
E
c
=0.1

E
0
(
N
g
)

E
1
(
N
g
)

E
2
(
N
g
)

0
( )
c N
1
( )
c N
2
0
( )

2
1
( )

N
g
=0.5

3
2
1
0
1
2
3
1
3
2
1
0
1
2
3
1
0
0
0.5
(charge
regime
)

QUBIT

0.0
0.2
0.4
0.6
0.8
1.0
1.0
0.5
0.0
0.5
1.0
1.5
2.0
From

( )
J C
E E

to

( )
J C
E E

Energy

N
g

E
J
/
E
c
=0.5

0.0
0.2
0.4
0.6
0.8
1.0
2
1
0
1
2
3
N
g

E
J
/
E
c
=2

0.0
0.2
0.4
0.6
0.8
1.0
4
2
0
2
Energy

N
g

E
J
/
E
c
=5

0.0
0.2
0.4
0.6
0.8
1.0
8
6
4
2
0
2
N
g

E
J
/
E
c
=10

STILL A QUBIT !

Two simple limits : (2)

( )
J C
E E

I.2) Cooper
-
Pair Box

(phase
regime
)

0.0
0.2
0.4
0.6
0.8
1.0
8
6
4
2
0
2
N
g

E
J
/
E
c
=10

4
2
0
2
4
1.0
0.5
0.0
0.5
1.0
4
2
0
2
4
1.0
0.5
0.0
0.5
1.0
0
( )
c N
1
( )
c N
0
0
0.5
0
0
0.5
2
0
( )

2
1
( )

J. Koch et al., PRA (2008)

Experimental

spectrum

of a
transmon

J.
Schreier

et al., PRB (2008)

N
g
(t)=
D
N
g
cos

t

One
-
qubit gates



TRANSMON QUBIT



ˆ ˆ
ˆ
( ) cos
t

2
C g J
H = E N-N -E
ˆ ˆ ˆ
ˆ
cos 2
g
N

 D
2
C J C
H = E N -E - E cos tN
transmon

drive

01
( )
2
z



 
cos
R x
t

 
Two
-
level

approximation

2 0 1
R C g
E N N
  D
I.2) Cooper
-
Pair Box

I.2) Cooper
-
Pair Box

One
-
qubit gates



TRANSMON QUBIT

f
0

Rotation :

X

|0>

|1>

X

Z

Y


=

01

I.2) Cooper
-
Pair Box

One
-
qubit gates



TRANSMON QUBIT

f
0

f
0
f

Rotation :

X
f

|0>

|1>

X

Z

Y

f

X
f

X


=

01

I.2) Cooper
-
Pair Box

One
-
qubit gates




TRANSMON QUBIT

f
0

f
0
f

Rotation :

X
f

|0>

|1>

X

Z

Y

f

X
f

X

Z
rotation


=

01

All rotations on Bloch
sphere

Fidelity

?

99%
J.M. Chow et al., PRL 102, 090502 (2009)

Decoherence



d

(t)

N
g
+dN
g
(t)

Noise in Hamiltonian parameters

DECOHERENCE

MAJOR OBSTACLE TO

QUANTUM COMPUTING

I.3)
Decoherence

1
0
01

Relaxation

(Spontaneous emission)

Decoherence in superconducting qubits

(Ithier et al., PRB 72, 134519, 2005)

1 2
1 1,01
( )
2
T D S
 




 G 
environmental
density of

modes

at qubit frequency

Pure dephasing

( )
i t
e

 

0
1
1
0
01
( )
i


t


i
(t)

1
1
2 2
2
T


G
 G  G
2
,
(0)
z
D S
  

G 
2
( )
( )
t
i t
f t e e


G
 
Low
-
frequency

noise

01
,
z
D






,
1
0 1
H
D






I.3)
Decoherence

Decoherence



d

(t)

N
g
+dN
g
(t)

Noise in Hamiltonian parameters

DECOHERENCE

Origin of the noise ???

I.3)
Decoherence

Decoherence



d

(t)

N
g
+dN
g
(t)

Noise in Hamiltonian parameters

DECOHERENCE

Origin of the noise ???

R

R

1)
ELECTROMAGNETIC

Low
-
frequency : Johnson
-
Nyquist due to thermal noise

High
-
frequency : spontaneous emission (quantum noise)


Under control

I.3)
Decoherence

Decoherence



d

(t)

N
g
+dN
g
(t)

Noise in Hamiltonian parameters

DECOHERENCE

Origin of the noise ???

R

R

2) MICROSCOPIC

e
-

Spin flips

Low
-
frequency noise

well studied :

Charge noise

3
2
(
10
)
( )
Ng
S



Flux noise

6
2
0
10
( )
( )
S





I.3)
Decoherence

High
-
frequency

(GHz)

microscopic

noise

TOTALLY UNKNOWN !!

I.3)
Decoherence

Decoherence



d

(t)

N
g
+dN
g
(t)

Noise in Hamiltonian parameters

DECOHERENCE

Origin of the noise ???

R

R

2) MICROSCOPIC

e
-

Spin flips

Low
-
frequency noise

well studied :

Charge noise

3
2
(
10
)
( )
Ng
S



Flux noise

6
2
0
10
( )
( )
S





High
-
frequency

(GHz)

microscopic

noise

TOTALLY UNKNOWN !!

CPB in charge regime

2
10 100
T ns

2
1 100
T s
m

Transmon

2
1 10
T ms

2
1 100
T s
m

State
-
of
-
the
-
art coherence times

T
1
=1
-
2
m
s

Schreier et al.,
PRB
77, 180502 (2008)

T
2
=1
-
3
m
s

I.3)
Decoherence

Very recent breakthrough: transmon in 3D cavity

T
1
=60
m
s

T
2
=15
m
s

H. Paik et al., quant
-
ph (2011)

ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET

I.3)
Decoherence

I.3)
Decoherence

SiO
2

PMMA
-
MAA

PMMA

e
-
O
2

Al
/
Al
2
O
3
/
Al

junctions

1) e
-
beam patterning

2) development

3) first evaporation

4) oxidation

5) second evap.

6) lift
-
off

7) electrical test

small junctions


Multi angle shadow evaporation

Fabrication techniques

small junctions


e
-
beam lithography

I.3)
Decoherence

gate


160 x160 nm

QUANTRONIUM (Saclay group)

FLUX
-
QUBIT (Delft group)

I.3)
Decoherence

I.3)
Decoherence

40
m
m

2
m
m

TRANSMON QUBIT (Saclay
group)

END OF FIRST LECTURE