josephson qubits
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Josephson qubits
P. Bertet
SPEC, CEA Saclay (France),Quantronics group
0 100 200 300 4000.0
0.2
0.4
0.6
0.8
1.0
11
00
01
switc
hing
pro
babi
lity
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
… ANDMUCH 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 box3) Decoherence of superconducting qubits
Ene
rgy
(a.u
)
x (a.u.)
Real atoms
Hydrogen atom
2
0
( )4eV rr
2ˆˆ ˆ( )2pH V rm
quantization ˆ ˆ,r p i
|0>
|1>|2>
|3>
« two-level atom »
E01=E1-E0=hn01
I.1) Introduction
Real atoms
Hydrogen atom
2
0
( )4eV rr
2ˆˆ ˆ( )2pH V rm
ˆ ˆ,r p i
Frequency n (a.u)
n01
FWHMG/2
P(1
) (a.
u)
Spectroscopy (weak field)
laser( ) ( )IH t d E t
0( ) cos 2E t E tn
+ spontaneous emission G
OpticalBloch
equations
P(1
)
0
1
Time (a.u)
Rabi oscillations (short pulses, strong field at n=n01)
0R
d Ef
h
|0> |0>
|1>
1 0 12
I.1) Introduction
quantization
X,f
E
0|0>|1>
|2>|3>
|4>
Quantum regime ??
ˆ ˆ,x p i ˆ ˆ, iQ
Electrical harmonic oscillator
k
m
x
2 2
( , )2 2kx pH x p
m
+qf
-q
2 2
( , )2 2
H QQL C
0km
01LC
( ) ( ') 't
t V t dt
( ) ( ') '
tQ t i t dt
I.1) Introduction
LC oscillator in the quantum regime ?
2 conditions :
X,f
E
|0>|1>
|2>|3>
|4>0kT
Typic : L nH C pF 0 5GHzn
At T=30mK : 0 8hkTn
X,f
E
|0>|1>
|2>|3>
|4>1Q
OK if dissipation negligible
Superconductors at T<<Tc
I.1) Introduction
T<<Tc : dissipation negligble at GHz frequencies
Q=4 1010
at 51GHzand at 1K
Microwave superconducting resonators
Tc(Nb)=9.2K
S. Kuhr et al., APL 90, 164101 (2006)
Quantumregime
I.1) Introduction
Necessity of anharmonicity
How to prepare |1> ?
X,f
E
0|0>|1>
|2>|3>
|4>+qf
-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 : 2d QV
e dt C
Josephson DC relation : sinCI I
R L= mod(2 )=θ -θ 0,/2e
θ 2
N=Q/2e
RθLθ
( ) ( ') 't
t V t dt
( ) ( ') '
tQ 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 IeI I I
POTENTIAL ENERGY ( ) cos cos2C
J JIE Ee
Josephson AC relation : 2d QV
e dt C
Josephson DC relation : sinCI I
R L= mod(2 )=θ -θ 0,/2e
θ 2
N=Q/2e
RθLθ
( ) ( ') 't
t V t dt
( ) ( ') '
tQ 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)
branchnode
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
eb
( ) ( ') '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
eb
4) Define node fluxes = sum of branch fluxes from ground
4= b+ d
51
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 iL L L
taking into account constraints imposed by external biases (fluxes or charges)
a
c d
eb
4= b+ d
51
3 2
ext
1 2
21 22 1cos
2ext
pot JL EL
Hamiltonian of an arbitrary circuit
=
=
Conjugate variables : ii
LQ
Classical Hamiltonian ( , )i i i iH Q Q L
Quantum Hamiltonian ˆˆ( , )i iH Q With ˆˆ ,i iQ i
ˆ ˆ( , )i iH n ˆ ˆ,i in i or with
ˆˆ / 2i in Q eˆ ˆ (2 / )i i e
a
c d
eb
4= b+ d
51
3 2
Different types of qubits
Cooper-pair boxes Phase qubits
CJ EE /0.1 1 50
Flux qubits
50 410
Junctions sizes
.01 to 0.04 mm2 .04mm2 100mm2
NISTSanta Barbara
TU DelftMIT
BerkeleyNEC
SaclayChalmers
NECYale
ETH Zurich
ShapeOf the
PotentialEnergy
-1 0 1
Ep (
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
CJ EE /0.1 1 50
Flux qubits
50 410
Junctions sizes
.01 to 0.04 mm2 .04mm2 100mm2
NISTSanta Barbara
TU DelftMIT
BerkeleyNEC
SaclayChalmers
NECYale
ETH Zurich
ShapeOf the
PotentialEnergy
-1 0 1
Ep (
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
Vg
The Cooper-Pair Box
1 degree of freedom1 knob
θ̂,N̂ i
ˆ ˆˆ cos2gC JEH = ( -E N N ) -
JCE ,E
2(2 )2ceEC
« charging energy »
Vg
20ˆ cosˆ
cosˆˆ ˆ
J2
gCH = ( -N ) -2
E E2L
N
1θ̂ , ˆ 1N i2 d° of freedom
JCE ,E1N
2N
2θ̂ , ˆ 2N i
θ1
θ2L
inductance
2 knobs
small
0
2
J
LE
2 11 2
ˆ ˆθ -θ ˆ ˆ ˆ2
,θ̂= =N N -N
ior
1 21 2
ˆ ˆˆˆ ˆ ˆ=θ +θ =2
, N +NK
i
θ
The split CPB
Vg
ˆ cos cosˆ ˆ2JgCH = (E N E-N ) -
2
θ̂,N̂ i1 d° of freedom
JCE ,E
N
2 knobs
0
δ=
tunable JE
The split CPB
Energy levels of the CPB
ˆ ( , ) ( )cosˆˆ C J2
g gH N = ( - EN ) -E N
Solve either in charge basis |N>
( , ), ( , )k g k gE N N
N ( ) ,k k NN
c N
ˆg
N
J2C
EE N N N N+1 NH = ( -N ) -2
N N+1
... ......
...
...
...
...
2
C Jg
2
C g
2
g
J J
CJ
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 ( , ) ( )cosˆˆ C J2
g gH N = ( - EN ) -E N
… or in phase basis |>
( , ), ( , )k g k gE N N
0,2 ( )2
0
( )k kd
ˆ ( , ) ( ) cos
2g gC JH N = ( -N ) -E1E
i
Solve Mathieu equation
( )cos
Jk kgC k k2 ( ) (( -N ) - =E)1 E
i(E )
I.2) Cooper-Pair Box
Two simple limits : (1) ( )J CE E
0 0.5 1 1.5 2 2.5 3
0
1
2
Ng
Ene
rgy
0 0.5 1 1.5 2 2.5 3
0
1
2E jE c 0
|N=0>
|N=1>
|N=2>
EJ/Ec=0
N
0 ( )c N
N
1( )c N3 2 1 0 1 2 3
1
3 2 1 0 1 2 3
1 0 0
0.5
0 0
0.5
20 ( )
21( )
(charge regime)
Two simple limits : (1) ( )J CE E
0 0.5 1 1.5 2 2.5 3
0
1
2
Ng
Ene
rgy
EJ/Ec=0.1
E0(Ng)E1(Ng)
E2(Ng)
3 2 1 0 1 2 3
1
3 2 1 0 1 2 3
1
0 ( )c N
1( )c N
0 0
0.52
0 ( )
21( )
0 0
0.5
Ng=0.01
(charge regime)
QUBIT
0 0
0.5
Two simple limits : (1) ( )J CE E
0 0.5 1 1.5 2 2.5 3
0
1
2
Ng
Ene
rgy
EJ/Ec=0.1
E0(Ng)
E1(Ng)
E2(Ng)
0 ( )c N
1( )c N
20 ( )
21( )
Ng=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
0.00.51.01.52.0
From ( )J CE E to ( )J CE E E
nerg
y
Ng
EJ/Ec=0.5
0.0 0.2 0.4 0.6 0.8 1.0 2
1
0
1
2
3
Ng
EJ/Ec=2
0.0 0.2 0.4 0.6 0.8 1.0
4
2
0
2
Ene
rgy
Ng
EJ/Ec=5
0.0 0.2 0.4 0.6 0.8 1.0 8
6
4
2
0
2
Ng
EJ/Ec=10
STILL A QUBIT !
Two simple limits : (2) ( )J CE 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
Ng
EJ/Ec=10
4 2 0 2 4 1.0 0.5
0.00.51.0
4 2 0 2 4 1.0 0.5
0 .00 .51 .0
0 ( )c N
1( )c N
0 0
0.5
0 0
0.5
20 ( )
21( )
J. Koch et al., PRA (2008)
Experimental spectrum of a transmon
J. Schreier et al., PRB (2008)
Ng(t)=DNgcost
One-qubit gates
TRANSMON QUBIT
ˆ ˆ ˆ( ) cost 2
C g JH = E N-N -E
ˆ ˆ ˆˆcos 2 gN D2C J CH = E N -E - E cos tN
transmon drive
01( )2 z
cosR xt Two-level
approximation
2 0 1R C gE N N DI.2) Cooper-Pair Box
I.2) Cooper-Pair Box
One-qubit gates
TRANSMON QUBIT
f0
Rotation : X
|0>
|1>
X
Z
Y
=01
I.2) Cooper-Pair Box
One-qubit gates
TRANSMON QUBIT
f0f0f
Rotation : Xf
|0>
|1>
X
Z
Y fXf
X
=01
I.2) Cooper-Pair Box
One-qubit gates
TRANSMON QUBIT
f0f0f
Rotation : Xf
|0>
|1>
X
Z
Y fXf
X
Z rotation
=01
All rotations on Bloch sphere
Fidelity ? 99%J.M. Chow et al., PRL 102, 090502 (2009)
Decoherence
d(t)
Ng+dNg(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 21 1 , 01( )
2T D S
G
environmental density ofmodes at qubit frequency
Pure dephasing
( )i te 0 1
1
0
01( )i
t
i(t)
1 12 2 2T
GG G
2, (0)zD S G
2( )( ) ti tf t e e
G
Low-frequencynoise
01,zD
,1 0 1HD
I.3) Decoherence
Decoherence
d(t)
Ng+dNg(t)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
I.3) Decoherence
Decoherence
d(t)
Ng+dNg(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)
Ng+dNg(t)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
R
R
2) MICROSCOPIC
e- Spin flips
Low-frequency noise well studied :
Charge noise3 2(10 )( )NgS
Flux noise6 2
010( )( )S
I.3) Decoherence
High-frequency (GHz)microscopic noise TOTALLY UNKNOWN !!
I.3) Decoherence
Decoherence
d(t)
Ng+dNg(t)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
R
R
2) MICROSCOPIC
e- Spin flips
Low-frequency noise well studied :
Charge noise3 2(10 )( )NgS
Flux noise6 2
010( )( )S
High-frequency (GHz)microscopic noise TOTALLY UNKNOWN !!
CPB in charge regime 2 10 100T ns 2 1 100T sm
Transmon 2 1 10T ms 2 1 100T sm
State-of-the-art coherence times
T1=1-2ms
Schreier et al., PRB 77, 180502 (2008)
T2=1-3ms
I.3) Decoherence
Very recent breakthrough: transmon in 3D cavity
T1=60msT2=15ms
H. Paik et al., quant-ph (2011)
ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET
I.3) Decoherence
I.3) Decoherence
SiO2
PMMA-MAAPMMA
e-
O2Al/Al2O3/Al junctions
1) e-beam patterning2) development3) first evaporation4) oxidation5) second evap.6) lift-off7) electrical test
small junctions Multi angle shadow evaporation
Fabrication techniquessmall junctions e-beam lithography
I.3) Decoherence
gate 160 x160 nm
QUANTRONIUM (Saclay group)
FLUX-QUBIT (Delft group)
I.3) Decoherence
I.3) Decoherence
40mm
2mm
TRANSMON QUBIT (Saclay group)
END OF FIRST LECTURE
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