1.0
10
01
switching probability
0.8
0.6
00
0.4
0.2
0.0
11
0
100
Josephson qubits
P. Bertet
SPEC, CEA Saclay (France),
Quantronics group
200
300
swap duration (ns)
400
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) Two-qubit gates : SWAP gate and Control-Phase gate
2) Two-qubit quantum processor : Grover algorithm
3) Towards a scalable quantum processor architecture
4) Perspectives on superconducting qubits
Requirements for QC
High-Fidelity
Single Qubit Operations
High-Fidelity Readout
of Individual Qubits
0
Deterministic, On-Demand
Entanglement between Qubits
III.1) Two-qubit gates
1
Coupling strategies
1) Fixed coupling
F
H int
Entanglement on-demand ???
« Tune-and-go » strategy
Coupling
effectively OFF
III.1) Two-qubit gates
Coupling activated
in resonance for t
Entangled qubits
Interaction effectively OFF
Coupling strategies
2) Tunable coupling
((tt)) OFF
ON
Hint (λ)

Entanglement on-demand ???
A) Tune ON/OFF the coupling with qubits on resonance
Coupling OFF
(OFF)
III.1) Two-qubit gates
Coupling activated
for t by ON
Entangled qubits
Interaction OFF (OFF)
Coupling strategies
2) Tunable coupling
 (t )  OFF
 cos 1  2  t
Hint (λ)
Entanglement on-demand ???
B) Modulate coupling
IN THIS LECTURE : ONLY FIXED COUPLING
Coupling OFF
(OFF)
III.1) Two-qubit gates
Coupling ON
by modulating 
Coupling OFF
(OFF)
How to couple transmon qubits ?
1) Direct capacitive coupling
FI
FII
Vg,II
Vg,I
coupling capacitor Cc
H  Ec , I ( Nˆ I  N g , I )  EJ , I (F I ) cos ˆI
 Ec , II ( Nˆ II  N g , II )  EJ , II (F II ) cos ˆII
2
Ec , I Ec , II
( Nˆ I  N g , I )( Nˆ II  N g , II )
Ecc
Cc
g  (2e)
0I Nˆ I 1I 0II Nˆ II 1II
CI CII
H q,I  
01I (F I )
 z,I
2
 (F II )
  01
 z , II
2
II
H q , II
H c  g x , I  x , II
g ( I  II    I  II  )
2
(note : idem
for phase qubits)
How to couple transmon qubits ?
2) Cavity mediated qubit-qubit coupling
J. Majer et al., Nature 449, 443 (2007)
R
D>>g
QI
g2
g1
QI
Q II
geff=g1g2/D
H eff  g eff  I  II    I  II  
III.1) Two-qubit gates
Q II
iSWAP Gate
H int
H/ 
01I
2
 zI 
01II
2
 zII  g  I  II   I II 
« Natural » universal gate :
On resonance, ( 01
I
 01II )
00 10
01 11
0
0
0
1
0 cos( gt ) i sin( gt ) 0 
 U ( )
U int (t )  
0 i sin( gt ) cos( gt ) 0  int 2 g


0
0
0
1


III.1) Two-qubit gates
iSWAP
0
1

0 1/ 2
0 i / 2

0
0
0

i / 2 0 
 iSWAP

1/ 2 0 
0
1 
0
Example : capacitively coupled transmons with individual readout
(Saclay, 2011)
fast flux line
ei
coupling capacitor
λ/4
λ/4
JJ
Readout Resonator
readout
resonator
i(t)
1 mm
qubits
50 µm
coupling
capacitor
Transmon
qubit
Josephsonfrequency
junction control
200 µm
Example : capacitively coupled transmons with individual readout
frequency
control
50 µm
III.1) Two-qubit gates
Spectroscopy
n01I
frequencies (GHz)
8
n01
7
5.16
II
ncI
2g/ = 9 MHz
5.14
ncII
5.12
6
5.10
5
0,0
III.1) Two-qubit gates
0,2
0,4
fI,II/f0
0,6
0.376
0.379
fI/f0
A. Dewes et al., in preparation
SWAP between two transmon qubits
6.82 GHz
QB I
Drive
X
6.67 GHz
QB II
6.42 GHz
f01
QB II
5.32GHz
QB I
5.13 GHz
6.03GHz
Swap Duration
1,0
Pswitch (%)
0,8
10
01
Raw data
0,6
00
0,4
0,2
11
0,0
III.1) Two-qubit gates
0
100
200
Swap duration (ns)
SWAP between two transmon qubits
6.82 GHz
QB I
Drive
X
6.67 GHz
QB II
6.42 GHz
f01
QB II
5.32GHz
QB I
5.13 GHz
6.03GHz
Swap Duration
1,0
10
Pswitch (%)
0,8
01
Data
corrected
from
readout
errors
0,6
0,4
0,2
00
0,0
0
0
III.1) Two-qubit gates
iSWAP
100
200
100
200
swap duration (ns)
Swap300
duration (ns)
How to quantify entanglement ??
Need to measure rexp
Quantum state tomography
|0>
Z
X
Pswitch
Y
|1>
III.1) Two-qubit gates
1 
z /2
How to quantify entanglement ??
Need to measure rexp
Quantum state tomography
|0>
Z
/2(X)
X
Pswitch
Y
|1>
III.1) Two-qubit gates
1    / 2
y
How to quantify entanglement ??
Need to measure rexp
Quantum state tomography
|0>
Z
/2(Y)
X
Pswitch
Y
1 
x / 2
|1>
III.1) Two-qubit gates
M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006)
How to quantify entanglement ??
readouts
X,Y
iSWAP
Z
tomo.
I
II
I,X,Y
I
II
0
20
40
60
80ns
3*3 rotations*3 independent probabilities (P00,P01,P10) = 27 measured numbers
Fit experimental density matrix rexp
Compute fidelity
III.1) Two-qubit gates
F  Tr  rth1/ 2 r exp rth1/ 2 
How to quantify entanglement ??
1,0
switching probability
|10>
|01>
0,8
0,6
|00>
0,4
0,2
|11>
0,0
0
|00>
100
200
300
400
swap duration (ns)
ideal
measured
|01>
F=98%
|10>
F=94%
|11>
III.1) Two-qubit gates
A. Dewes et al., in preparation
SWAP gate of capacitively coupled phase qubits
M. Steffen et al., Science 313, 1423 (2006)
F=0.87
III.1) Two-qubit gates
The Control-Phase gate
Another universal quantum gate : Control-Phase
00 01 10 11
1
0
U 
0

0
0
1 0 0

0 1 0

0 0 1
0 0
00
01
10
11
Surprisingly, also quite natural with superconducting circuits
thanks to their multi-level structure
III.1) Two-qubit gates
F.W. Strauch et al., PRL 91, 167005 (2003)
DiCarlo et al., Nature 460, 240-244 (2009)
Control-Phase with two coupled transmons
DiCarlo et al., Nature 460, 240-244 (2009)
H int /  g eff 1  1L 0 R 0 L1R  h.c   g eff 2  1L1R 0 L 2 R  h.c 
III.1) Two-qubit gates
Spectroscopy of two qubits + cavity
right qubit
Qubit-qubit swap interaction
left qubit
Cavity-qubit interaction
Vacuum Rabi splitting
cavity
V
R
Flux bias on right
transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)
One-qubit gates: X and Y rotations
fL
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
I
cos(2 f L t )
V
R
Flux bias on right
transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)
One-qubit gates: X and Y rotations
fR
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
I
cos(2 f R t )
V
R
Flux bias on right
transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)
One-qubit gates: X and Y rotations
fR
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
Q
sin(2 f R t )
VR
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
Fidelity = 99%
see
J. Chow et al., PRL (2009)
Two-qubit gate: turn on interactions
VR
Conditional
phase gate
Use control lines to push
qubits near a resonance
cavity
V
R
Flux bias on right
transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)
Two-excitation manifold of system
02
11
Two-excitation
manifold
• Avoided crossing (160 MHz)
11  02
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)
Adiabatic conditional-phase gate
02
11
tf
f01  f10
 a  2   f a (t )dt
t0
2-excitation
manifold

11  ei11 11
tf
11  10  01  2   (t )dt
t0
01
1-excitation
manifold
01  e
10
10  e
Flux bias on right transmon (a.u.)
i01
i10
01
10
(Courtesy Leo DiCarlo)
Implementing C-Phase
00
01
10
11
1 0
 0 ei01
U 
0 0

0 0
0
0
0  00
0  01

0  10
i11  11
e 
ei10
0
00 01 10 11
Adjust timing of flux pulse so that
only quantum amplitude of 11
acquires a minus sign:
1
0
U 
0

0
C-Phase11
III.1) Two-qubit gates
0  00
1 0 0  01

0 1 0  10

0 0 1 11
0 0
11
(Courtesy Leo DiCarlo)
Implementing Grover’s search algorithm
First implementation of q. algorithm with superconducting qubits (using Cphase gate)
DiCarlo et al., Nature 460, 240-244 (2009)
0, x  x0
f ( x)  
1, x  x0
“Find x0!”
Position:
0
I
II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)
Implementing Grover’s search algorithm
0, x  x0
f ( x)  
1, x  x0
“Find x0!”
Position:
0
I
II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)
Implementing Grover’s search algorithm
0, x  x0
f ( x)  
1, x  x0
“Find x0!”
Position:
0
I
II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)
Implementing Grover’s search algorithm
0, x  x0
f ( x)  
1, x  x0
“Find x0!”
Position:
0
I
II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)
Implementing Grover’s search algorithm
Classically, takes on average 2.25 guesses to succeed…
Use QM to “peek” inside all cards, find the queen on first try
Position:
0
I
II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)
0 0
 1 0 algorithm
Grover’s
0 1 0 0

Oˆ   
 0 0 1 0 


0 0 0 1
“unknown”
unitary
operation:
Challenge:
Find the location
of the -1 !!!
(= queen)
Previously implemented in NMR: Chuang et al. (1998)
Linear optics: Kwiat et al. (2000)
Ion traps: Brickman et al. (2005)
0
R y /2
R y /2
ij
0
R y /2
R y /2
00
R y /2
R y /2
oracle
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)
Grover step-by-step
 ideal  00
Begin in ground state:
oracle
R y /2
0
b
0
R y /2
c
R y /2
10 d
e
R y /2
R y /2
00 f
g
R y /2
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
 ideal
Grover step-by-step
1
  00  01  10  11 
2
Create a maximal
superposition:
look everywhere
at once!
oracle
R y /2
0
b
0
R y /2
c
R y /2
10 d
e
R y /2
R y /2
00 f
g
R y /2
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
 ideal
Grover step-by-step
1
  00  01  10  11 
2
Apply the “unknown”
function, and
mark the solution
1
0
cU10  
0

0
0
0

0 1 0 

0 0 1
0
1
0
0
oracle
R y /2
0
b
0
R y /2
c
R y /2
10 d
e
R y /2
R y /2
00 f
g
R y /2
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
 ideal
Grover step-by-step
1

00  11 

2
Some more 1-qubit
rotations…
Now we arrive in
one of the four
Bell states
oracle
R y /2
0
b
0
R y /2
c
R y /2
10 d
e
R y /2
R y /2
00 f
g
R y /2
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
 ideal
Grover step-by-step
1
  00  01  10  11 
2
Another (but known)
2-qubit operation now
undoes the entanglement
and makes an interference
pattern that holds
the answer!
oracle
R y /2
0
b
0
R y /2
c
R y /2
10 d
e
R y /2
R y /2
00 f
g
R y /2
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
Grover step-by-step
 ideal  10
Final 1-qubit
rotations reveal the
answer:
The binary
representation of “2”!
Fidelity >80%
oracle
R y /2
0
b
0
R y /2
c
R y /2
10 d
e
R y /2
R y /2
00 f
g
R y /2
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
Towards a scalable architecture ??
1) Resonator as quantum bus
|register>
….
|0>
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
2) Control-Phase Gate between any pair of qubits Qi and Qj
|register>
….
|0>
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
2) Control-Phase Gate between any pair of qubits Qi and Qj
A) Transfer Qi state to resonator
….
SWAP
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
2) Control-Phase Gate between any pair of qubits Qi and Qj
A) Transfer Qi state to resonator
B) Control-Phase between Qj and resonator
….
C-Phase
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
2) Control-Phase Gate between any pair of qubits Qi and Qj
A) Transfer Qi state to resonator
B) Control-Phase between Qj and resonator
C) Transfer back resonator state to Qi
….
SWAP
III.3) Architecture
Problems of this architecture
Uncontrolled phase errors
1) Off-resonant coupling Qk to resonator
….
III.3) Architecture
Problems of this architecture
Uncontrolled phase errors
1) Off-resonant coupling Qk to resonator
2) Effective coupling between qubits + spectral crowding
geff
geff
….
III.3) Architecture
RezQu (Resonator + zero Qubit) Architecture
damped
resonators
memory
resonators
qubits
q
q
q
q
q
coupling bus
resonator
frequency
zeroing
memory
single gate
coupled gate
measure
(tunneling)
(courtesy J. Martinis)
RezQu Operations
q1
Idling
 g2 
 2 
D 
2
q2
Transfers
& single
qubit gate
|g
|g
r
q
0
0
0
g
q
q'
g
0
0
0
q'
i-SWAP
• |g reduces off coupling
(>4th order)
• Store in resonator
(maximum coherence)
C-Z
(CNOT class)
time
• Intrinsic transfer 99.9999%
q
q'
q
g
0
g
r
i-SWAP
0 1
Measure
r'
1 2
11  i 02   11
e tunnel
(courtesy J. Martinis)
Perspectives on superconducting qubits
T1=60ms
T2=15ms
H. Paik et al., arxiv:quant-ph (2011)
1) Better qubits ?? Transmon in a 3D cavity
REPRODUCIBLE improvement of coherence time (5 samples)
Perspectives on superconducting qubits
Quantum feedback : retroacting on the qubit to stabilize a given
quantum state
Quantum information processing : better gates and more qubits !
« Non-linear » Circuit QED
Resonator made non-linear by incorporating JJ.
Parametric amplification, squeezing, back-action on qubit
Hybrid circuits
CPW resonators : versatile playground for coupling many systems :
Electron spins, nanomechanical resonators, cold atoms,
Rydberg atoms, qubits, …