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
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 architecture4) Perspectives on superconducting qubits
Requirements for QC
Deterministic, On-DemandEntanglement between Qubits
High-Fidelity Readoutof Individual Qubits
0 1
High-Fidelity Single Qubit Operations
III.1) Two-qubit gates
Coupling strategies
1) Fixed coupling
intH
Entanglement on-demand ???« Tune-and-go » strategy
Coupling effectively OFF
Entangled qubitsInteraction effectively OFF
Coupling activatedin resonance for t
F
III.1) Two-qubit gates
Coupling strategies
2) Tunable coupling
intH (λ)
Entanglement on-demand ???A) Tune ON/OFF the coupling with qubits on resonance
( ) OFFt
Coupling OFF(OFF)
Coupling activatedfor t by ON
( ) ONt
Entangled qubitsInteraction OFF (OFF)
( ) OFFt
III.1) Two-qubit gates
Coupling strategies
2) Tunable coupling
intH (λ)
Entanglement on-demand ???B) Modulate coupling
Coupling OFF(OFF)
( ) OFFt
Coupling ONby modulating
1 2cos t
Coupling OFF(OFF)
III.1) Two-qubit gates
IN THIS LECTURE : ONLY FIXED COUPLING
How to couple transmon qubits ?1) Direct capacitive coupling
coupling capacitor Cc
Vg,IIVg,I
, , ,
,
,
,
,
,
,
,
ˆˆ( ) (ˆˆ( ) (
ˆ
)co
ˆ2 (
s
)
s
( )
)co
c II II g II J
c I I g I J I I
c I c III
II II II
g I II g IIcc
I
E EN
E N N E
N N
N
E
E
N
E
H
N
F
F
FI FII
(note : idemfor phase qubits)
01, ,
( )2
II
q I z IH F
01, ,
( )2
IIII
q II z IIH F
, ,
( )c x I x II
I II I II
H g
g
2 ˆ0(2 0) 1ˆ 1I I I II II
cII
II I
g eC
NCC
N
J. Majer et al., Nature 449, 443 (2007)
D>>g
g1 g2
Q I Q IIR
How to couple transmon qubits ?2) Cavity mediated qubit-qubit coupling
geff=g1g2/DQ I Q II
eff eff I II I IIH g III.1) Two-qubit gates
iSWAP Gate
01 01/2 2
I III II I II I IIz zH g
intH
« Natural » universal gate : iSWAP
int
1 0 0 0
0 1/ 2 / 2 0( ) SWAP 2 0 / 2 1/ 2 0
0 0 0 1
iU i
g i
int
1 0 0 00 cos( ) sin( ) 0
( ) 0 sin( ) cos( ) 00 0 0 1
gt i gtU t
i gt gt
00 10 01 11
On resonance, 01 01I II ( )
III.1) Two-qubit gates
i(t)
1 mm
200 µm
qubitsreadout resonator
couplingcapacitor
50 µ
m Josephsonjunction
frequencycontrol
fast flux line
Transmonqubit
λ/4 λ/4JJ
coupling capacitor
Readout Resonator
ie
Example : capacitively coupled transmons with individual readout(Saclay, 2011)
qubit
50 µm
drive &readout
frequency control
Example : capacitively coupled transmons with individual readout
III.1) Two-qubit gates
0,0 0,2 0,4 0,6
5
6
7
8
frequ
enci
es (
GH
z)
fI,II/f0
n01I
n01II
ncI
ncII
fI/f00.376
5.14
5.10
5.12
5.16
0.379
2g/ = 9 MHz
Spectroscopy
A. Dewes et al., in preparation
III.1) Two-qubit gates
SWAP between two transmon qubits
5.13 GHz
5.32GHz
6.82 GHz
6.42 GHz
DriveQB I
QB II
QB IIQB I
f01Swap Duration
6.67 GHz
6.03GHz
X
0,0
0,2
0,4
0,6
0,8
1,0
11
00
01
10
Swap duration (ns)0 100 200
Psw
itch
(%) Raw data
III.1) Two-qubit gates
0 100 200 3000,0
0,2
0,4
0,6
0,8
1,0
swap duration (ns)
SWAP between two transmon qubits
5.13 GHz
5.32GHz
6.82 GHz
6.42 GHz
DriveQB I
QB II
QB IIQB I
f01Swap Duration
6.67 GHz
6.03GHz
X
Psw
itch
(%) Data
correctedfrom
readouterrors
Swap duration (ns)0 100 200
1001
00
iSWAPIII.1) Two-qubit gates
How to quantify entanglement ??
Need to measure rexp Quantum state tomography
|0>
|1>
X
Z
Y 1 / 2switch zP
III.1) Two-qubit gates
How to quantify entanglement ??
Need to measure rexp Quantum state tomography
|0>
|1>
X
Z
Y 1 / 2switch yP
/2(X)
III.1) Two-qubit gates
How to quantify entanglement ??
Need to measure rexp Quantum state tomography
|0>
|1>
X
Z
Y 1 / 2switch xP
/2(Y)
M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006) III.1) Two-qubit gates
0 20
iSWAPX,Y tomo.
readouts
III
I
II
Z
40 60 80ns
I,X,Y
How to quantify entanglement ??
3*3 rotations*3 independent probabilities (P00,P01,P10) = 27 measured numbers
Fit experimental density matrix rexp
Compute fidelity 1/ 2 1/ 2expth thF Tr r r r
III.1) Two-qubit gates
0 100 200 300 4000,0
0,2
0,4
0,6
0,8
1,0
|00>
|01>
|10>
|11>
measuredideal
|00>
|11>
|10> |01>
F=98% F=94%
swap duration (ns)
switc
hing
pro
babi
lity
How to quantify entanglement ??
A. Dewes et al., in preparationIII.1) Two-qubit gates
SWAP gate of capacitively coupled phase qubitsM. 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
1 0 0 00 1 0 00 0 1 00 0 0 1
U
00 1001 11
00
10
01
11
Surprisingly, also quite natural with superconducting circuitsthanks to their multi-level structure
F.W. Strauch et al., PRL 91, 167005 (2003)DiCarlo et al., Nature 460, 240-244 (2009) III.1) Two-qubit gates
Control-Phase with two coupled transmonsDiCarlo et al., Nature 460, 240-244 (2009)
int 1 2/ 1 0 0 1 1 0 21 . .eff L R Leff L R L R RH g h g cc h
III.1) Two-qubit gates
Spectroscopy of two qubits + cavity
Qubit-qubit swap interaction
cavity
left qubit
right qubit
Cavity-qubit interactionVacuum Rabi splitting
RVFlux bias on right transmon (a.u.)(Courtesy Leo DiCarlo)III.1) Two-qubit gates
Preparation1-qubit rotationsMeasurement
cavity I
One-qubit gates: X and Y rotations
RV
Lcos(2 )f t
Lf
x
y
z
Flux bias on right transmon (a.u.)(Courtesy Leo DiCarlo)III.1) Two-qubit gates
Preparation1-qubit rotationsMeasurement
cavity I
RV
Rcos(2 )f t
Rf
x
y
z
Flux bias on right transmon (a.u.)
One-qubit gates: X and Y rotations
(Courtesy Leo DiCarlo)III.1) Two-qubit gates
Preparation1-qubit rotationsMeasurement
cavity Q
Rf
RV
Rsin(2 )f t
x
y
z
Flux bias on right transmon (a.u.) seeJ. Chow et al., PRL (2009)
Fidelity = 99%
One-qubit gates: X and Y rotations
III.1) Two-qubit gates
cavity
Conditionalphase gate
Use control lines to push qubits near a resonance
RV
RVFlux bias on right transmon (a.u.)
Two-qubit gate: turn on interactions
(Courtesy Leo DiCarlo)III.1) Two-qubit gates
0211
Two-excitation manifold
Two-excitation manifold of system
• Avoided crossing (160 MHz)
11 20
Flux bias on right transmon (a.u.)
(Courtesy Leo DiCarlo)III.1) Two-qubit gates
Flux bias on right transmon (a.u.)
11 1e1 11 i
01 e01 01i
10 0e1 10 i
0
2 ( )ft
a at
f t dt
Adiabatic conditional-phase gate
10
01
11
2-excitationmanifold
1-excitationmanifold
0
11 10 01 2 ( )ft
t
t dt
0201 10f f
(Courtesy Leo DiCarlo)
1 0 0 00 1 0 00 0 1 00 0 0 1
U
00 1001 11
00
10
01
11
Adjust timing of flux pulse so that only quantum amplitude of acquires a minus sign:
11
01
10
11
1 0 0 00 0 00 0 00 0 0
i
i
i
ee
e
U
00 1001 11
00
10
01
11
Implementing C-Phase
11C-Phase11
(Courtesy Leo DiCarlo)III.1) Two-qubit gates
Position: I II III0
“Find the queen!”
Implementing Grover’s search algorithm
“Find x0!”
0
0
0,( )
1,x
f xx x
x
DiCarlo et al., Nature 460, 240-244 (2009)
First implementation of q. algorithm with superconducting qubits (using Cphase gate)
(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm
Position: I II III0
“Find x0!”
0
0
0,( )
1,x
f xx x
x
“Find the queen!”
Implementing Grover’s search algorithm
(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm
Position: I II III0
“Find x0!”
0
0
0,( )
1,x
f xx x
x
“Find the queen!”
Implementing Grover’s search algorithm
(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm
Position: I II III0
“Find x0!”
0
0
0,( )
1,x
f xx x
x
“Find the queen!”
Implementing Grover’s search algorithm
(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm
Position: I II III0
Classically, takes on average 2.25 guesses to succeed…
Use QM to “peek” inside all cards, find the queen on first try
“Find the queen!”
Implementing Grover’s search algorithm
(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm
Grover’s algorithm“unknown”unitary
operation:
Challenge:Find the location
of the -1 !!!(= queen)
1 0 0 00 1 0 0ˆ0 0 00 0 1
10
O
/2yR /2
yR
/2yR
0
0
ij/2
yR
/2yR
/2yR
00
oracle
Previously implemented in NMR: Chuang et al. (1998) Linear optics: Kwiat et al. (2000)
Ion traps: Brickman et al. (2005)
(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm
Begin in ground state:
ideal 00
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00b c d f
e
g
oracle
Grover step-by-step
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
Create a maximalsuperposition:look everywhere at once!
ideal 0 1 0 10 12
01 1
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00b c d f
e
g
oracle
Grover step-by-step
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
ideal 0 1 0 10 12
01 1
Apply the “unknown”function, and mark the solution
10
1 0 0 00 1 0 00 0 1 00 0 0 1
cU
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00b c d f
e
g
oracle
Grover step-by-step
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
Some more 1-qubitrotations…
Now we arrive in one of the four
Bell states
ideal1 112
00
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00
oracle
b c d f
e
g
Grover step-by-step
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
Another (but known)2-qubit operation now undoes the entanglement and makes an interferencepattern that holds the answer!
ideal 0 1 0 10 12
01 1
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00
oracle
b c d f
e
g
Grover step-by-step
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
Final 1-qubit rotations reveal theanswer:
The binary representation of “2”!
Fidelity >80%
ideal 10
/2yR /2
yR
/2yR
0
010
/2yR
/2yR
/2yR
00b c d f
e
g
oracle
Grover step-by-step
DiCarlo et al., Nature 460, 240 (2009)
(Courtesy Leo DiCarlo)
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
….
|0>
|register>
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
2) Control-Phase Gate between any pair of qubits Qi and Qj
….
|0>
|register>
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
A) Transfer Qi state to resonator2) Control-Phase Gate between any pair of qubits Qi and Qj
….
SWAP
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
A) Transfer Qi state to resonator2) Control-Phase Gate between any pair of qubits Qi and Qj
….
B) Control-Phase between Qj and resonator
C-Phase
III.3) Architecture
Towards a scalable architecture ??
1) Resonator as quantum bus
A) Transfer Qi state to resonator2) Control-Phase Gate between any pair of qubits Qi and Qj
….
B) Control-Phase between Qj and resonatorC) Transfer back resonator state to Qi
SWAP
III.3) Architecture
Problems of this architecture
….
1) Off-resonant coupling Qk to resonator Uncontrolled phase errors
III.3) Architecture
Problems of this architecture
….
1) Off-resonant coupling Qk to resonator Uncontrolled phase errors
2) Effective coupling between qubits + spectral crowding
geff geff
RezQu (Resonator + zero Qubit) Architecture
q q q q
memoryresonators
qubits
coupling busresonator
frequ
ency
q
single gate
memory
coupled gate
measure(tunneling)
dampedresonators
zeroing
(courtesy J. Martinis)
RezQu Operations
Idling
1q 2q
r
22
2
Dg
Transfers& singlequbit gate
• |g reduces off coupling (>4th order)• Store in resonator (maximum coherence)
q
g
0
0
q
0
0
'q
0
0
g
'qi-SWAPi-SWAP
C-Z(CNOT class)
q
g
r
'q
0
'r
110211 i
Measure
q
g
tunnele
time
10
21
• Intrinsic transfer 99.9999%
|g |g
(courtesy J. Martinis)
Perspectives on superconducting qubits
1) Better qubits ?? Transmon in a 3D cavity
H. Paik et al., arxiv:quant-ph (2011)
REPRODUCIBLE improvement of coherence time (5 samples)
T1=60msT2=15ms
Perspectives on superconducting qubits
Quantum feedback : retroacting on the qubit to stabilize a given quantum state
« Non-linear » Circuit QEDResonator made non-linear by incorporating JJ.Parametric amplification, squeezing, back-action on qubit
Quantum information processing : better gates and more qubits !
Hybrid circuits CPW resonators : versatile playground for coupling many systems :Electron spins, nanomechanical resonators, cold atoms,Rydberg atoms, qubits, …
PhDs, Postdocs, WANTED
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