1. SPEC, CEA Saclay (France),
Josephson qubits
P. Bertet
SPEC, CEA Saclay (France),
Quantronics group

2. 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

3. Single Qubit Operations High-Fidelity Readout of Individual Qubits
Requirements for QC
High-Fidelity
Single Qubit Operations
High-Fidelity Readout
of Individual Qubits 1
Deterministic, On-Demand
Entanglement between Qubits
III.1) Two-qubit gates

4. Coupling strategies 1) Fixed coupling Coupling activated
in resonance for t F
Entanglement on-demand ???
« Tune-and-go » strategy
Coupling
effectively OFF
Entangled qubits
Interaction effectively OFF
III.1) Two-qubit gates

5. Coupling strategies 2) Tunable coupling Coupling OFF (lOFF)
Coupling activated
for t by lON
Entangled qubits
Interaction OFF (lOFF) l
Entanglement on-demand ???
A) Tune ON/OFF the coupling with qubits on resonance
III.1) Two-qubit gates

6. Entanglement on-demand ???
Coupling strategies
2) Tunable coupling
Coupling OFF
(lOFF)
Coupling ON
by modulating l
Coupling OFF
(lOFF)
Entanglement on-demand ???
B) Modulate coupling
IN THIS LECTURE : ONLY FIXED COUPLING
III.1) Two-qubit gates

7. How to couple transmon qubits ?
1) Direct capacitive coupling FI
FII
Vg,II
Vg,I
coupling capacitor Cc
(note : idem
for phase qubits)

8. How to couple transmon qubits ?
2) Cavity mediated qubit-qubit coupling
J. Majer et al., Nature 449, 443 (2007) R
Q I
Q II
D>>g g1 g2
geff=g1g2/D
Q I
Q II
III.1) Two-qubit gates

9. « Natural » universal gate :
iSWAP Gate
« Natural » universal gate :
On resonance, ( )
III.1) Two-qubit gates

10. Example : capacitively coupled transmons with individual readout
(Saclay, 2011)
fast flux line
coupling capacitor
λ/4 JJ
λ/4
Transmon
qubit
i(t)
Readout Resonator
1 mm
readout
resonator
coupling
capacitor
200 µm
qubits
50 µm
Josephson
junction
frequency
control

11. Example : capacitively coupled transmons with individual readout
qubit
50 µm
drive &
readout
frequency
control
III.1) Two-qubit gates

12. fI,II/f0 Spectroscopy n01I fI/f0 0.376 5.14 5.10 5.12 5.16 0.379 n01II
2g/p = 9 MHz
n01II
ncI
frequencies (GHz)
ncII
fI,II/f0
A. Dewes et al., in preparation
III.1) Two-qubit gates

13. SWAP between two transmon qubits
5.13 GHz
5.32GHz
6.82 GHz
6.42 GHz
Drive
QB I
QB II
f01
Swap Duration
6.67 GHz
6.03GHz Xp
Swap duration (ns)
100
200
Pswitch (%)
Raw data
III.1) Two-qubit gates

14. SWAP between two transmon qubits
5.13 GHz
5.32GHz
6.82 GHz
6.42 GHz
Drive
QB I
QB II
f01
Swap Duration
6.67 GHz
6.03GHz Xp 01 10
Data
corrected
from
readout
errors
Pswitch (%) 00
100
Swap duration (ns)
200
III.1) Two-qubit gates

15. How to quantify entanglement ??
Need to measure rexp
Quantum state tomography
|0> Z X Y
|1>
III.1) Two-qubit gates

16. How to quantify entanglement ??
Need to measure rexp
Quantum state tomography
|0> Z
p/2(X) X Y
|1>
III.1) Two-qubit gates

17. How to quantify entanglement ??
Need to measure rexp
Quantum state tomography
|0> Z
p/2(Y) X Y
|1>
III.1) Two-qubit gates
M. Steffen et al., Phys. Rev. Lett. 97, (2006)

18. How to quantify entanglement ??
readouts
iSWAP Z
tomo.
X,Y I II
I,X,Y I II 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

19. How to quantify entanglement ??
|10>
|01>
|00>
switching probability
|11>
swap duration (ns)
ideal
|00>
measured
|01>
F=94%
F=98%
|10>
|11>
III.1) Two-qubit gates
A. Dewes et al., in preparation

20. SWAP gate of capacitively coupled phase qubits
M. Steffen et al., Science 313, 1423 (2006)
F=0.87
III.1) Two-qubit gates

21. The Control-Phase gate
Another universal quantum gate : Control-Phase
Surprisingly, also quite natural with superconducting circuits
thanks to their multi-level structure
F.W. Strauch et al., PRL 91, (2003)
DiCarlo et al., Nature 460, (2009)
III.1) Two-qubit gates

22. Control-Phase with two coupled transmons
DiCarlo et al., Nature 460, (2009)
III.1) Two-qubit gates

23. Spectroscopy of two qubits + cavity
right qubit
Qubit-qubit swap interaction
left qubit
Cavity-qubit interaction
Vacuum Rabi splitting
cavity
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)

24. One-qubit gates: X and Y rotationsz
Preparation
1-qubit rotations
Measurement x y
cavity I
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)

25. One-qubit gates: X and Y rotationsz
Preparation
1-qubit rotations
Measurement y x
cavity I
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)

26. One-qubit gates: X and Y rotationsz
Preparation
1-qubit rotations
Measurement y x
cavity Q
Fidelity = 99%
Flux bias on right transmon (a.u.)
see
J. Chow et al., PRL (2009)
III.1) Two-qubit gates

27. Two-qubit gate: turn on interactions
Conditional
phase gate
Use control lines to push qubits near a resonance
cavity
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)

28. Two-excitation manifold of system
• Avoided crossing (160 MHz)
Flux bias on right transmon (a.u.)
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)

29. Adiabatic conditional-phase gate
2-excitation
manifold
1-excitation
manifold
Flux bias on right transmon (a.u.)
(Courtesy Leo DiCarlo)

30. Implementing C-Phase 11 Adjust timing of flux pulse so that
only quantum amplitude of
acquires a minus sign: 11
C-Phase11
III.1) Two-qubit gates
(Courtesy Leo DiCarlo)

31. Implementing Grover’s search algorithm
First implementation of q. algorithm with superconducting qubits (using Cphase gate)
DiCarlo et al., Nature 460, (2009)
“Find x0!”
Position: I II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)

32. Implementing Grover’s search algorithm
“Find x0!”
Position: I II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)

33. Implementing Grover’s search algorithm
“Find x0!”
Position: I II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)

34. Implementing Grover’s search algorithm
“Find x0!”
Position: I II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)

35. 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: I II
III
“Find the queen!”
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)

36. Grover’s algorithm Previously implemented in NMR: Chuang et al. (1998)
Challenge:
Find the location
of the -1 !!! (= queen)
“unknown”
unitary
operation:
Previously implemented in NMR: Chuang et al. (1998)
Linear optics: Kwiat et al. (2000)
Ion traps: Brickman et al. (2005)
oracle
III.2) Two-qubit algorithm
(Courtesy Leo DiCarlo)

37. Grover step-by-step oracle Begin in ground state: b c d f g e
DiCarlo et al., Nature 460, 240 (2009) e
(Courtesy Leo DiCarlo)

38. Grover step-by-step oracle Create a maximal
superposition: look everywhere
at once!
oracle b c d f g
DiCarlo et al., Nature 460, 240 (2009) e
(Courtesy Leo DiCarlo)

39. Grover step-by-step oracle Apply the “unknown” function, and
mark the solution
oracle b c d f g
DiCarlo et al., Nature 460, 240 (2009) e
(Courtesy Leo DiCarlo)

40. Grover step-by-step oracle Some more 1-qubit rotations…
Now we arrive in one of the four Bell states
oracle b c d f g
DiCarlo et al., Nature 460, 240 (2009) e
(Courtesy Leo DiCarlo)

41. Grover step-by-step oracle Another (but known) 2-qubit operation now
undoes the entanglement and makes an interference pattern that holds
the answer!
oracle b c d f g
DiCarlo et al., Nature 460, 240 (2009) e
(Courtesy Leo DiCarlo)

42. Grover step-by-step oracle Final 1-qubit rotations reveal the answer:
The binary representation of “2”!
Fidelity >80%
oracle b c d f g
DiCarlo et al., Nature 460, 240 (2009) e
(Courtesy Leo DiCarlo)

43. |yregister> Towards a scalable architecture ?? |0>
1) Resonator as quantum bus
|yregister> ….
|0>
III.3) Architecture

44. |yregister> Towards a scalable architecture ?? |0>
1) Resonator as quantum bus
2) Control-Phase Gate between any pair of qubits Qi and Qj
|yregister> ….
|0>
III.3) Architecture

45. 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

46. 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

47. 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

48. Problems of this architecture
1) Off-resonant coupling Qk to resonator
Uncontrolled phase errors ….
III.3) Architecture

49. Problems of this architecture
1) Off-resonant coupling Qk to resonator
Uncontrolled phase errors
2) Effective coupling between qubits + spectral crowding
geff
geff ….
III.3) Architecture

50. RezQu (Resonator + zero Qubit) Architecture
damped
resonators
memory
resonators
qubits q q q q q
coupling bus
resonator
zeroing
memory
frequency
single gate
coupled gate
measure
(tunneling)
(courtesy J. Martinis)

51. RezQu Operations Transfers & single Idling qubit gate C-Z (CNOT class)
i-SWAP
i-SWAP
|g reduces off coupling
(>4th order)
Store in resonator
(maximum coherence)
time
Intrinsic transfer %
C-Z
(CNOT class)
Measure
tunnel
(courtesy J. Martinis)

52. T1=60ms T2=15ms Perspectives on superconducting qubits
H. Paik et al., arxiv:quant-ph (2011)
1) Better qubits ?? Transmon in a 3D cavity
REPRODUCIBLE improvement of coherence time (5 samples)

53. PhDs, Postdocs, WANTED 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
PhDs, Postdocs, WANTED
Hybrid circuits
CPW resonators : versatile playground for coupling many systems :
Electron spins, nanomechanical resonators, cold atoms,
Rydberg atoms, qubits, …