1. Josephson Flux Qubits in Charge-Phase Regime
D-Wave Systems Inc.
THE QUANTUM COMPUTING COMPANYTM
Josephson Flux Qubits in Charge-Phase Regime
M. H. S. Amin
D-Wave Systems Inc., Vancouver, Canada
Thanks to:
P. Echternach (JPL)
M. Grajcar (IPHT/Comenius)
E. Il’ichev (IPHT)
M. Kenyon (JPL)
A. Kleinsasser (JPL)
A. Maassen van den Brink (D-Wave)
G. Rose (D-Wave)
A. Shnirman (Kalsruhe)
A. Smirnov (D-Wave)
A. Zagoskin (D-Wave)

2. Charge Qubit vs Flux Qubit
r = EC /EJ
Flux qubits: r <<1
Charge qubits: r >>1
10-2
10-1 10
Il’ichev et al.
Van der Wal et al.
Nakamura et al.
Guillaume et al.
Pashkin et al.
Duty et al.
Chiorescu et al. r
Vion et al.
Large sensitivity to flux noise
Large sensitivity to charge noise
Charge-phase regime;
The most Interesting

3. Decoherence Time t t ~ ? Charge qubit Charge-phase regime
Saclay: t ~ 500 ns
NEC/Chalmers/JPL: t ~ 5 ns
3JJ flux qubit
Delft: t ~ 100 ns
D-Wave/IPHT: t ~ 2.5 ms
Phase-charge regime
t ~ ?

4. Problems with Flux Qubits
1. Single shot readout difficult
D-Wave/IPHT: t ~ 2.5 ms
-Characterization technique,
not readout
Delft/MIT: t ~ 100 ns
-Requires large L; Large coupling
to magnetic environment
-DC-SQUID is dissipative

5. Problems with Flux Qubits
1. Single shot readout difficult
2. Exponential dependence of D on
qubit parameters
3. Controllable coupling difficult
4. Large sensitivity to flux noise

6. Why Charge-Phase Regime?
The effects of both charge and flux noise
can be minimized
Readout can be easily switched on and off
Two degrees of freedom (instead of one) are
available for e.g. coupling and readout
Smaller sensitivity to system parameters

7. Phase can be used for readout
Quantronium Qubit
|n
|n+1 E
|1 EJ
|0 ng
1/2
|0 = 2-1/2 ( |n + |n+1)
Qubit States:
|1 = 2-1/2 ( |n - |n+1)
Uncertainty in Charge  Localization of phase
Phase can be used for readout

8. Quantronium Qubit persistent currents: F0 i1 i0 ¶ 2p E i = ¶ d E01j j F0
¶ d
Magic
point
E01
Ng = 1/2 ) i0 i1
d/2p
current (nA) 2 1 -
d/2p 2 1 ng 2 1 2 1 - 4 1 - 4 1 2 1

9. Dual of Quantronium What charge? Flux qubit: D |R |L E Energy Levels
1/2
Fe/ F0
Uncertainty in phase  Localization of charge
What charge?

10. Aharonov-Casher Effect
Aharonov-Bohm effect: e F
Interference F
Example: DC-SQUID

11. Aharonov-Casher EffectQ
Interference

12. Aharonov-Casher Effect
J.R. Friedman and D.A. Averin, PRL (2002). t1 t1 Cg F Vg t2 t2
 Two paths for flux to tunnel  Interference
Quasicharge
Island Voltage:
Island Charge:
State Dependent

13. Two Josephson Junction QubitCg Q Vg
To charge/voltage
detector F
Coupling can be switched off during the operation
 Large energy derivatives  Large coupling to
background charges
Problems:
 Large flux  Large coupling to
magnetic environment

14. Three Josephson Junction Qubit
h = t2 /t1 t1 t2
T.P. Orlando et al. PRB 60, (1999)
 Small flux, small coupling to environment
 Two islands available for coupling

15. Energy Eigenstates
Hamiltonian:

16. Energy Eigenstates
Effective Hamiltonian:

17. Energy Eigenstates Effective Hamiltonian: Eigenenergies: r = EC /EJ
h = t2 /t1
nA (=VgACg/2e)

18. Energy Eigenstates Effective Hamiltonian: Eigenenergies: r = EC /EJ
h = t2 /t1
nA (=VgACg/2e)

19. Island Voltages No Coupling Magic Point: nA = nB = f = 0  VA = VB = 0
At f = Fx/F0-1/2 = 0:
Magic Point: nA = nB = f = 0  VA = VB = 0
No Coupling

20. Island Voltages No Coupling Magic Point: nA = nB = f = 0  VA = VB = 0
At f = Fx/F0-1/2 = 0:
Magic Point: nA = nB = f = 0  VA = VB = 0
No Coupling
Charge/flux fluctuations
affect decoherence only
in the 2nd order

21. Island Voltages No Coupling Directional Coupling  VA = Max , VB = 0
At f = Fx/F0-1/2 = 0:
Magic Point: nA = nB = f = 0  VA = VB = 0
No Coupling
Coupled regime:
 VA = Max , VB = 0
Directional Coupling

22. Small sensitivity to system parameters at large r (= EC /EJ)
Some Numerics
Small sensitivity to system parameters at large r (= EC /EJ)

23. Readout Scheme Off: Vg = 0 during the operations
Switchable Readout:
Sensitive charge
(voltage) detector
Off: Vg = during the operations
On: Vg = e/2Cg at the time of readout

24. Qubits are coupled only if
Two Qubit Coupling
Switchable Coupling:
Qubits are coupled only if
V(1)gB  0 and V(2)gA  0.

25. Can couple every two qubits
Multi-Qubit Coupling
Coupling via a bus island:
Can couple every two qubits

26. Multi-Qubit Coupling
Nearest neighbors coupling:

27. QA large enough to be measured by rf-SET
Suggested Parameters
EC / EJ = 0.1, a = 0.8, Cg = 0.1 C
D  5.6 GHz, h  0.13
Island Voltage: VA  3.7 mV
Island Charge: QA  0.2e
QA large enough to be measured by rf-SET

28. Comparison with Other Qubits
3JJ flux qubit: Charge-phase qubit:
Needs finite L for readout
L can be small; Small coupling to magnetic environment

29. Comparison with Other Qubits
3JJ flux qubit: Charge-phase qubit:
Needs finite L for readout
L can be small; Small coupling to magnetic environment
D exponentially depends
on parameters
Significantly smaller parameter dependence

30. Comparison with Other Qubits
3JJ flux qubit: Charge-phase qubit:
Needs finite L for readout
L can be small; Small coupling to magnetic environment
D exponentially depends
on parameters
Significantly smaller parameter dependence
EJ/D0 ~ 350
EJ/D0 ~ 10
~3 orders of magnitude smaller kf ; smaller effect of flux fluctuations

31. Comparison with Other Qubits
Quantronium qubit: Charge-phase qubit:
kC ~ 1.3, CS ~ 5 fF
kC ~ 1.8, CS ~ 8 fF
Same sensitivity to background charge noise

32. Comparison with Other Qubits
Quantronium qubit: Charge-phase qubit:
kC ~ 1.3, CS ~ 5 fF
kC ~ 1.8, CS ~ 8 fF
Same sensitivity to background charge noise E1 E0 E2 E3 En
E10
E21
Anharmonicity: A = (E21- E10 )/ E10
Harmonic oscillator: A = 0
Ideal qubit: A = 

33. Comparison with Other Qubits
Quantronium qubit: Charge-phase qubit:
kC ~ 1.3, CS ~ 5 fF
kC ~ 1.8, CS ~ 8 fF
Same sensitivity to background charge noise
A = A = 1.7
~10 times better anharmonicity

34. Conclusion
-Operation in charge-phase regime is possible and desirable for flux qubits
-Single shot readout possible with no effect on other qubits
-Controlled coupling to other qubits easily achievable
Compared to the 3JJ qubit
-Three orders of magnitude less sensitive to flux fluctuations
-Smaller L, i.e. smaller coupling to the magnetic environment
-One order of magnitude less sensitive to system parameters
Compared to the quantronium:
- Same sensitivity to the background charge fluctuations
- 10 times larger anharmonicity

35. Conclusion
-Operation in charge-phase regime is possible and desirable for flux qubits
-Single shot readout possible with no effect on other qubits
-Controlled coupling to other qubits easily achievable
Compared to the 3JJ qubit
-Three orders of magnitude less sensitive to flux fluctuations
-Smaller L, i.e. smaller coupling to the magnetic environment
-One order of magnitude less sensitive to system parameters
Compared to the quantronium:
- Same sensitivity to the background charge fluctuations
- 10 times larger anharmonicity

36. Conclusion
-Operation in charge-phase regime is possible and desirable for flux qubits
-Single shot readout possible with no effect on other qubits
-Controlled coupling to other qubits easily achievable
Compared to the 3JJ qubit
-Three orders of magnitude less sensitive to flux fluctuations
-Smaller L, i.e. smaller coupling to the magnetic environment
-One order of magnitude less sensitive to system parameters
Compared to the quantronium:
- Same sensitivity to the background charge fluctuations
- 10 times larger anharmonicity

37. Conclusion
-Operation in charge-phase regime is possible and desirable for flux qubits
-Single shot readout possible with no effect on other qubits
-Controlled coupling to other qubits easily achievable
Compared to the 3JJ qubit
-Three orders of magnitude less sensitive to the flux fluctuations
-Smaller L, i.e. smaller coupling to the magnetic environment
-One order of magnitude less sensitive to system parameters
Compared to the quantronium:
- Same sensitivity to the background charge fluctuations
- 10 times larger anharmonicity

38. Conclusion
-Operation in charge-phase regime is possible and desirable for flux qubits
-Single shot readout possible with no effect on other qubits
-Controlled coupling to other qubits easily achievable
Compared to the 3JJ qubit
-Three orders of magnitude less sensitive to flux fluctuations
-Smaller L, i.e. smaller coupling to the magnetic environment
-One order of magnitude less sensitive to system parameters
Compared to the quantronium:
- Same sensitivity to the background charge fluctuations
- 10 times larger anharmonicity

39. Conclusion
-Operation in charge-phase regime is possible and desirable for flux qubits
-Single shot readout possible with no effect on other qubits
-Controlled coupling to other qubits easily achievable
Compared to the 3JJ qubit
-Three orders of magnitude less sensitive to flux fluctuations
-Smaller L, i.e. smaller coupling to the magnetic environment
-One order of magnitude less sensitive to system parameters
Compared to the quantronium:
- Same sensitivity to the background charge fluctuations
- 10 times larger anharmonicity