1. Operating in Charge-Phase Regime, Ideal for Superconducting Qubits M. H. S. Amin D-Wave Systems Inc. THE QUANTUM COMPUTING COMPANY TM D-Wave Systems Inc., Vancouver, Canada P. Echternach M. Grajcar E. Il’ichev A. Maassen van den Brink Thanks to: G. Rose A. Shnirman A. Smirnov A. Zagoskin

2. Spins as Qubits or Qubits as Spins Spin: E1E1 E0E0 Hamiltonian: H = B ·  A spin 1/2 system is a 2-level quantum system

3. Spins as Qubits or Qubits as Spins Effective Hamiltonian: H = B x  x + B z  z Qubit: A qubit is a multi-level quantum system. The first two states are separated from the rest E1E1 E0E0 E2E2 E3E3 EnEn

4. Spins as Qubits or Qubits as Spins E1E1 E0E0 E2E2 E3E3 EnEn Anharmonicity: A = (E 21  E 10 )  E 10 E ij = E i  E j Qubit: A qubit is a multi-level quantum system. The first two states are separated from the rest Harmonic oscillator: A = 0 Ideal qubit: A =  E10E10 E21E21

5. Superconducting Qubits Josephson Junctions They are all made of Josephson junctions Al Two degrees of freedom: 1. Phase difference  2. Charge q

6. Superconducting Qubits Josephson Junctions They are all made of Josephson junctions Al Two degrees of freedom: 1. Phase difference  2. Charge q Two relevant energy scales: For phase: Josephson energy E J =   I c /2  For charge: Charging energy E C = e 2 /2C

7. Superconducting Qubits Charge-Phase Uncertainty Relation:

8. Superconducting Qubits If the junction is inside a loop Flux through the loop: Charge-Flux Uncertainty Relation:  Charge-Phase Uncertainty Relation:

9. Tunnel Junction Superconducting Island Gate 1. Charge Qubit: |0  = n Cooper pairs on the island |1  = n+1 Cooper pairs on the island Y. Nakamura et al., Nature (1999) Charge Qubit vs Phase Qubit Quantum States: E J  E C

10. Quantum States: |0  = left rotating current |1  = right rotating current I. Chiorescu et al., Science (2003) E. Il’ichev et al., Phys. Rev. Lett. (2003) 2. Phase Qubit: Superconducting Loop Charge Qubit vs Phase Qubit E J  E C

11. Uncertainty in charge leads to localization of phase Superconducting Island Superconducting Loop D. Vion et al., Sicence (2002) 3. Charge-phase Qubit:    n  n     n  n  Quantum States: Charge Qubit vs Phase Qubit E J ~ E C

12. Decoherence Time  Charge qubit  NEC/Chalmers/JPL:   ns 3JJ flux qubit D-Wave/IPHT:    s Delft:    ns Charge-phase regime  Saclay:   ns Phase-charge regime     Charge Qubit vs Phase Qubit

13. 3JJ flux qubit: Problem with Single Shot Readout

14. 3JJ flux qubit: D-Wave/IPHT:    s Problem with Single Shot Readout -Characterization technique, not readout

15. 3JJ flux qubit: D-Wave/IPHT:    s Problem with Single Shot Readout Delft/MIT:    ns -Characterization technique, not readout -Requires large L; Large coupling to magnetic environment -DC-SQUID is dissipative

16.  ( @ N g = 1/2 ) i0i0 i1i1  current ( nA ) persistent currents:  22 j j E i   Quantronium Qubit ngng  Magic point

17.  ext Uncertainty in phase  Localization of charge What charge? QAQA QBQB 3JJ qubit Dual of Quantronium E  ext    |R  |L  Energy Levels M.H.S. Amin, cond-mat/0311220

18. Dual of Quantronium Quantronium: Island VgVg CgCg Loop VgVg CgCg Island phase detector charge detector Dual of Quantronium:

19. Hamiltonian:  = t 2 /t 1 t1t1 t2t2 Accessing the Charges

20. n A (=V gA C g /2e) r  = E C /E J Accessing the Charges Eigenenergies:

21. n A (=V gA C g /2e) r  = E C /E J Accessing the Charges

22. Hamiltonian: Eigenenergies:  = t 2 /t 1 t1t1 t2t2 Energy Eigenvalues

23. Magic Point: n A = n B = f = 0  V A = V B = 0 No Coupling Island Voltages

24. Magic Point: n A = n B = f = 0  V A = V B = 0 No Coupling Charge/flux fluctuations affect decoherence only in the 2nd order Island Voltages

25. No Coupling Island Voltages Coupled regime:  V A = Max, V B = 0 Directional Coupling Magic Point: n A = n B = f = 0  V A = V B = 0

26. Small sensitivity to system parameters at large r  = E C /E J  Some Numerics

27. V g = 0 during the operations V g = e/2C g at the time of readout Sensitive charge (voltage) detector Readout Scheme Switchable Readout

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

29. Multi-Qubit Coupling Can coupled every two qubits

30. E C / E J = 0.05,  = 0.85, C g / C = 0.02   4.4 GHz,   0.1 Island Voltage: V A  2.1  V Island Charge: Q A  0.1e Large enough to be measured by rf-SET Suggested Parameters

31. Comparison with Other Qubits 3JJ flux qubit: Charge-phase qubit:

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

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

34. Comparison with Other Qubits 3JJ flux qubit: Charge-phase qubit: Needs finite L for readout L can be small; Small coupling to magnetic environment  exponentially depends on parameters Significantly smaller parameter dependence E J   E J   Two orders of magnitude smaller k f ; smaller effect of charge fluctuations

35. Comparison with Other Qubits Quantronium qubit: Charge-phase qubit: k C ~ 5, C ~ 0.5 fF k C ~ 1.6, C ~ 4 fF ~25 times less sensitive to the background charges

36. Comparison with Other Qubits Quantronium qubit: Charge-phase qubit: Anharmonicity: A = (E 21  E 10 )  E 10 Harmonic oscillator: A = 0 Ideal qubit: A =  E1E1 E0E0 E2E2 E3E3 EnEn E10E10 E21E21 ~25 times less sensitive to the background charges k C ~ 5, C ~ 0.5 fF k C ~ 1.6, C ~ 4 fF

37. Comparison with Other Qubits A = 0.2 A = 8.4 ~40 times better anharmonicity Quantronium qubit: Charge-phase qubit: ~25 times less sensitive to the background charges k C ~ 5, C ~ 0.5 fF k C ~ 1.6, C ~ 4 fF

38. Conclusion Compared to the 3JJ qubit -Two 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: - 25 times less sensitive to background charge fluctuations - 40 times larger anharmonicity -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

39. Conclusion Compared to the 3JJ qubit -Two 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: - 25 times less sensitive to background charge fluctuations - 40 times larger anharmonicity -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

40. Conclusion Compared to the 3JJ qubit -Two 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: - 25 times less sensitive to background charge fluctuations - 40 times larger anharmonicity -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

41. Conclusion Compared to the 3JJ qubit -Two 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: - 25 times less sensitive to background charge fluctuations - 40 times larger anharmonicity -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

42. Conclusion Compared to the 3JJ qubit -Two 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: - 25 times less sensitive to background charge fluctuations - 40 times larger anharmonicity -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

43. Conclusion Compared to the 3JJ qubit -Two 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: - 25 times less sensitive to background charge fluctuations - 40 times larger anharmonicity -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