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Josephson qubits

Josephson qubits. P. Bertet. SPEC, CEA Saclay (France ), Quantronics group. Outline. Lecture 1: Basics of superconducting qubits. Lecture 2: Qubit readout and circuit quantum electrodynamics. Lecture 3: 2- qubit gates and quantum processor architectures.

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Josephson qubits

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

  2. Outline Lecture 1: Basics of superconductingqubits Lecture 2: Qubitreadout and circuit quantum electrodynamics Lecture 3: 2-qubitgates and quantum processor architectures 1) Two-qubitgates : SWAP gate and Control-Phase gate 2) Two-qubit quantum processor : Groveralgorithm 3) Towards a scalable quantum processor architecture 4) Perspectives on superconductingqubits

  3. Requirementsfor QC High-Fidelity Single Qubit Operations High-FidelityReadout of IndividualQubits 0 1 Deterministic, On-Demand EntanglementbetweenQubits III.1) Two-qubitgates

  4. Couplingstrategies 1) Fixedcoupling F Entanglement on-demand ??? « Tune-and-go » strategy Couplingactivated in resonance for t Entangledqubits Interaction effectively OFF Coupling effectively OFF III.1) Two-qubitgates

  5. Couplingstrategies 2) Tunablecoupling l Entanglement on-demand ??? A) Tune ON/OFF the couplingwithqubits on resonance Couplingactivated for t by lON Entangledqubits Interaction OFF (lOFF) Coupling OFF (lOFF) III.1) Two-qubitgates

  6. Couplingstrategies 2) Tunablecoupling Entanglement on-demand ??? B) Modulatecoupling IN THIS LECTURE : ONLY FIXED COUPLING Coupling ON by modulatingl Coupling OFF (lOFF) Coupling OFF (lOFF) III.1) Two-qubitgates

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

  8. How to couple transmonqubits ? 2) Cavitymediatedqubit-qubitcoupling J. Majer et al., Nature 449, 443 (2007) R Q II Q I D>>g g2 g1 Q II Q I geff=g1g2/D III.1) Two-qubitgates

  9. iSWAPGate « Natural » universalgate : On resonance, ) ( III.1) Two-qubitgates

  10. Example : capacitivelycoupledtransmonswithindividualreadout (Saclay, 2011) 1 mm readout resonator qubits fast flux line couplingcapacitor frequency control Josephson junction coupling capacitor 200 µm 50 µm λ/4 λ/4 JJ Transmon qubit i(t) ReadoutResonator

  11. Example : capacitivelycoupledtransmonswithindividualreadout drive & readout qubit frequency control 50 µm III.1) Two-qubitgates

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

  13. SWAP betweentwotransmonqubits 6.82 GHz QB I Drive Xp 6.67 GHz QB II 6.42 GHz 5.32GHz QB II f01 6.03GHz QB I 5.13 GHz Swap Duration Raw data Pswitch (%) 0 100 Swap duration (ns) 200 III.1) Two-qubitgates

  14. SWAP betweentwotransmonqubits 6.82 GHz QB I Drive Xp 6.67 GHz QB II 6.42 GHz 5.32GHz QB II f01 6.03GHz QB I 5.13 GHz Swap Duration 01 10 Data corrected from readout errors Pswitch (%) 00 Swap duration (ns) 0 100 200 III.1) Two-qubitgates

  15. How to quantifyentanglement ?? Need to measurerexp Quantum state tomography |0> Z X Y |1> III.1) Two-qubitgates

  16. How to quantifyentanglement ?? Need to measurerexp Quantum state tomography |0> Z p/2(X) X Y |1> III.1) Two-qubitgates

  17. How to quantifyentanglement ?? Need to measurerexp Quantum state tomography |0> Z p/2(Y) X Y |1> M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006) III.1) Two-qubitgates

  18. How to quantifyentanglement ?? readouts tomo. iSWAP Z X,Y I II I,X,Y I II 60 80ns 20 40 0 3*3 rotations*3 independentprobabilities (P00,P01,P10) = 27 measurednumbers Fit experimentaldensitymatrixrexp Computefidelity III.1) Two-qubitgates

  19. How to quantifyentanglement ?? |10> |01> |00> switchingprobability |11> swap duration (ns) ideal |00> measured |01> F=94% F=98% |10> |11> A. Dewes et al., in preparation III.1) Two-qubitgates

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

  21. The Control-Phase gate Anotheruniversal quantum gate : Control-Phase Surprisingly, alsoquitenaturalwithsuperconducting circuits thanks to theirmulti-level structure F.W. Strauch et al., PRL 91, 167005 (2003) DiCarlo et al., Nature 460, 240-244 (2009) III.1) Two-qubitgates

  22. Control-Phase withtwocoupledtransmons DiCarlo et al., Nature 460, 240-244 (2009) III.1) Two-qubitgates

  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.) (Courtesy Leo DiCarlo) III.1) Two-qubitgates

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

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

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

  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.) (Courtesy Leo DiCarlo) III.1) Two-qubitgates

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

  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: C-Phase11 (Courtesy Leo DiCarlo) III.1) Two-qubitgates

  31. Implementing Grover’s search algorithm First implementation of q. algorithmwithsuperconductingqubits (usingCphasegate) DiCarlo et al., Nature 460, 240-244 (2009) “Find x0!” 0 II III I Position: “Find the queen!” (Courtesy Leo DiCarlo) III.2) Two-qubitalgorithm

  32. Implementing Grover’s search algorithm “Find x0!” 0 II III I Position: “Find the queen!” (Courtesy Leo DiCarlo) III.2) Two-qubitalgorithm

  33. Implementing Grover’s search algorithm “Find x0!” 0 II III I Position: “Find the queen!” (Courtesy Leo DiCarlo) III.2) Two-qubitalgorithm

  34. Implementing Grover’s search algorithm “Find x0!” 0 II III I Position: “Find the queen!” (Courtesy Leo DiCarlo) III.2) Two-qubitalgorithm

  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 0 II III I Position: “Find the queen!” (Courtesy Leo DiCarlo) III.2) Two-qubitalgorithm

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

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

  38. Grover step-by-step 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 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 Some more 1-qubitrotations… 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 Another (but known) 2-qubit operation now undoes the entanglement and makes an interferencepattern 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 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. Towards a scalable architecture ?? 1) Resonator as quantum bus |yregister> …. |0> III.3) Architecture

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

  45. Towards a scalable architecture ?? 1) Resonator as quantum bus 2) Control-Phase Gatebetweenany pair of qubitsQi 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 Gatebetweenany pair of qubitsQi and Qj A) Transfer Qi state to resonator B) Control-Phase betweenQj and resonator …. C-Phase III.3) Architecture

  47. Towards a scalable architecture ?? 1) Resonator as quantum bus 2) Control-Phase Gatebetweenany pair of qubitsQi and Qj A) Transfer Qi state to resonator B) Control-Phase betweenQj and resonator C) Transfer back resonator state to Qi …. SWAP III.3) Architecture

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

  49. Problems of this architecture Uncontrolled phase errors 1) Off-resonantcouplingQk to resonator 2) Effective couplingbetweenqubits + 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 single gate frequency coupled gate measure (tunneling) (courtesy J. Martinis)

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