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SPEC, CEA Saclay (France),

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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 rotations
z 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 rotations
z 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 rotations
z 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, …


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