Electrical quantum engineering with superconducting circuits

1.0 10 0.8 01 switching probability 0.6 0.4 0.2 00 P. Bertet & R. Heeres SPEC, CEA Saclay (France), Quantronics group 11 0.0 0 100 200 300 400 swap duration (ns) Electrical quantum engineering with superconducting circuits

Outline Lecture 1: Basics of superconducting qubits 1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) Readout using a resonator 2) Amplification & Feedback 3) Quantum state engineering Lecture 3: Multi-qubit gates and algorithms 1) Coupling schemes 2) Two-qubit gates and Grover algorithm 3) Elementary quantum error correction Lecture 4: Introduction to Hybrid Quantum Devices

Requirements for QC High-Fidelity Single Qubit Operations High-Fidelity Readout of Individual Qubits 0 1 Deterministic, On-Demand Entanglement between Qubits III.1) Two-qubit gates

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

2) Tunable coupling Coupling strategies λ () t = λ OFF ON H int (λ) λ Entanglement on-demand??? A) Tune ON/OFF the coupling with qubits on resonance Coupling OFF (λ OFF ) III.1) Two-qubit gates Coupling activated for τ by λ ON Entangled qubits Interaction OFF (λ OFF )

2) Tunable coupling Coupling strategies H int (λ) λ() t = λoff + δλ cos ω ± ω ( ) 1 2 t Entanglement on-demand??? B) Modulate coupling IN THIS LECTURE : ONLY FIXED COUPLING Coupling OFF (λ OFF ) III.1) Two-qubit gates Coupling ON by modulating λ Coupling OFF (λ OFF )

How to couple transmon qubits? 1) Direct capacitive coupling Φ I Φ II V g,i V g,ii coupling capacitor C c HH = EE cc,ii NN II NN gg,ii 2 EEJJ,II Φ II +EE cc,iiii NN IIII NN gg,iiii 2 EEJJ,IIII Φ IIII g cos θθ II cos θθ IIII +2 EE cc,iiee cc,iiii EE cccc (NN II NN gg,ii )(NN IIII NN gg,iiii ) C = (2e) 0 1 2 c ˆ 1 0 ˆ I NI I II NII II CICII ω ( Φ ) = 2 II ω01 ( Φ II ) = σ 2 I 01 I H qi, σ zi, H q, II z, II H = gσ σ c xi, xii, g( σ + σ + σ σ + ) I II I II

How to couple transmon qubits? 2) Cavity mediated qubit-qubit coupling Q I R >>g J. Majer et al., Nature 449, 443 (2007) S. Osnaghi et al., PRL (2001) Q II g 1 g 2 Q I g eff =g 1 g 2 / Q II III.1) Two-qubit gates ( + + ) H = g σ σ + σ σ eff eff I II I II

iswap Gate H / H int ω ω = + + 2 2 I II 01 I 01 II I II I II σz σz g σσ σσ ( ) + + «Natural» universal gate : iswap I ω = ω On resonance, ( 01 01 ) II U int () t 00 10 01 11 1 0 0 0 0 cos( gt) i sin( gt) 0 = 0 i sin( gt) cos( gt) 0 0 0 0 1 U int 1 0 0 0 π 0 1/ 2 i / 2 0 ( ) = iswap 2g 0 i / 2 1/ 2 0 = 0 0 0 1 III.1) Two-qubit gates

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

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

Spectroscopy frequencies (GHz) 8 7 6 5 ν 01 ΙΙ ν c Ι ν c ΙΙ ν 01 Ι 2g/π = 9 MHz 5.16 5.14 5.12 5.10 0,0 0,2 0,4 0,6 0.376 0.379 φ I /φ 0 φ I,II /φ 0 A. Dewes et al., PRL (2011) III.1) Two-qubit gates

SWAP between two transmon qubits Drive QB I QB II X π 6.82 GHz 6.67 GHz 6.42 GHz f 01 QB II QB I 5.32GHz 5.13 GHz Swap Duration 6.03GHz 1,0 Pswitch (%) 0,8 0,6 0,4 10 01 00 Raw data 0,2 0,0 III.1) Two-qubit gates 0 100 200 11 Swap duration (ns)

SWAP between two transmon qubits Drive QB I QB II X π 6.82 GHz 6.67 GHz 6.42 GHz f 01 QB II QB I 5.32GHz 5.13 GHz Swap Duration 6.03GHz Occupation proba 1,0 0,8 0,6 0,4 0,2 10 00 01 Data corrected from readout errors III.1) Two-qubit gates 0,0 0 100 100 200 200 300 swap duration (ns) iswap Swap duration (ns)

How to quantify entanglement?? Need to measure ρ exp Quantum state tomography 0> Z Y X P switch ( + σ ) 1 /2 z 1> III.1) Two-qubit gates

How to quantify entanglement?? Need to measure ρ exp Quantum state tomography 0> Z π/2(x) Y X P switch ( + σ ) y 1 /2 1> III.1) Two-qubit gates

How to quantify entanglement?? Need to measure ρ exp Quantum state tomography 0> Z π/2(y) Y X P switch ( + σ ) 1 /2 x 1> III.1) Two-qubit gates M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006)

How to quantify entanglement?? readouts I II X,Y iswap Z tomo. I,X,Y I II 0 20 40 60 80ns 3*3 rotations*3 independent probabilities (P 00,P 01,P 10 ) = 27 measured numbers III.1) Two-qubit gates Fit experimental density matrix ρ exp Compute fidelity ( 1/2 1/2 F = Tr ρ ) th ρexpρth

Occupation proba How to quantify entanglement?? 1,0 10> 01> 0,8 0,6 00> 0,4 0,2 0,0 11> 0 100 200 300 400 swap duration (ns) F=98% 00> 01> 10> ideal measured F=94% 11> III.1) Two-qubit gates A. Dewes et al., PRL (2011)

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

Other universal two-qubit gates The control-phase gate U 00 01 10 11 1 0 0 0 0 1 0 0 = 0 0 1 0 0 0 0 1 00 01 10 11 The controlled-not gate 00 01 10 11 αα 0 + ββ 1 0 αα 00 + ββ 11 U CCCCCCCC = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 00 01 10 11

One-qubit Hadamard gate Decomposition of CNOT gate

Control-Phase with two coupled transmons DiCarlo et al., Nature 460, 240-244 (2009) ( 1 +. c) + ( h. ) H / = g 10 0 h g 11 0 2 + c III.1) Two-qubit gates int eff 1 L R L R eff 2 L R L R

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

One-qubit gates: X and Y rotations Preparation 1-qubit rotations Measurement f L z y x cavity I cos(2 π ft) L III.1) Two-qubit gates V R Flux bias on right transmon (a.u.) (Courtesy Leo DiCarlo)

One-qubit gates: X and Y rotations f R Preparation 1-qubit rotations Measurement z y x cavity I cos(2 π ft) R III.1) Two-qubit gates V R Flux bias on right transmon (a.u.) (Courtesy Leo DiCarlo)

One-qubit gates: X and Y rotations f R Preparation 1-qubit rotations Measurement z y x cavity Q sin(2 π ft) R III.1) Two-qubit gates V R Flux bias on right transmon (a.u.) Fidelity = 99% see J. Chow et al., PRL (2009)

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

Two-excitation manifold of system 11 02 Two-excitation manifold Avoided crossing (160 MHz) 11 02 Flux bias on right transmon (a.u.) III.1) Two-qubit gates (Courtesy Leo DiCarlo)

Adiabatic conditional-phase gate 02 11 f + f 01 10 t f ϕa = 2 π δ fa() t dt t 0 01 10 2-excitation manifold 1-excitation manifold ζ iϕ 11 e 1 11 1 t f ϕ11 = ϕ10 + ϕ01 2 π ζ() t dt 01 01 e iϕ 01 iϕ 10 e 1 10 0 t 0 Flux bias on right transmon (a.u.) (Courtesy Leo DiCarlo)

Implementing C-Phase U 00 01 10 11 1 0 0 0 iϕ01 0 e 0 0 = iϕ10 0 0 e 0 iϕ 11 0 0 0 e 00 01 10 11 00 01 10 11 Adjust timing of flux pulse so that only quantum amplitude of 11 acquires a minus sign: U 1 0 0 0 0 1 0 0 = 0 0 1 0 0 0 0 1 00 01 10 11 C-Phase 11 11 III.1) Two-qubit gates (Courtesy Leo DiCarlo)

The Grover Search Algorithm: A Benchmark for Quantum Speed-Up A search oracle marks a state i Four possible oracles. Which one have we got? i { 00 01 10 11} What is the probability to give correct answer after one call of the oracle? CLASSICALLY Try one state. Probability ¼ to be the state marked by oracle. If it is not marked, guess randomly amongst 3 remaining possible states. Total maximal classical probability of success = ¼ + ¾*1/3 = 1/2

The Grover Search Algorithm: A Benchmark for Quantum Speed-Up A search oracle marks a state i Four possible oracles. Which one have we got? In one call : i { 00 01 10 11} In one call : prepare a test state call the quantum Oracle O analyze the result 0 0 Y π/2 Y π/2 x O x i=00? i=01 i=10 i=11 i iswap X π/2 X π/2 1 1 00 i 2 Oi i i i 2 Grover, Proc. 28th Ann. ACM Sym. on the Theory of Computing (1996) i

Implementation of the Grover Algorithm preparation quantum oracle O 00 analysis readout 0 0 x Y π/2 Y π/2 x O00 x iswap Z π/2 Z π/2 iswap X π/2 X π/2 0 0 1 1 00 01 10 11 after calling the oracle (i = 00) final result of algorithm single run probabilities 67 % 00> 01> 10> 11> <00 <01 <10 <11 π 0 fidelity F = 87 98% % fidelity F = 70 % Dewes et.al. PRB (2012)

Results for Different Oracle Functions 00 01 10 11 π 0 67 % 55 % 62 % 52 % F i > 50 % for all four oracles Demonstration of quantum speed-up Dewes et.al. PRB (2012)

Steps towards quantum computer Quantum Error Correction R. Schoelkopf and M. Devoret, Science (2013)

Basics of Quantum Error Correction ψψ = αα 0 + ββ 1 Bit-flip errors Phase-flip errors αα 1 + ββ 0 αα 1 + ee iiii ββ 0 Detecting and correcting these errors? Difficulty : Quantum measurement!

Correcting bit-flip errors (1) : encoding «Physical» qubit «Logical» qubit D. Mermin, lecture notes on quantum computation, chap. 5

Correcting bit-flip errors (2) : detecting the error Key property of logical qubit state αα 000 + ββ 111 Each two-qubit pair is in an eigenstate of the parity operators PP 12, PP 23, PP 13 with value +1 : PP 12 = +1, PP 23 = +1, PP 13 = +1 In case of one bit-flip error, the occurrence and the position of the errors can be detected by measuring the parities of each pair. If there is no error, the parity measurements do not perturb the state But an error can be detected. For instance on qubit 2 would yield αα 010 + ββ 101 which would result in PP 12 = 1, PP 23 = 1, PP 13 = +1 The challenge of QEC is thus to repititvely and non-destructively measure parity operators of pairs of qubits

Correcting bit-flip errors (2) : detecting an error «Ancilla Qubits» useful for the parity measurements D. Mermin, lecture notes on quantum computation, chap. 5

Correcting bit-flip errors (2) : detecting an error with a 9-qubit chip (Nature, 2015) («Ancilla»)

Correcting bit-flip errors (2) : detecting an error with a 9-qubit chip

Correcting bit-flip errors (2) : implementation

Correcting bit-flip errors (2) : implementation CNOTs1 CNOTs2

Correcting bit-flip errors (2) : detecting an error with a 9-qubit chip

Correcting bit-flip errors (2) : detecting errors with a 9-qubit chip

Conclusions Multi-qubit operations possible thanks to improvements in coherence times and high-fidelity single-qubit gates and readout First implementations of algorithms, and even elementary Quantum Error Correction schemes (only bit-flip errors) Real quantum error correction still requires major improvements in gate fidelity, coherence times, control electronics, fabrication, But Still very far from a working quantum processor