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

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 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

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

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

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

Coupling strategies 2) Tunable coupling λ (t ) = λoff +δλ cos ( ω1 ﾱ ω2 ) t H int (λ) Entanglement on-demand??? B) Modulate coupling IN THIS LECTURE : ONLY FIXED COUPLING Coupling OFF (λοφφ) III.1) Two-qubit gates Coupling ON by modulating λ Coupling OFF (λοφφ)

How to couple transmon qubits? 1) Direct capacitive coupling ΦI ΦII Vg,II Vg,I coupling capacitor Cc H = Ec, I ( Nˆ I N g, I ) EJ, I (Φ I ) cos θˆi + Ec, II ( Nˆ II N g, II ) EJ, II (Φ II ) cos θˆii +2 Ec, I Ec, II ( Nˆ I N g, I )( Nˆ II N g, II ) Ecc Cc hg = (2e) I Nˆ I 1I II Nˆ II 1II CI CII 2 hω1i (Φ I ) H q,i = σ z,i 2 II hω (Φ II ) H q, II = 1 σ z, II 2 H c = hgσ x, I σ x, II ; hg (σ I +σ II + σ I σ II + ) (note : idem for phase qubits)

How to couple transmon qubits? 2) Cavity mediated qubit-qubit coupling J. Majer et al., Nature 449, 443 (27) R QI g2 g1 QI III.1) Two-qubit gates Q II >>g geff=g1g2/ Q II

iswap Gate H int ω1i I ω1ii II H / h= σz σ z + g ( σ +I σ II + σ I σ +II ) 2 2 «Natural» universal gate : On resonance, ( ω1 I = ω1ii ) 1 1 11 1 cos( gt ) i sin( gt ) U ( π ) = U int (t ) = i sin( gt ) cos( gt ) int 2 g 1 III.1) Two-qubit gates iswap 1 1/ 2 i / 2 i / 2 = iswap 1/ 2 1

Example : capacitively coupled transmons with individual readout (Saclay, 211) fast flux line eiϕ λ/4 coupling capacitor λ/4 JJ Readout Resonator readout resonator i(t) 1 mm qubits 5 µm coupling capacitor Transmon qubit Josephsonfrequency junction control 2 µm

Example : capacitively coupled transmons with individual readout dri v rea e & do ut qu b it frequency control 5 µm III.1) Two-qubit gates

Spectroscopy ν1ι frequencies (GHz) 8 5.16 ν1ιι 7 2g/π = 9 MHz νcι νcιι 5.14 5.12 6 5.1 5, III.1) Two-qubit gates,2,4 φi,ii/φ,6.376.379 φi/φ A. Dewes et al., in preparation

SWAP between two transmon qubits 6.82 GHz QB I Drive Xπ 6.67 GHz QB II 6.42 GHz f 1 QB II 5.32GHz QB I 5.13 GHz 6.3GHz Swap Duration 1, Pswitch (%),8 1 1 Raw data,6,4,2, III.1) Two-qubit gates 11 1 2 Swap duration (ns)

SWAP between two transmon qubits 6.82 GHz QB I Drive Xπ 6.67 GHz QB II 6.42 GHz f 1 QB II 5.32GHz QB I 5.13 GHz 1, 1 Pswitch (%),8 6.3GHz Swap Duration 1 Data corrected from readout errors,6,4,2, III.1) Two-qubit gates iswap 1 2 1 2 swap duration (ns) Swap3 duration (ns)

How to quantify entanglement?? Need to measure ρexp Quantum state tomography > Z X Y 1> III.1) Two-qubit gates Pswitch ; ( 1 + σ z ) /2

How to quantify entanglement?? Need to measure ρexp Quantum state tomography > Z π/2(x) X Y 1> III.1) Two-qubit gates ( Pswitch ; 1 + σ y ) /2

How to quantify entanglement?? Need to measure ρexp Quantum state tomography > Z π/2(y) X Y Pswitch ; ( 1 + σ x ) /2 1> III.1) Two-qubit gates M. Steffen et al., Phys. Rev. Lett. 97, 552 (26)

How to quantify entanglement?? readouts X,Y I II iswap Z tomo. I,X,Y I II 2 4 6 8ns 3*3 rotations*3 independent probabilities (P,P1,P1) = 27 measured numbers Fit experimental density matrix ρexp Compute fidelity III.1) Two-qubit gates F = Tr ( ρth1/ 2 ρexp ρth1/ 2 )

How to quantify entanglement?? 1, switching probability 1> 1>,8,6 >,4,2 11>, > 1 2 3 4 swap duration (ns) ideal measured 1> F=98% 1> F=94% 11> III.1) Two-qubit gates A. Dewes et al., in preparation

SWAP gate of capacitively coupled phase qubits M. Steffen et al., Science 313, 1423 (26) F=.87 III.1) Two-qubit gates

The Control-Phase gate Another universal quantum gate : Control-Phase 1 1 11 1 U= 1 1 1 1 11 1 Surprisingly, also quite natural with superconducting circuits thanks to their multi-level structure III.1) Two-qubit gates F.W. Strauch et al., PRL 91, 1675 (23) DiCarlo et al., Nature 46, 24-244 (29)

Control-Phase with two coupled transmons DiCarlo et al., Nature 46, 24-244 (29) H int / h = g eff 1 ( 1L R L1R + h.c ) + g eff 2 ( 1L1R L 2 R + h.c ) III.1) Two-qubit gates

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

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

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

One-qubit gates: X and Y rotations fr z Preparation 1-qubit rotations Measurement y x cavity Q sin(2π f R t ) VR Flux bias on right transmon (a.u.) III.1) Two-qubit gates Fidelity = 99% see J. Chow et al., PRL (29)

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

Two-excitation manifold of system 11 2 Two-excitation manifold Avoided crossing (16 MHz) 11 ﾱ 2 Flux bias on right transmon (a.u.) III.1) Two-qubit gates (Courtesy Leo DiCarlo)

Adiabatic conditional-phase gate 2 11 tf f 1 + f1 ϕa = 2π ﾱδ f a (t )dt t 2-excitation manifold ζ 11 ﾱ eiϕ11 11 tf ϕ11 = ϕ1 + ϕ1 2π ﾱζ (t ) dt t 1 1-excitation manifold 1 ﾮ e 1 1 ﾱ e Flux bias on right transmon (a.u.) iϕ1 iϕ1 1 1 (Courtesy Leo DiCarlo)

Implementing C-Phase 1 1 11 1 eiϕ1 U = eiϕ1 1 1 iϕ11 e 11 1 1 11 Adjust timing of flux pulse so that only quantum amplitude 11 of acquires a minus sign: 1 U= C-Phase11 III.1) Two-qubit gates 1 1 1 1 111 11 (Courtesy Leo DiCarlo)

Implementing Grover s search algorithm First implementation of q. algorithm with superconducting qubits (using Cphase gate) DiCarlo et al., Nature 46, 24-244 (29) ﾱ, x ﾱ x f ( x) = ﾱ ﾱ1, x = x Find x! Position: I II III Find the queen! III.2) Two-qubit algorithm (Courtesy Leo DiCarlo)

Implementing Grover s search algorithm ﾮ, x ﾮ x f ( x) = ﾮ ﾮ 1, x = x Find x! Position: I II III Find the queen! III.2) Two-qubit algorithm (Courtesy Leo DiCarlo)

Implementing Grover s search algorithm ﾮ, x ﾮ x f ( x) = ﾮ ﾮ 1, x = x Find x! Position: I II III Find the queen! III.2) Two-qubit algorithm (Courtesy Leo DiCarlo)

Implementing Grover s search algorithm ﾮ, x ﾮ x f ( x) = ﾮ ﾮ 1, x = x Find x! Position: I II III Find the queen! III.2) Two-qubit algorithm (Courtesy Leo DiCarlo)

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)

1 algorithm Grover s unknown unitary operation: 1 ˆ Oψ = ψ 1 1 Challenge: Find the location of the -1!!! (= queen) Previously implemented in NMR: Chuang et al. (1998) Linear optics: Kwiat et al. (2) Ion traps: Brickman et al. (25) ij oracle III.2) Two-qubit algorithm (Courtesy Leo DiCarlo)

Grover step-by-step ψ ideal = Begin in ground state: oracle b c 1 d e f g DiCarlo et al., Nature 46, 24 (29) (Courtesy Leo DiCarlo)

ψ ideal Grover step-by-step 1 = ( + 1 + 1 + 11 ) 2 Create a maximal superposition: look everywhere at once! oracle b c 1 d e f g DiCarlo et al., Nature 46, 24 (29) (Courtesy Leo DiCarlo)

ψ ideal Grover step-by-step 1 = ( + 1 1 + 11 ) 2 Apply the unknown function, and mark the solution 1 cu1 = 1 1 1 oracle b c 1 d e f g DiCarlo et al., Nature 46, 24 (29) (Courtesy Leo DiCarlo)

ψ ideal 1 = ( 2 Grover step-by-step + 11 ) Some more 1-qubit rotations Now we arrive in one of the four Bell states oracle b c 1 d e f g DiCarlo et al., Nature 46, 24 (29) (Courtesy Leo DiCarlo)

ψ ideal Grover step-by-step 1 = ( 1 + 1 11 ) 2 Another (but known) 2-qubit operation now undoes the entanglement and makes an interference pattern that holds the answer! oracle b c 1 d e f g DiCarlo et al., Nature 46, 24 (29) (Courtesy Leo DiCarlo)

Grover step-by-step ψ ideal = 1 Final 1-qubit rotations reveal the answer: The binary representation of 2! Fidelity >8% oracle b c 1 d e f g DiCarlo et al., Nature 46, 24 (29) (Courtesy Leo DiCarlo)

Towards a scalable architecture?? 1) Resonator as quantum bus ψregister>. > III.3) Architecture

Towards a scalable architecture?? 1) Resonator as quantum bus 2) Control-Phase Gate between any pair of qubits Qi and Qj ψregister>. > III.3) Architecture

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

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

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

Problems of this architecture Uncontrolled phase errors 1) Off-resonant coupling Qk to resonator. III.3) Architecture

Problems of this architecture Uncontrolled phase errors 1) Off-resonant coupling Qk to resonator 2) Effective coupling between qubits + spectral crowding geff geff. III.3) Architecture

RezQu (Resonator + zero Qubit) Architecture damped resonators memory resonators qubits q q q q q coupling bus resonator frequency zeroing memory single gate coupled gate measure (tunneling) (courtesy J. Martinis)

RezQu Operations g2 2 Idling 2 q2 Transfers & single qubit gate gﾮ gﾮ r g g i-swap gﾱreduces off coupling (>4th order) Store in resonator (maximum coherence) i-swap time g r q q' q C-Z (CNOT class) Intrinsic transfer 99.9999% Measure 1 g r' 1 2 e tunnel (courtesy J. Martinis)

Perspectives on superconducting qubits T1=6µs T2=15µs H. Paik et al., arxiv:quant-ph (211) 1) Better qubits?? Transmon in a 3D cavity REPRODUCIBLE improvement of coherence time (5 samples)

Perspectives on superconducting qubits Quantum feedback : retroacting on the qubit to stabilize a given quantum state Quantum information processing : better gates and more qubits! N A,W D E T «Non-linear» Circuit QED Resonator made non-linear by incorporating JJ. Parametric amplification, squeezing, back-action on qubit P, s td s o s c o Hybrid circuits CPW resonators : versatile playground for coupling many systems : Electron spins, nanomechanical resonators, cold atoms, Rydberg atoms, qubits, D h P