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Girvin Seminar Metrology of Entangled States Expt. in Circuit QED Leo DiCarlo Andrew Houck Applied Physics + Physics David Schuster Yale University Hannes Majer PI’s: Jerry Chow Rob Schoelkopf Joe Schreier Michel Devoret Blake Johnson Steven Girvin Luigi Frunzio Theory Lev Bishop Jens Koch IARPA Jay Gambetta Alexandre Blais Florian Marquardt Eli Luberoff Lars Tornberg Terri Yu Metrology of Entangled States Expt. in Circuit QED Leo DiCarlo Princeton Andrew Houck Applied Physics + Physics David Schuster Yale University Vienna Hannes Majer PI’s: Jerry Chow Rob Schoelkopf Joe Schreier Michel Devoret Blake Johnson Steven Girvin Luigi Frunzio Theory Lev Bishop Jens Koch IARPA IQC/Waterloo Jay Gambetta U. Sherbrooke Alexandre Blais Munich Florian Marquardt Eli Luberoff Göteborg Lars Tornberg Terri Yu Recent Reviews ‘Wiring up quantum systems’ R. J. Schoelkopf, S. M. Girvin Nature 451, 664 (2008) ‘Superconducting quantum bits’ John Clarke, Frank K. Wilhelm Nature 453, 1031 (2008) Quantum Information Processing 8 (2009) ed. by A. Korotkov Overview • ‘Transmon’ qubit, insensitive to charge noise • Circuit QED: using cavity bus to couple qubits • Two qubit gates and generation of Bell’s states • “Metrology of entanglement” – using joint cQED msmt. • Demonstration of Grover and Deutsch-Josza algorithms DiCarlo et al., cond-mat/0903.2030 Nature, (in press, June 2009) Quantum Computation and NMR of a Single ‘Spin’ Electrical circuit with two quantized energy levels is like a spin -1/2. Single Spin ½ Quantum Measurement Vds C C C gb Box c ge Vgb Vge SET (After Konrad Lehnert) ‘Transmon’ Cooper Pair Box: Charge Qubit that Works! EJ >> EC Josephson junction e (SQUID loop): 300 µm g plasma oscillation of 2 or 3 Cooper pairs: almost no static dipole Added metal = capacitor & antenna Transmon qubit insensitive to 1/f electric fields * Theory: J. Koch et al., PRA (2007); Expt: J. Schreier et al., PRB (2008) Flux qubit + capacitor: F. You et al., PRB (2006) ‘Transmon’ Cooper Pair Box: Charge Qubit that Works! Josephson junction plasma oscillations are anharmonic: e g 300 µm EJ >> EC -4 -3 -2 -1 0 +1 +2 +3 +4 n = Number of pairs that have tunneled 2 ω = − [ ]+ − 2 6.1 GHz HEJ tunneling 4EC ( nˆ n offset ) 12 1 Transition frequency tunable via ω 6.5 GHz 01 SQUIDAlmost flux. a harmonic oscillator except coordinate n is integerℏω valued.≈ 0 018EE J C Outsmarting Noise: Sweet Spot 1st coherence strategy: optimize design sweet spot Energy transition freq. 1st order insensitive to gate noise Offset Charge n_offset Strong sensitivity of frequency to charge noise But T2 still < 500 ns due to second-order noise! Vion et al., Science 296, 886 (2002) “Eliminating” Charge Noise with Better Design EJ/EC = 1 EJ/EC = 25 - 100 Energy exponentially suppresses 1/f! Cooper-pair Box “Transmon” Koch et al., 2007; Houck et al., 2008 Single Qubit Rotations Fidelity = 99% J. Chow et al., PRL (2009): 2 6.1 GHz 1 ω 01 6.5 GHz 0 V= Ωx ( t )cos(ω t ) σ x Rabi 01 T =1.5µs Ω y ω σ y 1 +Rabi (t )sin(01 t ) π delay measure time Ramsey Fringe and Qubit Coherence 2 Fidelity = 99% 6.1 GHz 1 J. Chow et al., PRL (2009): ω 01 6.5 GHz x 0 = Ω ω σ VRabi ( t )cos(01 t ) * = µ T2 3.0 s π / 2 delay π / 2 measure time Coherence in Transmon Qubit = µ Random benchmarking of 1-qubit ops T1 1.5 s Chow et al. PRL 2009: Technique from Knill et al. for ions * = = µ TT22 1 3.0 s Error per gate = 1.2 % 1= 1 + 1 ⇒ > µ * Tϕ 35 s TTT22 1 ϕ Similar error rates in phase qubits (UCSB): Lucero et al. PRL 100, 247001 (2007) Cavity Quantum Electrodynamics (cQED) 2g = vacuum Rabi freq. κ = cavity decay rate γ = “transverse” decay rate Strong Coupling = g > κ , γ Jaynes-Cummings Hamiltonian ℏω − + ˆ † 1 a † H = ℏω ()a a + − σˆ − ℏg()aσ +σ a + HHκ + γ r 2 2 z Quantized Field 2-level system Electric dipole Dissipation Interaction Coupling SC Qubits: Use a Circuit Element a capacitor entangled states! Concurrence ~ 55% Charge qubits: NEC 2003 Phase qubits: UCSB 2006 an inductor tunable (SQUID) element Flux qubits: Delft 2007 Flux qubits: Berkeley 2006, NEC 2007 Or tunable bus, Chalmers Qubits Coupled with a Quantum Bus use microwave photons guided on wires! “Circuit QED” Blais et al., Phys. Rev. A (2004) transmission out line “cavity” Josephson-junction 7 GHz in qubits Expts: Majer et al., Nature 2007 (Charge qubits / Yale) Sillanpaa et al., Nature 2007 (Phase qubits / NIST) flux bias lines A Two-Qubit Processor control qubit frequency 1 ns resolution DC - 2 GHz T = 10 mK cavity: “entanglement bus,” driver, & detector transmon qubits How do we entangle two qubits? −π RY ( / 2) rotation on each qubit yields superposition: 1 Ψ =()0 + 1 ⊗()0 + 1 2 1 = ( 00 +10 +01 + 1 1 ) 2 ‘Conditional Phase Gate’ entangler: +1 0 0 0 0+ 1 0 0 1 Ψ =( 00 + 1 0 + 01 − 1 1 ) 0 0+ 1 0 2 0 0 0 −1 No longer a product state! How do we entangle two qubits? +1 0 0 0 0+ 1 0 0 1 1 Ψ =( 00 + 0 1 + 10 −11 ) =( 0→ + 1 ← ) 0 0+ 1 0 2 2 0 0 0 −1 +π RY ( / 2) rotation on LEFT qubit yields: 1 Bell =()00 + 1 1 2 Other 3 Bell states similarly achieved. Entanglement on Demand 1 ( + ) 2 L’état quantique c’est Moi! How do we realize the conditional phase gate? +1 0 0 0 0+ 1 0 0 1 Ψ =( 00 + 0 1 + 10 − 1 1 ) 0 0+ 1 0 2 0 0 0 −1 Use control lines to push qubits near a resonance: A controlled z-z interaction also à la NMR Key is to use 3rd level of transmon (outside the logical subspace) Coupling turned off. 2 Coupling turned on: Near resonance with 3rd level 1 ω ≈ ω 01 12 2 2 1 1 Energy is shifted if and only if both qubits are in excited state. 0 0 Adiabatic Conditional Phase Gate • Avoided crossing (160 MHz) • A frequency shift On/off ratio ≈ 100:1 Use large on-off ratio of ζ to implement 2-qubit phase gates. Strauch et al. PRL (2003): proposed use of excited states in phase qubits Adjust timing so that amplitude for both qubits to be excited acquires a minus sign: +1 0 0 0 0+ 1 0 0 1 Ψ =( 00 + 1 0 + 01 − 1 1 ) 0 0+ 1 0 2 0 0 0 −1 Entanglement on Demand Bell state Fidelity Concurrence 00+ 11 91% 88% 00− 11 94% 94% + Re(ρ ) 01 10 90% 86% 01− 10 87% 81% UCSB: Steffen et al., Science (2006) ETH: Leek et al., PRL (2009) How do we read out the qubit state and measure the entanglement? Two Qubit Joint Readout via Cavity Cavity frequency pulled by qubits Cavity transmission Cavity Frequency “Strong dispersive cQED”: Schuster et al., 2007 Two Qubit Joint Readout via Cavity Initial polarization of qubit? > 99.7% (Bishop et al., 2009) -> reset fidelity is high! Cavity Pull is linear in spin polarizations ∆ω =σ z + σ z aL b R Cavity transmission Cavity Frequency Complex transmitted amplitude is non-linear in cavity pull: κ / 2 t = ω− ω − ∆ ω + κ drive cavity i / 2 Most general non-linear function of two Ising spin variables: =β +β σ z +β σ z +β σ z ⊗σ z t 0 1 L 2 R 12 L R Joint Readout = βσ L + βσ R + β σL ⊗ σ R VH ~ M 1 z 2 z 12 z z β β β Ω 1 ~ 1;2 ~ 0.8;12 ~ 0.5 Lt Ω Rt Joint Readout = βσ L + βσ R + β σL ⊗ σ R VH ~ M 1 z 2 z 12 z z Ω L − σ L = t/TL Ω z e co s( Lt) Ω R − σ R = t/TR Ω z e co s( Rt) Ω − Ω Ω + Ω LR LR State Tomography = βσ L + βσ R + β σL ⊗ σ R VH ~ M 1 z 2 z 12 z z Combine joint readout with one-qubit “analysis” rotations σ L + R π z ~VH ( Ident .) VH (Yπ ) -pulse on right σ R + L π z ~VH ( Ident .) VH (Yπ ) -pulse on left σ Lσ R + R L π z z ~VH ( Ident .) VH (Yπ ,Yπ ) on both Possible to acquire correlation information even with single, ensemble averaged msmt.! Rotate qubits to map other correlations onto z-z. See similar from Zurich group: Fillip et al., PRL 102, 200402 (2009). Measuring the Two-Qubit State Total of 16 msmts.: LLL IYXY,,,π π/2 π /2 and combinations RRR IYXY,,,π π/2 π /2 (almost) raw data left right correlations qubit qubit σ z =σz =σ z σ z = Ground state: ψ =00 = ↑↑ L R L R 1 Measuring the Two-Qubit State Apply π-pulse to invert state of right qubit σ z = + L 1 One qubit excited: ψ =01 = ↑↓ σz =σ z σ z = − R L R 1 Measuring the Two-Qubit State Now apply a two-qubit gate to entangle the qubits 1 Entangled state: ψ =()00 − 1 1 2 σ z =σ z = L R 0 σ zσ z = + L R 1 σ yσ y = + L R 1 σ xσ x = − L R 1 Witnessing Entanglement z z’ x’ CHSH operator = entanglement witness CHSH = XXZX ′′′−Z +X + ZZ′ θ x If variables take on the values ±1 Clauser, Horne, and exist even independent of Shimony & Holt (1969) measurement then CHSH = XZ()()XZ′′′− + X + Z′ Either: =0 =±2 Or: =±2 =0 Classically: CHSH ≤ 2 Witnessing Entanglement z z’ x’ CHSH operator = entanglement witness CHSH = XXZX ′′′−Z +X + ZZ′ θ ′′′′− + + x XXZXZ XZZ XXZXZ′′′′+ −XZ + Z Clauser, Horne, Shimony & Holt (1969) Bell state Separable bound: 2.44± 0.05 CHSH ≤ 2 not ? Bell’s violation! (loopholes abound) but state is clearly highly entangled! (and no likelihood req.) Control: Analyzing Product States z CHSH operator = entanglement witness x’ z’ CHSH = XXXZZ′ − Z′ + X ′ + Z′ XXZXZ′′′′− +XZ + Z θ XXZXZ′′′′+ −XZ + Z x product state Clauser, Horne, Shimony & Holt (1969) no entanglement! Using entanglement on demand to run first quantum algorithm on a solid state quantum processor Skip to Summary General Features of a Quantum Algorithm Qubit register Working M qubits create encode function process measure initialize superposition in a unitary will involve entanglement between qubits Maintain quantum coherence 1) Start in superposition: all values at once! 2) Build complex transformation out of one-qubit and two-qubit “gates” 3) Somehow* make the answer we want result in a definite state at end! *use interference: the magic of the properly designed algorithm The Search Problem −1, x ≠ x f() x = 0 = 1, x x0 “Find x0!” Position: 0 I II III “Find the queen!” The Search
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