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Entangled States and Quantum Algorithms in Circuit QED Expt Entangled States and Quantum Algorithms in Circuit QED Expt. 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 Jay Gambetta Alexandre Blais Florian Marquardt Eli Luberoff Lars Tornberg Terri Yu Entangled States and Quantum Algorithms in Circuit QED Expt. 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 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 • Noise and how to ignore it • 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., Nature 460, 240 (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) Decoherence Time: Superposition is Lost 1 500s TOTAL ACQUISITION TIME 0.5 BILLION SHOTS 100/1 population) SIGNAL/NOISE = ≠ ( contrast=61%, "visibility" > 87% Devoret group at Yale: 1 POLE FIT Ramsey fringes 0 6 Time in nanoseconds Different types of SC qubits ► Nonlinearity from Josephson junctions NEC, Chalmers, Nakamura et al., NEC Labs C Saclay, Yale charge Vion et al., Saclay qubit = E Devoret et al., Schoelkopf et al., Yale, J (CPB) E Delsing et al., Chalmers C TU Delft,UCB Lukens et al., SUNY flux Mooij et al., Delft qubit = 40-100E Orlando et al., MIT J Clarke, UC Berkeley E C Martinis et al., UCSB NIST,UCSB Simmonds et al., NIST phase Wellstood et al., U Maryland qubit Koch et al., IBM = 10,000E J …and more… E Reviews: Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001) M. H. Devoret, A. Wallraff and J. M. Martinis, cond-mat/0411172 (2004) J. Q. You and F. Nori, Phys. Today, Nov. 2005, 42 State of the Art in Superconducting Qubits • Nonlinearity from Josephson junctions (Al/AlOx/Al) Charge Charge/Phase Flux Phase TU Delft/SUNY NIST/UCSB NEC/Chalmers Saclay/Yale Junction size EJ = EC # of Cooper pairs •1st qubit demonstrated in 1998 (NEC Labs, Japan) • “Long” coherence shown 2002 (Saclay/Yale) • Several experiments with two degrees of freedom • C-NOT gate (2003 NEC, 2006 Delft and UCSB ) • CHSH Bell inequality violation (2009, UCSB, Yale [w/meas. Loophole]) • 2 qubit Grover search and Deutsch-Josza algorithms (2009, Yale) Progress in Superconducting QC… ?? 9 WHY SUPERCONDUCTIVITY? E few electrons N ~ 109 total number of electrons “forest” of states N even ~ 1eV 2Δ ~ 1meV superconducting gap ATOM SUPERCONDUCTING NANOELECTRODE Collective Quantization easiest (?) to understand for charge qubits An isolated superconductor has definite charge. For an even number of electrons there are Al, Nb no low energy degrees of freedom! Unique non-degenerate quantum ground state. Normal State Superconducting State 2Δ Ne(2 ) 11 To have any degrees of freedom, we need a junction. charge qubits ()(2)M − ne Josephson tunnel barrier ()(2)Nn+ e 2Δ 2Δ 2Δ 2Δ 2Δ n =−2 n =−1 n = 0 n =+1 n =+2 Tunnel coupling: −+++ EnnnJ ∑ 11 n n Charging energy: + 2 4Ennnc ∑ 12 n First Rabi oscillations: 1999 13 First SC charge qubit works! But… coherence time extremely short. Generic asymmetry means qubit has different static dipole moments in ground and excited states. Ground Excited state state Environment can measure the qubit state via stray electric fields. What to do? 14 Outsmarting noise: CPB sweet spot ◄ charge fluctuations sweet spot only sensitive nd energy to 2 order energy fluctuations in gate charge! Vion et al., ng (gate charge) Science 296, 886 (2002) ng 15 First Ramsey Fringe Experiment Proves True Coherence of Superpositions Science 296, 886 (2002) T2* rises to 300-500 ns Double sweet spot! 16 ‘Transmon’ Cooper Pair Box: Charge Qubit that Beats Charge Noise Josephson junction: EJ >> EC 300 μm e g Added metal = capacitor & antenna plasma oscillation of 2 or 3 Cooper pairs: exponentially small static dipole Transmon qubit insensitive to 1/f electric fields * Theory: J. Koch et al., PRA (2007); Expt: J. Schreier et al., P 17 Flux qubit + capacitor: F. You et al., PRB (2006) Transmon Qubit: Nota Bene! Sweet Spot Everywhere! unlike phase qubit Ψ=Ψ+()ϕϕ ( 2)π EJ/EC = 1 EJ/EC = 5 EJ/EC = 10 EJ/EC = 50 Energy Ng (gate charge) charge Exponentially small charge dispersion! dispersion ωω−≈− 18 Adequate anharmonicity: 12 01E c 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 TT2123.0s Error per gate = 1.2 % 111=+ > μ * ⇒ Tϕ 35 s TTT212 ϕ Similar error rates in phase qubits (UCSB): Lucero et al. PRL 100, 247001 (2007) ‘Moore’s Law’ for Charge Qubit Coherence Times ?? (No Echo) ≥ μ 20 T2 now limited largely by T1 Tϕ 30 s 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=()01 + ⊗()01 + 2 1 =+++()00 10101 1 2 ‘Conditional Phase Gate’ entangler: ⎛⎞+1000 ⎜⎟ 0100+ 1 ⎜⎟Ψ=()01000 + +11 −1 ⎜⎟00+ 10 2 ⎜⎟ ⎝⎠000 −1 No longer a product state! How do we realize the conditional phase gate? ⎛⎞+1000 ⎜⎟ 0100+ 1 ⎜⎟Ψ=()00101 + +01 −1 ⎜⎟00+ 10 2 ⎜⎟ ⎝⎠000 −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 Entanglement on demand using controlled phase gate 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) Yale: DiCarlo et al., Nature (2009) How do we read out the qubit state and measure the entanglement? Two Qubit Joint Readout via Cavity Reverse of Haroche et al.: photons measure qubits χ χ R 2 L χ ~/g Δ χ + χ L R Cavity transmission Frequency “Strong dispersive cQED”: Schuster et al., 2007 Basic two-qubit readout demo’d in Majer et al., 2007 Cavity Pull is linear in spin polarizations Δ=ω σ z +σ z abL R Cavity transmission Frequency Complex transmitted amplitude is non-linear in cavity pull: κ /2 t = ωω−−Δ+ ωκ drive cavity i /2 Most general non-linear function of two Ising spin variables: =+β β σσz +βσz +β z ⊗σ z t 011LL 2 RR2 State Tomography = ββσ L ++σ R βσσL ⊗ R VMH12~ z z 12 zz Combine joint readout with one-qubit “analysis” rotations Do nothing: β σ L ++ββσ R σ LR⊗σ 1 z 21z 2zz + π -pulse on β σ L −−ββσ R σ LR⊗σ right qubit 1 z 21z 2zz β σ L 2 1 z See similar from Zurich group: Fillip et al., PRL 102, 200402 (2009). State Tomography = ββσ L ++σ R βσσL ⊗ R VMH12~ z z 12 zz Combine joint readout with one-qubit “analysis” rotations Do nothing: β σ L ++ββσ R σ LR⊗σ 1 z 21z 2zz + π -pulse on −ββσ L −+σ R βσσL ⊗ R both qubits 12z z 12 zz β σ LR⊗σ 2 12 zz Possible to acquire correlation info., even with single, ensemble averaged msmt.! (single-shot fidelity ~ 10% here) Measuring the Full Two-Qubit State Total of 16 msmts.: L LL IY,,πππ X/2 , Y /2 and combinations R RR IY,,πππ X/2 , Y /2 max. likelihood (almost) raw data (nonlinear!) Re(ρ ) left right correlations qubit qubit Density matrix Ground state: ψ = 00 Measuring the Two-Qubit State Apply π-pulse to invert state of left qubit Re(ρ ) One qubit excited: ψ = 10 Measuring the Two-Qubit State Now apply a two-qubit gate to entangle the qubits 1 Entangled state: ψ =+()0101 2 Re(ρ ) C = 0.94 +-?? “Concurrence”: What’s the ρ = { λλλλ−−−} entanglement C( )max0,1234 metric? λ are e-values of ρρ% Witnessing Entanglement z z’ x’ CHSH operator = entanglement witness CHSH = XXZXZ′′′−++Z X Z ′ θ XX ′′′′−++XZZ X ZZ x XX ′′′′+−+XZZ X ZZ Clauser, Horne, Shimony & Holt (1969) Bell state Separable bound: CHSH ≤ 2 2.61± 0.04 not test of hidden variables… (loopholes abound) but state is clearly highly entangled! Chow et al., arXiv:0908.1955 Control: Analyzing Product States z CHSH operator = entanglement witness x’ z’ CHSH = XZXX′ −+Z′ + ZX ′ Z′ XX ′′′′−++XZZ X ZZ θ XX ′′′′+−+XZZ X ZZ x product state Clauser, Horne, Shimony & Holt (1969) control experiment shows no entanglement Using entanglement on demand to run first quantum algorithm on a solid state quantum processor Skip to Summary Grover’s Algorithm Challenge: “unknown” Find the location unitary of the -1 !!! operation: Previously implemented in NMR: Chuang et al., 1998 Optics: Kwiat et al., 2000 Ion traps: Brickman et al., 2003 ORACLE 10 pulses w/ nanosecond resolution, total 104 ns duration
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