Entangled States and Quantum Algorithms in Circuit QED

Transcription

1 Entangled States and Quantum Algorithms in Circuit QED Applied Physics + Physics Yale University PI s: Rob Schoelkopf Michel Devoret Steven Girvin Expt. Leo DiCarlo Andrew Houck David Schuster Hannes Majer Jerry Chow Joe Schreier Blake Johnson Luigi Frunzio Theory Lev Bishop Jens Koch Jay Gambetta Alexandre Blais Florian Marquardt Eli Luberoff Lars Tornberg Terri Yu

2 Entangled States and Quantum Algorithms in Circuit QED Applied Physics + Physics Yale University PI s: Rob Schoelkopf Michel Devoret Steven Girvin Princeton Vienna IQC/Waterloo U. Sherbrooke Munich Göteborg Expt. Leo DiCarlo Andrew Houck David Schuster Hannes Majer Jerry Chow Joe Schreier Blake Johnson Luigi Frunzio Theory Lev Bishop Jens Koch Jay Gambetta Alexandre Blais Florian Marquardt Eli Luberoff Lars Tornberg Terri Yu

3 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

4 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)

5 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 V gb C gb C c C ge V ds Box SET V ge (After Konrad Lehnert)

6 ( population) 1 Decoherence Time: Superposition is Lost 500s TOTAL ACQUISITION TIME 0.5 BILLION SHOTS SIGNAL/NOISE = 100/1 contrast=61%, "visibility" > 87% 0 1 POLE FIT Devoret group at Yale: Ramsey fringes Time in nanoseconds 6

7 Different types of SC qubits Nonlinearity from Josephson junctions E J = E C NEC, Chalmers, Saclay, Yale charge qubit (CPB) Nakamura et al., NEC Labs Vion et al., Saclay Devoret et al., Schoelkopf et al., Yale, Delsing et al., Chalmers E J = 10,000E C E J = E C TU Delft,UCB NIST,UCSB Reviews: flux qubit phase qubit Lukens et al., SUNY Mooij et al., Delft Orlando et al., MIT Clarke, UC Berkeley Martinis et al., UCSB Simmonds et al., NIST Wellstood et al., U Maryland Koch et al., IBM and more Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001) M. H. Devoret, A. Wallraff and J. M. Martinis, cond-mat/ (2004) J. Q. You and F. Nori, Phys. Today, Nov. 2005, 42

8 State of the Art in Superconducting Qubits NEC/Chalmers Nonlinearity from Josephson junctions (Al/AlO x /Al) Charge Charge/Phase Saclay/Yale Flux TU Delft/SUNY Phase NIST/UCSB Junction size E J = E C # of Cooper pairs 1 st 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)

9 Progress in Superconducting QC?? 9

10 WHY SUPERCONDUCTIVITY? E few electrons N ~ 10 9 forest of states total number of electrons N even ~ 1 ev 2Δ ~ 1meV superconducting gap ATOM SUPERCONDUCTING NANOELECTRODE

11 Collective Quantization easiest (?) to understand for charge qubits An isolated superconductor has definite charge. Al, Nb For an even number of electrons there are no low energy degrees of freedom! Unique non-degenerate quantum ground state. Normal State Superconducting State 2Δ To have any degrees of freedom, we need a junction. N(2 e) 11

12 charge qubits ( M n)(2 e) ( N + n)(2 e) Josephson tunnel barrier 2Δ 2Δ 2Δ 2Δ 2Δ n = 2 n = 1 n = 0 n =+1 n =+2 Tunnel coupling: E J n n+ 1 + n+ 1 n Charging energy: n Ec n n n n 12

13 First Rabi oscillations:

14 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 state Excited state Environment can measure the qubit state via stray electric fields. What to do? 14

15 Outsmarting noise: CPB sweet spot charge fluctuations energy energy sweet spot only sensitive to 2 nd order fluctuations in gate charge! n g n g (gate charge) Vion et al., Science 296, 886 (2002) 15

16 First Ramsey Fringe Experiment Proves True Coherence of Science 296, 886 (2002) Superpositions T 2 * rises to ns Double sweet spot! 16

17 Transmon Cooper Pair Box: Charge Qubit that Beats Charge Noise Josephson junction: E J >> E C 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)

18 Transmon Qubit: Sweet Spot Everywhere! Nota Bene! unlike phase qubit Ψ ( ϕ) =Ψ ( ϕ+ 2 π ) E J /E C = 1 E J /E C = 5 E J /E C = 10 E J /E C = 50 Energy N g (gate charge) Exponentially small charge dispersion! ω ω E Adequate anharmonicity: c charge dispersion 18

19 Coherence in Transmon Qubit T1 = 1.5μs Random benchmarking of 1-qubit ops Chow et al. PRL 2009: Technique from Knill et al. for ions * 2 1 T = 2T = 3.0μs * = + Tϕ > 35μs T 2T T ϕ Error per gate = 1.2 % Similar error rates in phase qubits (UCSB): Lucero et al. PRL 100, (2007)

20 Moore s Law for Charge Qubit Coherence Times?? (No Echo) T2 now limited largely by T1 T ϕ 30μs 20

21 Qubits Coupled with a Quantum Bus use microwave photons guided on wires! Circuit QED Blais et al., Phys. Rev. A (2004) transmission line cavity out 7 GHz in Josephson-junction qubits Expts: Majer et al., Nature 2007 (Charge qubits / Yale) Sillanpaa et al., Nature 2007 (Phase qubits / NIST)

22 A Two-Qubit Processor flux bias lines control qubit frequency 1 ns resolution DC - 2 GHz T = 10 mk cavity: entanglement bus, driver, & detector transmon qubits

23 RY How do we entangle two qubits? ( π /2) rotation on each qubit yields superposition: 1 Ψ = ( ) ( ) 2 1 = ( ) 2 Conditional Phase Gate entangler: Ψ = ( ) No longer a product state!

24 How do we realize the conditional phase gate? Ψ = ( ) Use control lines to push qubits near a resonance: A controlled z-z interaction also à la NMR

25 Key is to use 3 rd level of transmon (outside the logical subspace) Coupling turned off Coupling turned on: Near resonance with 3 rd level ω ω Energy is shifted if and only if both qubits are in excited state.

26 Entanglement on demand using controlled phase gate Re( ρ) Bell state Fidelity Concurrence % 88% 94% 94% 90% 86% 87% 81% UCSB: Steffen et al., Science (2006) ETH: Leek et al., PRL (2009) Yale: DiCarlo et al., Nature (2009)

27 How do we read out the qubit state and measure the entanglement?

28 Two Qubit Joint Readout via Cavity Reverse of Haroche et al.: photons measure qubits Cavity transmission χl χ L + χ R χ R χ 2 ~ g / Δ Strong dispersive cqed : Schuster et al., 2007 Basic two-qubit readout demo d in Majer et al., 2007 Frequency

29 Cavity Pull is linear in spin polarizations Cavity transmission z L Δ ω = aσ + b σ z R Frequency Complex transmitted amplitude is non-linear in cavity pull: t = κ /2 ω ω Δ ω+ iκ /2 drive cavity Most general non-linear function of two Ising spin variables: t β σ z βσ z β σ z 0 1 L 2 R 12 L = β σ z R

30 State Tomography V M σ z L R L R H ~ = β1 σ z + β2 + β12 σz σz Combine joint readout with one-qubit analysis rotations Do nothing: + π-pulse on right qubit β σ + β σ + β σ σ 1 1 L z L z R L R 2 z 12 z z β σ β σ β σ σ R L R 2 z 12 z z L 2β1 σ z See similar from Zurich group: Fillip et al., PRL 102, (2009).

31 State Tomography V M σ z L R L R H ~ = β1 σ z + β2 + β12 σz σz Combine joint readout with one-qubit analysis rotations Do nothing: + π-pulse on both qubits β σ + β σ + β σ σ 1 L z R L R 2 z 12 z z β σ β + β σ σ L R L R 1 z 2 σ z 12 z z L R 2β12 σ z σ z Possible to acquire correlation info., even with single, ensemble averaged msmt.! (single-shot fidelity ~ 10% here)

32 Measuring the Full Two-Qubit State Total of 16 msmts.: (almost) raw data IY,, X, Y L L L π π/2 π/2 R R R π π/2 π/2 IY,, X, Y and combinations max. likelihood (nonlinear!) Re( ρ) left qubit right qubit correlations Ground state: ψ = 00 Density matrix

33 Measuring the Two-Qubit State Apply π-pulse to invert state of left qubit Re( ρ) One qubit excited: ψ = 10

34 Measuring the Two-Qubit State Now apply a two-qubit gate to entangle the qubits Entangled state: 1 ψ = + 2 ( ) Re( ρ) What s the entanglement metric? Concurrence : C( ρ) = max 0, λ λ λ λ λ are e-values of C = ?? { } ρρ%

35 z z θ Witnessing Entanglement x Clauser, Horne, Shimony & Holt (1969) Separable bound: CHSH 2 x not test of hidden variables (loopholes abound) but state is clearly highly entangled! Chow et al., arxiv: CHSH operator = entanglement witness CHSH = XX XZ + ZX + ZZ Bell state XX XZ + ZX + ZZ XX + XZ ZX + ZZ 2.61± 0.04

36 Control: Analyzing Product States z z θ x Clauser, Horne, Shimony & Holt (1969) x CHSH operator = entanglement witness CHSH = XX XZ + ZX + ZZ product state XX XZ + ZX + ZZ XX + XZ ZX + ZZ control experiment shows no entanglement

37 Skip to Summary Using entanglement on demand to run first quantum algorithm on a solid state quantum processor

38 Grover s Algorithm unknown unitary operation: Challenge: Find the location of the -1!!! 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

39 ψ ideal = 00 Grover Step-by-Step Begin in ground state:

40 1 ψ ideal = Create a maximal superposition: look everywhere at once! Grover Step-by-Step ( )

41 1 ψ ideal = Apply the unknown function, and mark the solution Grover Step-by-Step ( ) cu =

42 ideal ( 11 ) ψ = + Grover Step-by-Step Some more 1-qubit rotations Now we arrive in one of the four Bell states

43 Grover Step-by-Step 1 ψ ideal = ( ) Another (but known) 2-qubit operation now undoes the entanglement and makes an interference pattern that holds the answer!

44 ψ ideal = 10 Grover Step-by-Step Final 1-qubit rotations reveal the answer: The binary representation of 2! The correct answer is found >80% of the time!

45 Grover search in action Grover in action with Other Oracles Oracle Ô= cu cu cu 10 cu 11 F = 81% 80% 82% 81% F = ψ ρ ψ Fidelity to ideal output ideal (average over 10 repetitions) ideal

46 The cost of entanglement 1 Cryogenic HEMT amp 2 Room Temp Amps 1 Two-channel digitizer 1 Two-channel AWG 1 Four-channel AWG 2 Scalar signal generators 2 Vector signal generators 1 Low-frequency generator 1 Rubidium frequency standard 2 Yokogawa DC sources 1 DC power supply 1 Amp biasing servo 1 Computer 10 3 Coffee pods

47 Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs 30 tons, computers in the future may have only 1,000 vacuum tubes and perhaps only weigh one and a half tons. Popular Mechanics, 1949 Many thanks to: Jay Gambetta, Jens Koch, Ren-Shou Huang, Alexandre Blais, Krishnendu Sengupta, Aashish Clerk, Andreas Wallraff, David Schuster, Andrew Houck, Florian Marquardt, Jens Koch, Terri Yu, Lev Bishop, Robert Schoelkopf, Michel Devoret

48 We still have a long way to go. Theorist Experimentalist 48

49 Joe Schreier Circuit QED Team Members Blake Johnson Jared Schwede Jens Koch Jay Gambetta Steve Girvin Lev Bishop Hannes Majer Leo DiCarlo Emily Chan David Schuster Jerry Chow Luigi Frunzio Andrew Houck Michel Devoret Funding:

50 Summary Rudimentary two-qubit processor Grover algorithm with Fidelity Adiabatic C-phase gate Entanglement on demand CHSH as entanglement witness DiCarlo et al., cond-mat , Nature in press

51 Additional Slides Follow

52 Multiplexed Qubit Control and Read-Out Single Oscillation 52

53 Witnessing Entanglement Bell state Product state

54 Measuring the Two-Qubit State Now apply a two-qubit gate to entangle the qubits Concurrence directly: (for pure states) C = Q Q= XX + XY + XZ + YX + YY + K+ ZZ

55 Single shot readout fidelity Histograms of single shot msmts. Integrated probabilities e g e g Measurement with ~ 5 photons in cavity; SNR ~ 4 in one qubit lifetime (T 1 ) T1 ~ 300 ns, low Q cavity on sapphire

56 Projective measurement e g e + g Measurement after pi/2 pulse bimodal, halfway between

57 Deutsch-Jozsa Algorithm f ( x ) = 0 f ( x ) = Constant functions 10 f ( x) = x f ( x) = x Answer is encoded in the state of left qubit Balanced functions 00 The correct answer is found >84% of the time.

58 On/Off Ratio for Two-Qubit Coupling > 100 MHz 1 MHz 3-level transmons ζ = 2g g L R L R R L R L L L R R ( ω01 ωc )( ω01 ωc ) ( ω01 ωc )( ω01 ωc ) ( ω01 ω12 )( ω01 ωc ) ( ω01 ω12 )( ω01 ωc ) 4 th -order in qubit-cavity coupling! Diverges at Point II

59 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

60 Adjust timing so that amplitude for both qubits to be excited acquires a minus sign: Ψ = ( )

61 General Features of a Quantum Algorithm Qubit register Working qubits initialize create superposition encode function in a unitary process M measure 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