Entangled States and Quantum Algorithms in Circuit QED Expt. Leo DiCarlo Andrew Houck Applied + Physics David Schuster Hannes Majer PI’s: Jerry Chow Rob Schoelkopf Joe Schreier Blake Johnson 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

• Two 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 ?

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 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 ‘’ 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 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 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 Grover Step-by-Step ψ = ideal 00

Begin in ground state: Grover Step-by-Step 1 ψ =+++()0101010 1 ideal 2

Create a maximal superposition: look everywhere at once! Grover Step-by-Step 1 ψ =+−+()0101010 1 ideal 2

Apply the “unknown” function, and mark the solution

⎛⎞10 0 0 ⎜⎟01 0 0 cU = ⎜⎟ 10 ⎜⎟00− 10 ⎜⎟ ⎝⎠00 0 1 Grover Step-by-Step 1 ψ =+()00 11 ideal 2

Some more 1-qubit rotations…

Now we arrive in one of the four Bell states Grover Step-by-Step 1 ψ =−+−()0101010 1 ideal 2

Another (but known) 2-qubit operation now undoes the entanglement and makes an interference pattern that holds the answer! Grover Step-by-Step ψ = ideal 10

Final 1-qubit rotations reveal the answer:

The binary representation of “2”!

The correct answer is found >80% of the time! Grover search in actionGrover in action with Other Oracles

Oracle ˆ = OcU00 cU01 cU10 cU11

F = 81% 80% 82% 81%

= ψ ρ ψ Fidelity F ideal ideal to ideal output

(average over 10 repetitions) 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 103 Coffee pods “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, , David Schuster, Andrew Houck, Florian Marquardt, Jens Koch, Terri Yu, Lev Bishop, Robert Schoelkopf, Michel Devoret We still have a long way to go.

Theorist

Experimentalist

48 Circuit QED Team Members Jared Steve Lev Blake Schwede Jens Jay Girvin Bishop Joe Johnson Koch Gambetta Schreier

Hannes Majer David Schuster Jerry Andrew Leo Chow Luigi Houck Michel DiCarlo Emily Frunzio Devoret Chan

Funding: Summary

Rudimentary two-qubit processor Entanglement on demand Adiabatic C-phase gate • Grover algorithm with Fidelity

CHSH as entanglement witness

DiCarlo et al., cond-mat 0903.2030, Nature in press Additional Slides Follow Multiplexed Qubit Control and Read-Out

Single Oscillation

52 Witnessing Entanglement

Bell state

Product state Measuring the Two-Qubit State Now apply a two-qubit gate to entangle the qubits

Q −1 Concurrence directly: C = (for pure states) 2 22222 2 QXXXYXZYXYY=++++++K ZZ 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 (T1) T1 ~ 300 ns, low Q cavity on sapphire Projective measurement

g e

e + g

• Measurement after pi/2 pulse bimodal, halfway between Deutsch-Jozsa Algorithm

= = fx0 () 0 fx1() 1

Constant functions 10

Answer is encoded in = =− the state of left qubit fx0 () x fx1() 1 x

Balanced functions 00 The correct answer is found >84% of the time. On/Off Ratio for Two-Qubit Coupling

> 100 MHz

1 MHz

3-level ⎛ ⎞ ζ =− 22⎜ 111+ + + 1⎟ 2ggLR 22 2 2 ⎜ ()()ωωωωLR−− ωωωω()() RL −− ωωωω()() RLLLRR − ωωωω −()() − −⎟ ⎝ 01 C 01 C 01 C 01 C 01 12 01 C 01 12 01 C ⎠

Diverges at Point II 4th-order in qubit-cavity coupling! 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:

⎛⎞+1000 ⎜⎟ 0100+ 1 ⎜⎟Ψ=()01000 + +11 −1 ⎜⎟00+ 10 2 ⎜⎟ ⎝⎠000 −1 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