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) Eiiaig ChargeNoise Better with Design “Eliminating” Energy Cooper-pair Box Cooper-pair E J /E C = 1 Koch et al., 2007; Houck et al., 2008 exponentially suppresses 1/f! E “Transmon” J /E C 5-100 = 25 - 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 σ t ) 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 + HH γ κ + 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 κ ω − − ∆ + 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 β= β β σ L +σ R + σ σ L ⊗ 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
β= β β σ L +σ R + σ σ L ⊗ 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
β= β β σ L +σ R + σ σ L ⊗ 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 π /2 /2 and combinations RRR IYXY,,, π π π /2 /2
(almost) raw data
left right correlations qubit qubit
σ z = σσ z =σ z 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 = ↑↓ σσ z =σ z 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 Problem −1, x ≠ x f() x = 0 = 1, x x0
“Find x0!”
Position: 0 I II III
“Find the queen!” The Search Problem −1, x ≠ x f() x = 0 = 1, x x0
“Find x0!”
Position: 0 I II III
“Find the queen!” The Search Problem −1, x ≠ x f() x = 0 = 1, x x0
“Find x0!”
Position: 0 I II III
“Find the queen!” The Search Problem
Classically, takes on average 2.25 guesses to succeed…
Use QM to “peek” under all the cards, find queen on first try!
Position: 0 I II III
“Find the queen!” Grover’s Algorithm
Challenge: “unknown” Find the location unitary of the -1 !!! operation: (= queen)
Previously implemented in NMR: Chuang et al., 1998 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 =(00 +0 1 + 1 0 + 1 1 ) ideal 2
Create a maximal superposition: look everywhere at once! Grover Step-by-Step 1 ψ =(00 +0 1 − 1 0 + 1 1 ) ideal 2
Apply the “unknown” function, and mark the solution
1 0 0 0 0 1 0 0 cU = 10 0 0− 1 0 0 0 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 =(00 −0 1 + 1 0 − 1 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 = O cU00 cU01 cU10 cU11
F = 81% 80% 82% 81%
= ψ ρ ψ Fidelity F ideal ideal to ideal output
(average over 10 repetitions) 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
56 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 Q= XX2 + XY2 + XZ2 + YX2 + YY2 +… + ZZ 2 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
= = f0 ( x ) 0 f1( x ) 1
Constant functions 10
Answer is encoded in = = − the state of left qubit f0 () x x f1( x ) 1 x
Balanced functions 00 The correct answer is found >84% of the time. 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 One-Qubit Gates
01
Preparation 1-qubit rotations Measurement
10
cavity I Cos(ωΡ t)
Apply microwave pulse resonant with qubit Spectroscopy of Qubits Interacting with Cavity
right qubit
Qubit-qubit swap interaction Majer et al., Nature (2007)
left qubit
Cavity-qubit interaction Vacuum Rabi splitting Wallraff et al., Nature (2004) cavity
Flux bias line controls qubit transition frequency Spectroscopy of Qubits Interacting with Cavity
right qubit Qubits mostly separated 01 and non-interacting due to frequency difference Preparation 1-qubit rotations Measurement
10 left qubit
cavity One-Qubit Gates
right qubit 01
Preparation 1-qubit rotations Measurement
10 left qubit
cavity Q Sin(ωΡ t)
J. Chow et al., PRL (2009): Fidelity = 99% Two-Qubit Gate: Turn On Interactions
01
Use control lines to push Conditional qubits near a resonance: phase gate A controlled z-z 10 interaction also ala NMR
cavity Two-Excitation Manifold of System
“Qubits” and cavity both have multiple levels… On/Off Ratio for Two-Qubit Coupling
> 100 MHz
1 MHz
3-level transmons ζ = − 2 2 1 + 1 + 1 + 1 2gLR g 2 2 2 2 ()()ω ω ω ω ωL ω ω ω ω− ω R −()()R −L −()()RLL − −()()LRR − − 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! State Tomography
β= β β σ L +σ R + σ σ L ⊗ 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 info., even with single, ensemble averaged msmt.!
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 π /2 /2 and combinations RRR IYXY,,, π π π /2 /2 max. likelihood (nonlinear!) (almost) raw data Re(ρ )
left right correlations qubit qubit
Ground state: ψ = 00 Density matrix Measuring the Two-Qubit State
Apply π-pulse to invert state of right qubit Re(ρ )
One qubit excited: ψ = 01 Measuring the Two-Qubit State Now apply a two-qubit gate to entangle the qubits 1 Entangled state: ψ =()00 − 1 1 2 Re(ρ )
C = 0.94 +-?? “Concurrence”: What’s the λ λ λ λρ = { − − − } entanglement C( ) max 0, 1 2 3 4 metric? λ are e-values of ρρɶ Microwave pulses at ω 01 can be used for single qubit rotations.