Matt Reed Yale University Boston, MA - February 28, 2012

1. Three-qubit quantum error correction with superconducting circuits Matt Reed Yale University Boston, MA - February 28, 2012 Leo DiCarlo Simon Nigg Luyan Sun Luigi Frunzio Steven Girvin Robert Schoelkopf

2. Outline • Why is QEC necessary? • Repetition codes • Our architecture: cQED • Adiabatic and sudden two-qubit phase gates • GHZ states • Efficient Toffoli gate using third-excited state • Bit- and phase-flip error correction Reed, et al.Nature482, 382 (2012)

3. Why do we need to correct? Classical bit z Quantum bit State value y “1” “0” x Control signal Small control fluctuations do not change the system state – compressed phase space Small control fluctuations do cause a change in the system state! p ~ 10-2 - 10-5 Error probability ~ 10-15 To get p~10-15would need T1 ~ 1 year

4. Classical repetition code • No cloning theorem • Measurements project qubits • Errors are continuous 1-p Probability p of having a bit flipped “Binary symmetric channel” 0 0 p 0 000 1 111 Received Sent p 1 1 1-p Repetition code: send each bit three times, then vote Reduces classical error rate to 3p2 – 2p3 Quadratic! Can we do this for quantum computing? Some reasons to think no:

5. GHZ-like states It is not possible to go from But we can make All ZiZj correlations are +1, independentof and Qubits 1 and 2 are either: both 0 both 1 or Z1Z2 = = +1 = “I don’t know where they are pointing, but I know they’re pointing in the same direction”

6. Flipping GHZs What happens when we flip one of the qubits in a GHZ-like state? Z1Z2 = -1 Z2Z3= +1 Z1Z2 = +1 Z2Z3= +1 Independentof and Each error has a different observable! - The basis for the bit flip code Four errors = two classical bits

7. Circuit quantum electrodynamics Our system: superconducting qubits coupled to a microwave resonator • Protection from spontaneous emission • Qubit readout • Multiplexed qubit drives (single-qubit gates) • Mediate qubit coupling (multi-qubit gates) Transmon qubits Transmission-line resonator bus In analogy to cavity QED: cQED: WallraffNature 431, 162 (2004) Bus: Majer Nature 449, 443 (2007) Readout: Reed PRL105, 173601 (2010)

8. Four-qubit cQED device • Four transmon qubits coupled to single 2D microwave resonator cavity Q3 • Three qubits biased at 6, 7, and ~8 GHz • Fourth qubit above cavity and unused • T1 ~ 1 μs, T2 ~ 0.5 μs • Flux bias linesto control frequency • Nanosecond speed - two qubit gates Frequency (GHz) Q2 Q1 Flux bias on Qubit 1 (a.u.) DiCarlo, et al. Nature 467 574 (2010)

9. Adiabatic multiqubit phase gates A two qubit phase gate can be written: Entanglement! Interactions on two excitation manifold give entangling two-qubit conditional phases Top qubit flux bias (a.u.) DiCarlo, et al. Nature460, 240 (2009)

10. Adiabatic multiqubit phase gates A two qubit phase gate can be written: Entanglement! Interactions on two excitation manifoldgive entangling two-qubit conditional phases Can give a universal “Conditional Phase Gate” Top qubit flux bias (a.u.) DiCarlo, et al. Nature460, 240 (2009)

11. Sudden multiqubit phase gates Suddenly move into resonance with Crossing measurement: • Jump to a flux • Wait some time • Jump back • Measure if in 11 (black) or 02 (white) Or transfer to in 6 ns! Previously proposed: Strauchet al., PRL91, 167005 (2003)

12. Entangled states on demand State Tomography 01 DiCarlo, et al. Nature 467 574 (2010)

13. GHZ states on demand 01 State Tomography 10 Can simply change the preparation of Q2 to encode any state DiCarlo, et al. Nature 467 574 (2010)

14. Error correction with GHZ states Measurements force finite rotations to full flips error encode diagnose fix nose or or Logic Works for any single error GHZ state for Nielsen & Chuang NMR:Cory et al. PRL81, 2152 (1998) Ions:Chiaverini et al. Nature432, 602 (2004)

15. Measurement-free QEC Feed-forward measurement hard in this first expt - Measurement-free version of the code Toffoliimplements classical logic • only acts on flipped subspace encode diagnose fix Reset (potentially) Toffoli (CCNot) gate Toffoli can be constructed with five two-qubit gates, but that’s expensive How can we do better? Nielsen & Chuang Cambridge Univ. Press Ions: P. Schindler et al. Science 332, 1059 (2011)

16. Toffoli gate with noncomputational states Two-qubit gate requires two excitations The essence! This interaction is small, so use intermediary Three-qubit interaction: third excited state Adiabatic interaction: Sudden transfer: Identical for: Three-qubit phase here!

17. Classical truth table How do we prove the gate works? First, measure classical action Classically, a phase gate does nothing. So we dress it up to make it a CCNOT F= 86% (>50% the time of an equivalent construction) 000 010 100 110 111 101 Classical input state 001 011 011 001 110 101 Optics: Lanyon Nat. Phys. 5, 134 (2009) Ions: MonzPRL102, 040501 (2009) SCQs: MariantoniScience334, 61 (2011) Fedorov Nature481, 170 (2012) 100 010 111 000 Classical output state

18. Quantum process tomography of CCPhase Want to know the action on superpositions: (but now with 64 basis states) Experiment 0.6 Theory Invert to find 0.3 0.0 Input operator Input operator Output operator Output operator F= 78% 4032 Pauli correlation measurements (90 minutes)

19. Protection from single qubit bit-flip errors “Error” rotation by some angle Correct subspace with error Prepare Encode in three-qubit state Decode syndromes Measure single-qubit state fidelity to Ideally, there should be no dependence of fidelity on the error rotation angle

20. Correction fidelity vs. bit-flip error rotation Encode, single known error, decode, fix, and measure resulting state fidelity No correction Error on Q2 Error on Q1 Error on Q3 (No-correction curve has finite fidelity because its duration is the same as the corrected curves)

21. Error syndromes Is the algorithm really doing what we think? Look at two-qubit density matrices ofancillas after a full flip Bottom flip No error 1 0 -1 II Top flip Protected flip ZI ZZ IZ IZ ZI ZZ II

22. Phase-flip error correction code Bit-flips are not common errors, but phase flips are – modify code Differs from bit-flip code by single qubit rotations; e.g. change of coordinate system Apply errors and measure fidelity to the prepared state as a function of p More realistic error model: Simultaneous flips on each qubit happen with probability Code only works for single errors. P(2 or 3 errors) = 3p2 – 2p3 Expect quadratic dependenceon p

23. Simultaneous phase-flip errors To measure the effect of the code on any state, test with four one-qubit basis states Depends only quadratically on error probability! No correction Corrected For better coherence, see 3D Cavity session L39 (room 109B)

24. Summary • Realized the simplest version of gate-based QEC • Both bit- and phase-flip correction • Not fault-tolerant (gate based, un-encoded) • Based on new three-qubit phase gate • Adiabatic interaction with transmon third excited state • Works for any three nearest-neighbor qubits • 86% classical fidelity and 78% quantum process fidelity Reed, et al.Nature482, 382 (2012)

25. Questions? Reed, et al.Nature482, 382 (2012)

26. CCNot gate pulse sequence More than two times faster than equivalent two-qubit gate sequence

27. Three qubit state tomography Joint Readout Example: extract no pre-rotation: on Q1 and Q2: on Q1 and Q3: on Q2 and Q3: DiCarlo, et al. Nature 467 574 (2010)