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

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. Nature 482, 382 (2012)

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

Classical repetition code 1 p Sent Received Probability p of having a bit flipped “Binary symmetric channel” 0 000 1 111 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: No cloning theorem Measurements project qubits Errors are continuous

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

Flipping GHZs Z1Z2 = +1 Z1Z2 = -1 Z2Z3 = +1 Z2Z3 = +1 What happens when we flip one of the qubits in a GHZ-like state? Z1Z2 = +1 Z2Z3 = +1 Z1Z2 = -1 Z2Z3 = +1 Independent of and Flipped State Z1Z2 Z2Z3 None +1 Q1 -1 Q2 Q3 Each error has a different observable! - The basis for the bit flip code Four errors = two classical bits

Circuit quantum electrodynamics Our system: superconducting qubits coupled to a microwave resonator Transmon qubits Transmission-line resonator bus In analogy to cavity QED: Protection from spontaneous emission Qubit readout Multiplexed qubit drives (single-qubit gates) Mediate qubit coupling (multi-qubit gates) cQED: Wallraff Nature 431, 162 (2004) Bus: Majer Nature 449, 443 (2007) Readout: Reed PRL 105, 173601 (2010)

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

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. Nature 460, 240 (2009)

Adiabatic multiqubit phase gates A two qubit phase gate can be written: Entanglement! Interactions on two excitation manifold give entangling two-qubit conditional phases Can give a universal “Conditional Phase Gate” Top qubit flux bias (a.u.) DiCarlo, et al. Nature 460, 240 (2009)

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) Previously proposed: Strauch et al., PRL 91, 167005 (2003) Or transfer to in 6 ns!

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

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

Error correction with GHZ states Measurements force finite rotations to full flips encode error diagnose nose fix or GHZ state for Logic Works for any single error Nielsen & Chuang NMR: Cory et al. PRL 81, 2152 (1998) Ions: Chiaverini et al. Nature 432, 602 (2004)

Measurement-free QEC encode diagnose fix Feed-forward measurement hard in this first expt - Measurement-free version of the code Toffoli implements classical logic • only acts on flipped subspace Toffoli (CCNot) gate encode diagnose fix Reset (potentially) 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)

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

How do we prove the gate works? First, measure classical action 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 111 101 011 001 110 100 010 000 Classical input state Classical output state F = 86% (>50% the time of an equivalent construction) Optics: Lanyon Nat. Phys. 5, 134 (2009) Ions: Monz PRL 102, 040501 (2009) SCQs: Mariantoni Science 334, 61 (2011) Fedorov Nature 481, 170 (2012)

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

Protection from single qubit bit-flip errors Prepare “Error” rotation by some angle Correct subspace with error 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

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)

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

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 More realistic error model: Simultaneous flips on each qubit happen with probability Apply errors and measure fidelity to the prepared state as a function of p Code only works for single errors. P(2 or 3 errors) = 3p2 – 2p3 Expect quadratic dependence on p

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

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. Nature 482, 382 (2012)

Questions? Reed, et al. Nature 482, 382 (2012)

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

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)