Three-qubit quantum error correction with superconducting circuits Matt Reed Yale University December 6, 2011 - 10:00 am Leo DiCarlo Luigi Frunzio Simon Nigg Steven Girvin Luyan Sun Robert Schoelkopf
Outline • Introduction to superconducting qubits • Adiabatic and sudden two-qubit phase gates • GHZ states • Efficient Toffoli gate using third-excited state • Bit- and phase-flip error correction • Outlook
Superconducting transmon qubits ω 01 ≠ ω 12 Energy L = L J 2 C ω 12 cos δ 1 ω 01 0 Nonlinear LC Oscillator Junction phase δ 300 µ m Theory: Koch , et al. PRA (2007) Experiment: Schreier , et al. PRB (2009) Review: Houck , et al. Quant. Int. Proc. (2009)
Circuit quantum electrodynamics Cavity QED Circuit QED Couple transmon qubits to superconducting microwave resonator • Protection from spontaneous emission • Multiplexed qubit drives ( single-qubit gates ) • Couple qubits together ( multi-qubit gates ) • Qubit readout Jaynes-Cummings Hamiltonian: H = � ω r ( a † a + 1 / 2) + � ω a σ z + � g ( a † σ − + a σ + )
Qubit readout H = � ω r ( a † a + 1 / 2) + � ω a σ z + � g ( a † σ − + a σ + ) g ⌧ ∆ = ω a � ω r Dispersive limit: ⇤ ⌅ � ω r + g 2 a † a + 1 / 2 ⇥ + � ω a σ z / 2 H ≈ � ∆ σ z Homodyne voltage (µV) Homodyne voltage (µV) | 0 i 60 60 | 1 i 40 40 20 20 Three qubits: multiplexed 0 0 9.10 9.10 9.12 9.12 9.14 9.14 9.16 9.16 M = 000 000 Frequency (GHz) Frequency (GHz) State tomography Reed , et al. Phys. Rev. Lett. 105, 173601 (2010)
Four qubit cQED device • Four transmon qubits coupled to single microwave resonator • Three qubits biased at 6 , 7 , and ~8 GHz (and one above) • Each has a flux bias line to control frequency in nanoseconds • Two qubit gates DiCarlo, et al. Nature 467 574 (2010)
Adiabatic multiqubit phase gates | 02 � A two qubit phase gate can be written: | 11 � | 00 ⇤ � ⇥ | 00 ⇤ | 01 ⇥ � e i φ 01 | 01 ⇥ Entanglement! | 10 ⇥ � e i φ 10 | 10 ⇥ | 11 ⇥ � e i ( φ 01 + φ 10 + φ 11 ) | 11 ⇥ Interactions on two excitation manifold give entangling two-qubit conditional phases | 01 � Z φ 01 = ∆ ω 01 ( t ) dt | 10 � Top qubit flux bias (a.u.)
Adiabatic multiqubit phase gates | 02 � A two qubit phase gate can be written: | 11 � | 00 ⇤ � ⇥ | 00 ⇤ | 01 ⇥ � e i φ 01 | 01 ⇥ Entanglement! | 10 ⇥ � e i φ 10 | 10 ⇥ | 11 ⇥ � e i ( φ 01 + φ 10 + φ 11 ) | 11 ⇥ ζ Interactions on two excitation manifold give entangling two-qubit conditional phases | 01 � Z φ 11 = − 2 π ζ ( t ) dt Z φ 01 = ∆ ω 01 ( t ) dt Can give a universal “Conditional Phase Gate” | 00 ⇥ � | 00 ⇥ φ 01 = φ 10 = 0 | 01 ⇥ � | 01 ⇥ | 10 � | 10 ⇥ � | 10 ⇥ φ 11 = π | 11 ⇤ ⇥ � | 11 ⇤ Top qubit flux bias (a.u.) DiCarlo , et al. Nature 460 , 240 (2009)
Sudden multiqubit phase gates τ Frequency Suddenly move into resonance with | 11 i | 02 i | 11 i ! | ψ ( t = 0) i = | + i + | �i 11 Time | + � 02 02 | �⇥ 11 | ψ ( t ) i = e i ∆ t/ 2 | + i + e − i ∆ t/ 2 | �i τ = 12 ns | ψ ( t = 2 π / ∆ ) i = � ( | + i + | �i ) ! � | 11 i Or, transfer to in 6 ns! | 02 i Strauch et al ., PRL (2003): proposed this approach
Entangled states on demand /2 R π /2 0 R π Tomography y y 01 1 State ( ) 0 0 0 1 1 /2 0 ψ = ⊗ + R π T y 2 0 F 94% = ψ ρ ψ = T T DiCarlo, et al. Nature 467 574 (2010)
GHZ states /2 R π /2 0 R π Tomography y y 01 ψ target = 1 State ( ) 000 + 111 /2 0 R π y 2 10 /2 /2 0 R π R π y y GHZ GHZ F 88% = ρ = DiCarlo, et al. Nature 467 574 (2010)
Properties of GHZ-like states ψ GHZ = 1 ( ) 000 + 111 All Z i Z j correlations are +1 2 All Z i Z j correlations are still ψ QEC = α 000 + β 111 +1, independent of and β α Flipped State Z 1 Z 2 Z 2 Z 3 qubit α 000 + β 111 None +1 +1 α 100 + β 011 Q 1 -1 +1 α 010 + β 101 Q 2 -1 -1 α 001 + β 110 Q 3 +1 -1 Each error has a different observable!
Bit-flip error correction circuit (measurement-free implementation) “Toffoli” gate is hard diagnose & fix encode error decode 0 | junk � X or 0 1 0 1 α + β X α + β or | junk � 0 X Toffoli can be constructed with five two-qubit gates, but that’s expensive α | 000 � + β | 111 � α | 010 � + β | 101 � α | 111 � + β | 101 � = ( α | 1 ⇥ + β | 0 ⇥ ) � | 11 ⇥ GHZ state for | α | = | β | Can we do better? P. Schindler et al. Science 332 1059 (2011) Nielsen & Chuang Cambridge Univ. Press
Toffoli gate with noncomputational states | 11 � � | 02 � Two-qubit gate requires two excitations | 111 � � | 003 � Three-qubit interaction: third excited state The essence! | 111 i ! | 102 i � | 003 i This interaction is small, so use intermediary � � 1 | | 1 ⊗ ⊗ | 012 � � 0 � 1 0 | 1 0 | 002 � 2 | | 021 � | 003 � | 100 � ⊗ | 1 � � 1 1 1 | | 030 � | 011 � | 020 � � 2 | 201 � 0 1 | | 120 � | 300 � | 210 � (m Φ 0 ) (m Φ 0 ) Adiabatic interaction: | 102 i � | 003 i Sudden transfer: | 011 i ! | 002 i Three-qubit phase here! | 111 i ! | 102 i
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 | input ⇥ � ˆ O � | output ⇥ R π / 2 R − π / 2 x x 1 0.9 O | ψ in ⇥ | 2 0.8 0.7 0.6 | � ψ out | ˆ 0.5 0.4 0.3 0.2 0.1 0 000 1 010 2 100 3 110 4 001 8 7 5 011 F = 86% 6 Classical output state 6 101 5 4 7 3 111 2 8 1
Quantum process tomography of CCPhase Want to know the action on superpositions: | input ⇥ � ˆ O � | output ⇥ (but now with 64 basis states) 4 N X χ m,n A m ρ in A † ρ out = P ( ρ in ) = Invert to find χ n m,n =1 Theory Experiment 0.6 0.3 0.0 F = 77% 4032 Pauli correlation measurements (90 minutes)
Bit-flip error correction with fast Toffoli Prepare state “Error” rotation Correct | +X i by some angle 0 | junk � 0 1 0 1 α + β Y α + β | junk � 0 Decode Measure single- Encode in error qubit state three-qubit syndromes fidelity to state | +X i Ideally, there should be no dependence of fidelity on the error rotation angle
Correction fidelity vs. error rotation Encode, single known error, decode, fix, and measure resulting state fidelity 1.0 1.0 1.0 1.0 Protected qubit state fidelity Protected qubit state fidelity Protected qubit state fidelity Protected qubit state fidelity 0.8 0.8 0.8 0.8 No correction Y Error on Q2 uncorrected 0.6 0.6 0.6 0.6 Error on Q2 Y Error on Q2 Error on Q1 Z Error on Q1 Error on Q3 Z Error on Q3 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 -2 -1 0 1 2 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 Error angle [Pi] Error angle [Pi] Error angle [Pi] Error angle [Pi] Error rotation (pi)
Error syndromes T | junk � Is the algorithm really doing what we think? P 0 1 α + β Look at two-qubit density matrices of after a full flip | junk � B | junk � Bottom flip - | 01 � No error - | 00 � 1 1 1 � 0.5 0.5 0 � 0 0 − 0.5 − 0.5 -1 � Top flip - | 10 � Protected flip - | 11 � − 1 − 1 1 1 2 2 4 1 4 1 3 3 3 3 2 2 0.5 0.5 4 4 1 1 0 0 − 0.5 − 0.5 − 1 − 1 1 1 2 2 4 4 3 3 3 3 2 2 4 4 1 1 It is also clear here why you need at least three qubits!
Simultaneous phase -flip errors More realistic error model: Flip happens with probability p =sin 2 ( /2) θ Correction only works for single errors . Probability of two or three errors: 3 p 2 – 2 p 3 Not corrected: Depends only quadratically on error probability! Corrected : +Z +X +Y -Z 1.0 0.0 Phase flip probability
Conclusions • Demonstrated 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 77% quantum process fidelity Preprint available at arXiv:1109.4948 (accepted to Nature )
Outlook • Concatenating bit and phase flip codes gives full QEC • But requires nine qubits • A logical qubit per cavity with intra-cavity coupling? • Planar qubits are not coherent enough • But we’ve made huge progress on that front with a parallel experiment (Paik, et. al. arXiv:1105.4652, in the press at PRL ) • Three-dimensional architecture yields ~ 40 times longer qubit lifetimes • Need to re-integrate control knobs (e.g. FBLs) and scale up b a 250 m 50 mm c g / 2
Questions? Preprint: Reed, et al. arXiv:1109.4948 (accepted to Nature )
CCNot gate pulse sequence a b π -CCPhase π / 2 π / 2 π More than two times faster than equivalent two-qubit gate sequence
Recommend
More recommend
