PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON CIRCUITS

Transcription

1 PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON CIRCUITS

2 PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON CIRCUITS Michel Devoret, Yale University Acknowledgements to Yale quantum information team members: M. Hays C. Axline R. Heeres d G. de Lange S. Mundhada J. Blumoff Y. Liu A. Grimm L. Frunzio A. Narla K. Chou W. Pfaff A. Kou S. Girvin K. Serniak Y. Gao N. Ofek Z. Leghtas b L. Glazman V. Sivak A. Petrenko c U. Vool M. Hatridge a L. Jiang K. Sliwa c P. Rheinhold C. Wang f S. Shankar M. Mirrahimi S. Touzard A. Roy f Z. Wang R. Schoelkopf Now at: a U. Pittsburgh, b ENS Paris, c QCI, d CEA Saclay, f U. M. Amherst, f U. Aachen Sponsors: Capri Spring School 27

3 PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON CIRCUITS LECTURE I: INTRODUCTION AND OVERVIEW LECTURE II: CAT CODES LECTURE III: PARITY MONITORING AND CORRECTION

4 PROTECTING CLASSICAL INFORMATION ONE BIT OF INFORMATION ENCODED IN A SWITCH energy of switch ENERGY BARRIER PROTECTS INFORMATION error exp U kt B U k T B switch position

5 ELEMENT OF QUANTUM INFORMATION: QUBIT Two basis quantum states: Z Generic superposition: cos exp i sin 2 2 X Y Bloch sphere MUST PRESERVE ALL SUPERPOSITIONS, I.E. THE ENTIRE SPHERE

6 CARDINAL POINTS OF THE BLOCH SPHERE Z Z Z X X Y Y X Y

7 DECOHERENCE: LOSS OF INFORMATION Z Y X "T " "T f "

8 BY CAREFUL ENCODING IN A LARGE ENOUGH SPACE, QUANTUM INFORMATION CAN BE PROTECTED SIMPLE EXAMPLE: IN GENERAL: i cos e sin 2 2 X AND Y COMPONENTS IN M = MANIFOLD ARE PROTECTED FROM NOISE IN COMMON MODE MAGNETIC FIELD B X Z Y E ELEMENTARY ERRORS E ONLY MOVE OCTAHEDRON ALONG A 3+n th DIMENSION, WITHOUT CHANGING ITS VOLUME

9 3 STRATEGIES FOR PROTECTING QUANTUM INFORMATION - Encode using defects in topological quantum matter Example: Majorana fermions - Encode into a register of qubits, monitor multi-qubit parities Examples: Steane & surface codes - Encode into a quantum cavity resonator and monitor photon number parity Example: cat codes

10 L L REGISTER ERROR CORRECTION: STEANE CODE

11 STEANE CODE ERROR CORRECTION PROTOCOL i = Z Z Z X X X X ee e2 Z e3e4e5 Z Z X X X ee e2 e3e4e5 Z Z Z X X X ee e2 e3e4e 5 Z q i Z Z X X X ee e2 e3e4e 5 Z Z X X ee e2 e3e4e 5 Z Z X X ee e2 e3e4e 5 Z i = 6 Z X X ee e2 e3e4e 5 Z j = j = 5 e j

12 SUPERCONDUCTING QUBIT PARADIGM: CONSIDER A SINGLE, DRIVEN ELECTROMAGNETIC MODE... DRIVE SIGNAL in cos A t A t REFLECTED SIGNAL out cos f A t A t MICROWAVE FREQUENCIES kt B

13 ATOM vs CIRCUIT Hydrogen atom Superconducting LC oscillator L C electron condensate metallic lattice positive ions

14 ATOM vs CIRCUIT Hydrogen atom Superconducting LC oscillator L C electron condensate metallic lattice positive ions

15 ATOM vs CIRCUIT Hydrogen atom Superconducting LC oscillator L C unique electron whole superconducting condensate velocity of electron current through inductor force on electron voltage across capacitor

16 ATOM vs CIRCUIT Hydrogen atom Superconducting LC oscillator.nm mm L C unique electron whole superconducting condensate velocity of electron current through inductor force on electron voltage across capacitor

17 LC CIRCUIT AS A QUANTUM HARMONIC OSCILLATOR F L +Q -Q C E F = LI Q = CV r F F ˆ,Qˆ i

18 F WAVEFUNCTIONS OF LC CIRCUIT E I r In every energy eigenstate, (microwave photon state) current flows in opposite directions simultaneously! Y F Y Y F F

19 EFFECT OF DAMPING E F important: as little resistance as possible E n F dissipation broadens energy levels i r n 2 2 RC r

20 CAN PLACE CIRCUIT IN ITS GROUND STATE F E I r F some cold dissipation is actually needed: provides reset of circuit! r kt -5 GHz mk B

21 UNLIKE ATOMS IN VACUUM, TWO CIRCUITS EASILY COUPLE THROUGH THEIR WIRES: COUPLING CAN BE ARBITRARILY STRONG! HAMILTONIAN BY DESIGN!

22 CAVEAT: THE QUANTUM STATES OF A PURELY LINEAR CIRCUIT CANNOT BE FULLY CONTROLLED! F E NO STEERING TO AN ARBITRARY STATE IF SYSTEM PERFECTLY LINEAR r F

23 NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate

24 NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate Emission high T frequency

25 NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate absolute non-linearity is ratio of peak distance to peak width Emission spectrum frequency

26 JOSEPHSON TUNNEL JUNCTION: A NON-LINEAR INDUCTOR WITH NO DISSIPATION nm S I S superconductorinsulatorsuperconductor tunnel junction L J C J - nm MIGHT BE SUPERSEDED SOME DAY BY ANOTHER DEVICE (BASED ON ATOMIC POINT CONTACTS, NANOWIRES,???)

27 JOSEPHSON TUNNEL JUNCTION: A NON-LINEAR INDUCTOR WITH NO DISSIPATION nm S I S superconductorinsulatorsuperconductor tunnel junction L J C J I f t ' V t dt ' I I I f / L J L J f I f I I sin f/ f f 2e

28 PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON CIRCUITS LECTURE I: INTRODUCTION AND OVERVIEW LECTURE II: CAT CODES LECTURE III: PARITY MONITORING AND CORRECTION

29 JOSEPHSON TUNNEL JUNCTION: A NON-LINEAR INDUCTOR WITH NO DISSIPATION nm S I S superconductorinsulatorsuperconductor tunnel junction L J C J I U E cos f/ f J L J f E 2 J f t ' V t dt ' 2E J f bare Josephson potential f 2e

30 ELECTROMAGNETIC OSCILLATOR ENDOWED WITH JOSEPHSON NON-LINEARITY Potential energy Position coordinate Emission high T Kerr frequency frequency

31 QUANTUM STATES OF A HARMONIC OSCILLATOR REPRESENTED BY THEIR QUASI-PROBABILITY WIGNER FUNCTION IN PHASE SPACE vacuum: n=> I iq W D PD ; P aa photon: n=> Q Q I coherent state squeezed vacuum I cat state Q Q Q I 2 I I 2

32 QUANTUM STATES OF A HARMONIC OSCILLATOR REPRESENTED BY THEIR QUASI-PROBABILITY WIGNER FUNCTION IN PHASE SPACE vacuum: n=> I iq W D PD ; P aa photon: n=> Q Q I coherent state squeezed vacuum I cat state Q Q Q I 2 I I 2

33 CAT ENCODING, AND INFORMATION DECAY 3 even odd L ' ' even â typical time evolution t odd L i i i i ' i '

34 BLOCH SPHERE OF 4-LEG CAT CODE +Z +Y +X Leghtas et al. PRL (23) Vlastakis et al. Science (23) Mirrahimi et al. NJP (24) Sun et al. Nature (24)

35 TWO TYPES OF CAT ERROR MUST BE ADDRESSED: - COHERENT STATE SHRINKING, DEPHASING & DEFORMING - PHOTON NUMBER PARITY JUMPING HOW DO WE FULLY CONTROL THE RESONATOR?

36 .2 cm DEVICES 3.4 cm 9 μm Josephson Ring Modulator Al Si 2.5 mm N. Bergeal et al. Nature (28) H. Paik et. al. PRL (2) G. Kirchmair et. al. Nature (23)

37 EXPERIMENTAL SETUP.5 m 77 K 4 K K mk 2 mk quantum circuits microwave generators arbitrary waveform generator atomic clock amplifier electronics acquisition card dilution refrigerator instrument rack

38 junction 2 OSCILLATORS COUPLED THROUGH A TUNNEL JOSEPHSON JUNCTION storage dump & readout H SIMPLIFIED MODEL TE modes Qs Fs Qr Fr EJ cos Fr Fs / f 2C 2L 2C 2L s s r s E L C rs, r, s r, s L s Cs C r Lr

39 A POWERFUL DETERMINISTIC GATE COMBINING MICROWAVE LIGHT AND A QUBIT cavity oscillator superconducting qubit H a a z c q T 2 strong dispersive regime: Stark shift per photon > linewidths Can combine two cavities, a qubit and a quantum limited Josephson amplifier: storage cavity superconducting qubit S S readout cavity Leghtas et al., Phys. Rev. A87, 4235 (23) R amplification chain

40 junction 2 OSCILLATORS COUPLED THROUGH A TUNNEL JOSEPHSON JUNCTION storage dump & readout E TE modes s q r

41 junction 2 OSCILLATORS COUPLED THROUGH A TUNNEL JOSEPHSON JUNCTION storage dump & readout E TE modes H / sas as rar ar qaq a - c qq q 2 a q ( ) 2 a q ( ) 2 - c qs a q a q a s a s - c qr a q a q a r a r c rs ( 4 a ) 2 ( s a ) 2 r + h.c.+...

42 NON-DEMOLITION MEASUREMENT OF SUPERCONDUCTING ATOM Cavity QED: Pioneering expts in Haroche's and Kimble's groups HEMT-amp Q I ref JPC Circuit QED: Wallraff et al. (24), Houck et al., (27), Gambetta et al. (28), Vijay et al., (22). qubit + resonator Josephson pre-amp Readout amplitude I Q 2 2 e f c dispersive shift c g f readout frequency 9º Readout phase Q I -9º tan / e q ~ p /2 resonator linewidth ~ c g f f c

43 5 I m /σ 7.5 σ QUANTUM JUMPS OF QUBIT STATE Time (ms) ~ 9% of fluctuations of signal at 3K are quantum noise! ~2 bits per qubit lifetime

44 TWO TYPES OF CAT ERRORS MUST BE ADDRESSED: - COHERENT STATE SHRINKING, DEPHASING & DEFORMING - PHOTON NUMBER PARITY JUMPING

45 REVIEW THE USUAL DRIVEN-DAMPED HARMONIC OSCILLATOR

46 REVIEW THE USUAL DRIVEN-DAMPED HARMONIC OSCILLATOR

47 REVIEW THE USUAL DRIVEN-DAMPED HARMONIC OSCILLATOR p x

48 REVIEW THE USUAL DRIVEN-DAMPED HARMONIC OSCILLATOR p x

49 REVIEW THE USUAL DRIVEN-DAMPED HARMONIC OSCILLATOR Wigner function W(β) Im(β) Re(β) Hamiltonian : loss operator : H = e * a + e a k a a = -2ie /k

50 AVERAGE EVOLUTION OF DENSITY MATRIX FOR AN OPEN SYSTEM d dt, c i H c c c c c c c loss operator 2

51 A SPECIAL DRIVEN-DAMPED OSCILLATOR

52 A SPECIAL DRIVEN-DAMPED OSCILLATOR Force ~ X 2

53 A SPECIAL DRIVEN-DAMPED OSCILLATOR Force ~ X 2 Damping ~ V 2

54 A SPECIAL DRIVEN-DAMPED OSCILLATOR Im(β) Re(β) Wigner function W(β)

55 A SPECIAL DRIVEN-DAMPED OSCILLATOR Im(β) Re(β) Wigner function W(β) Hamiltonian : loss operator : H = e 2 * a 2 + e(a ) 2 k 2 a 2 a = ± -2ie 2 /k 2

56 PROTECTING QUANTUM SUPERPOSITIONS IN JOSEPHSON CIRCUITS LECTURE I: INTRODUCTION AND OVERVIEW LECTURE II: CAT CODES LECTURE III: PARITY MONITORING AND CORRECTION

57 A SPECIAL DRIVEN-DAMPED OSCILLATOR Force X 2 Im(β) Damping X 2 Re(β) Wigner function W(β) Hamiltonian : loss operator : H = e 2 * a 2 + e(a ) 2 k 2 a 2 Mirrahimi et al., New J. Phys. 6, 454 (24) a = ± -2ie 2 /k 2

58 junction 2 OSCILLATORS COUPLED THROUGH A TUNNEL JOSEPHSON JUNCTION storage dump & readout E TE modes s q r

59 junction 2 OSCILLATORS COUPLED THROUGH A TUNNEL JOSEPHSON JUNCTION storage dump & readout E TE modes H / sas as rar ar qaq a - c qq q 2 a q ( ) 2 a q ( ) 2 - c qs a q a q a s a s - c qr a q a q a r a r c rs ( 4 a ) 2 ( s a ) 2 r + h.c.+...

60 TWO-PHOTON EXCHANGE BETWEEN THE 2 CAVITIES density of states dump storage p 2 s d drive pump ω ω d ω p ω s ω s ω s ω s ω p ω d effective drive Hamiltonian: e ( 2 a ) 2 s + h.c. effective loss operator: k 2 a s 2

61 CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) = Q Re(β) = I

62 CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β)

63 CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β)

64 CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β)

65 CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β)

66 CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β)

67 Im(β) CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β) Re(β)

68 Im(β) CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β) Re(β)

69 Im(β) CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) Im(β) π/2 W(β) Re(β) Re(β)

70 Im(β) CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) π/2 W(β) Re(β)

71 Im(β) CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL) π/2 W(β) Re(β) see also Puri and Blais arxiv:65.948

72 Im(β) CAT SQUEEZES OUT OF VACUUM! pumping time raw data μs 2 μs 7 μs 9 μs quantum interference! W(β) reconstruction numerical simulation Re(β)

73 Im(β) CAT SQUEEZES OUT OF VACUUM! Data of two years ago pumping time raw data μs 2 μs 7 μs 9 μs quantum interference! W(β) reconstruction numerical simulation Re(β) Leghtas et. al. Science (25)

74 Im(β) PRESENT CAT-PUMPING COHERENCE NOW: κ 2 κ = with /κ = 92 μs M. Reagor et al., APL (23), C. Axline et al. APL (26) Wigner qubit separate from storage ancilla W Re(β) Touzard, Grimm, et al. arxiv:75.24

75 Im(β) PRESENT CAT-PUMPING COHERENCE NOW: κ 2 κ = with /κ = 92 μs M. Reagor et al., APL (23), C. Axline et al. APL (26) Wigner qubit separate from storage ancilla W Re(β) Touzard, Grimm, et al. arxiv:75.24

76 Im(β) RABI OSCILLATIONS OF CAT WHILE PUMPING L = α + α 2.5 Movie slowed down by -6 W.6.3 α + α L = α α Re(β) observation of quantum Zeno dynamics of protected information! Misra & Sudarshan (977); Itano et al., (99); Fisher, Gutierrez-Medina & Raizen (2); Facchi & Pascazio (22); Raimond et al., (22)

77 Im(β) MEASUREMENT OF THE 4 PAULI OPERATORS OF THE PROTECTED QUBIT MANIFOLD - W Re(β)

78 Encode +Z +X +X L log +Z L PROCESS TOMOGRAPHY OF QUANTUM ZENO RABI ROTATION / : N( α ± α ) +Y L / : N( α ± i α ) +Y log X π/2 +Z L / : ±α +X L +Y L

79 Identity +X +Z +Y

80 +X +Z DURATION OF LOGICAL GATES:.8 ms +Y X π/2 Zeno π/2 rotation around X GATE DONE WHILE CORRECTING!

81 TWO TYPES OF CAT ERRORS MUST BE ADDRESSED: - COHERENT STATE SHRINKING, DEPHASING & DEFORMING - PHOTON NUMBER PARITY JUMPING

82 SYNDROME MEASUREMENT cavity transmon transmon cavity

83 SYNDROME MEASUREMENT cavity transmon transmon cavity

84 SYNDROME MEASUREMENT cavity transmon n even n odd transmon cavity

85 SYNDROME MEASUREMENT cavity transmon n even n odd transmon cavity

86 SYNDROME MEASUREMENT cavity transmon n even n odd transmon cavity

87 SYNDROME MEASUREMENT cavity transmon n even transmon cavity

88 REAL-TIME QUANTUM CONTROLLER FOR FEEDFORWARD CORRECTION FPGA ADCs DACs Analog Inputs Analog Outputs

89 4-CAT ENCODING SLOWS DOWN QUANTUM INFORMATION DECAY slide courtesy of R. Schoelkopf and A. Petrenko Ofek & Petrenko et al., Nature (26) Bare Transmon Qubit after Fidelity =.26 4Cat Code NO QEC Qubit after Fidelity =.58 4Cat Code W/ QEC Qubit after Fidelity =.72 QEC = Quantum Error Correction from Parity Monitoring using Field Programmable Gate Array

90 MULTI-CAVITY ARCHITECTURES M. Reagor et al., APL (23), C. Axline et al. (in preparation) Wang and Gao et al., Science (26) other experiments currently underway in the lab with 5 to qubits and 5 to cavities The Y-mon!!

91 JOINT WIGNER FUNCTION OF THE TWO-MODE CAT W J (b A, b B ): a function in a 4D space States live in Hilbert space of dimension ~ 5! y - = a A a B - -a A -a B (a =.92) Quantum coherence Predicted Wigner for ideal state: Photon distribution P. Millman et al. Eur. Phys. J. D 32, 233 (25)

92 Ideal: Measured: Wang and Gao et al., Science (26)

93 SUMMARY AND PERSPECTIVES Quantum information protection can be static and/or dynamic Cat code encodes information in dynamical states: hardware efficiency Demonstration of operation while protecting Attained break-even point in quantum error correction Entanglement of two logical qubits Combine protection of cat energy loss and parity monitoring Demonstrate hierarchical layers of quantum error correction

94 MEET THE TEAM! Y. Liu E. Zalys-Geller M. Hatridge S. Mundhada K. Serniak N. Frattini A. Narla Z. Minev S. Shankar A. Roy M. Hays K. Sliwa Z. Leghtas V. Sivak S. Touzard C. Smith U. Vool A. Kou A. Grimm V. Albert R. Schoelkopf L. Frunzio M. Mirrahimi L. Glazman L. Jiang S. Girvin