Lecture 2 Version: 14/08/29. Frontiers of Condensed Matter San Sebastian, Aug , Dr. Leo DiCarlo dicarlolab.tudelft.

Transcription

1 Introduction to quantum computing (with superconducting circuits) Lecture 2 Version: 14/89 Frontiers of Condensed Matter San Sebastian, Aug. 28-3, 214 Dr. Leo DiCarlo l.dicarlo@tudelft.nl dicarlolab.tudelft.nl

2 Summar of Lecture #1 θ Quantum states ϕ Quantum gates X H Rn ˆ ( θ ) QFT U f Quantum msm t ˆM

3 Anatom of a simple quantum algorithm Qubit register Ancilla qubits initialize create maximal superposition encode function in a unitar clever process will involve entanglement and disentanglement between qubits M measure Maintain quantum coherence (DiVincenzo criteria) 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 result in a computational-basis state at end! *use quantum interference: the magic of properl designed algorithm

4 Outline of Lecture #2 Quantum computing hardware based on superconducting circuits One- & two- qubit gates Quantum measurement Simple quantum algorithms (games) achieving speedup.

5 Several architectures have ran quantum algorithms Nuclear Spins (NMR) Photons NV centers Trapped Ions 212 Superconducting circuits 23 29

6 The appeal of integrated electrical circuits Modular architecture: few building blocks Parallel fabrication Intel

7 LC circuit as a quantum harmonic oscillator φ L ( aa ˆ ˆ ) +Q -Q Hˆ = ω 1 r + 2 ˆ φ Qˆ ˆ φ Qˆ ˆ = + ; ˆ = φ Q φ Q a i a i C ZPF ZPF ZPF ZPF E ω r φ annihilation and creation operators aa ˆ ˆ =, 1 7 φ = 2 ω LQ ; = 2 ω C ZPF r ZPF r trapped photons! M. Devoret, Les Houches Session LXIII (1995)

8 Wh microwave frequencies + crogenic temp? CAN PLACE CIRCUIT IN ITS QUANTUM GROUND STATE E 5 GHz ω r ω r 1 kt B φ Pn r/ kt B ( = ) = 1 e ω 2 GHz ~ 1 K

9 Wh superconducting? φ E ω r 1 φ important: as little dissipation as possible dissipation broadens energ levels, enhancing energ relaxation

10 Wh Josephson? generator = µwave LASER E ω r φ IN LINEAR CIRCUITS, ALL TRANSITIONS BETWEEN QUANTUM LEVELS ARE DEGENERATE! Cannot confine sstem to a two-level subspace = LEAKAGE

11 Wh Josephson? NEED NONLINEARITY TO FULLY REVEAL QM E Position coordinate Emission spectrum frequenc ω 34 ω 23 ω 12 ω 1

12 Josephson tunnel junction A NON-LINEAR INDUCTOR WITH NO DISSIPATION Ι ~1nm S I S superconductorinsulatorsuperconductor tunnel junction φ t ( ') = V t dt ' φ U = E J cos 2π φ 2E J I φ = sin 2 2πLJ φ φ φ π φ = h 2e bare Josephson potential

13 Josephson junctions in real life: imperfectl beautiful Al/AlOx/Al credit L. Frunzio and D. I. Schuster credit I. Siddiqi and F.Pierre E J ~ 5 K L J ~ 15 ph 1nm E J ~.5 K L J ~ 15 nh

14 Superconducting circuits: artificial atoms Josephson junctions C E Transmon qubit: Koch et al., PRA (28), Schreier et al., PRB (28) φ

15 Flux control of qubit frequencies I 1 Φ 1 Fast & Local LJ ( Φ) C I 2 Φ 2 ω 1 1 π ~4 8 L ( Φ) C ( ω ω ) / 2π 3 MHz 1 12 J GHz

16 Cavit QED with wires Circuit QED Wallraff et al., Nature (24) Blais et al., Phs. Rev. A (24) Josephson-junction qubits out in Transmission-line resonator mediates interaction between qubits protects qubits from continuum allows qubit readout Expts: Sillanpää et al., Nature (27) Majer et al., Nature (27) (Phase qubits / NIST) (Transmon qubits / Yale)

17 First quantum processors 29 model DiCarlo et al., Nature (29) 21 model 1 mm DiCarlo et al., Nature (21) Reed et al., Nature (212)

18 212 model J. P. Groen et al., Phs. Rev. Lett. (213) 213 model O.P. Saira et al., Phs. Rev. Lett. (214)

19 Latest Delft processor (214 Model)

20 Similar developments at Chalmers, UCSB, ETH 214 Model: 2D+ connectivit flux controls feedline Control and readout b frequenc multiplexing CPW cross-overs Air bridges

21 3D circuit QED A Schoelkopf group breakthrough: H. Paik et al., PRL (211) 2 nm 25 µm

22 A quantum computing roadmap Review: Devoret & Schoelkopf, Science (213)

23 Single-qubit control and measurement 1 X = 1 = X H = = H Y i = i = Y Z 1 = 1 = Z Rotations R ( ) n θ ˆ ˆ Rn ˆ ( θ) = cos( θ / 2) I isin( θ / 2) n σ σ = XYZ ˆ, ˆ, ˆ { }

24 One-qubit gates: X and Y rotations Preparation 1-qubit rotations Measurement f L z x cavit I cos(2 π ft) L V R Flux bias on right transmon (a.u.)

25 One-qubit gates: X and Y rotations f R Preparation 1-qubit rotations Measurement z x cavit I cos(2 π ft) R V R Flux bias on right transmon (a.u.)

26 One-qubit gates: X and Y rotations f R Preparation 1-qubit rotations Measurement z x cavit Q sin(2 π ft) R V R Flux bias on right transmon (a.u.) see Fidelit > 99% J. Chow et al., PRL (29)

27 Individual qubit readouts NbTiN Coupling resonator - QUANTUM BUS Readout Q1 Readout Q2 Qubit 1 Qubit 2 Feedline 2 mm Readout 1 Readout

28 Characterizing individual readout fidelit NbTiN Readout 1 Readout Q Qubit 1 Feedline Qubit 2 2 mm averaged transients single-shot histograms 1i i

29 Characterizing individual readout fidelit NbTiN Readout 1 Readout Q Qubit 1 Feedline Qubit 2 2 mm averaged transients single-shot histograms 1i i

30 Characterizing individual readout fidelit NbTiN Readout 1 Readout Q Qubit 1 Feedline Qubit 2 2 mm error budget i 96% +1 Fidelit = 84% 1i 88% -1

31 A quantum computing roadmap C-NOT/C-Phase, Deutsch-Jozsa, Grover s, Measurement-free error correction (repetition code) Review: Devoret & Schoelkopf, Science (213)

32 A universal set of gates 1 X = 1 Y = X i = i = Y Z 1 = 1 = Z H = = H Rotations R ( ) n θ ˆ ˆ Rn ˆ ( θ) = cos( θ / 2) I isin( θ / 2) n σ σ = XYZ ˆ, ˆ, ˆ { } U 1 1 = 1 1 = U 1 1 = 1 1 Conditional phase gate Controlled-NOT

33 Spectroscop of two qubits + cavit V R right qubit Dispersive qubit-qubit swap interaction left qubit cavit Cavit-qubit interaction Vacuum Rabi splitting V R Flux bias on right transmon (a.u.) Background: Majer et al., Nature (27) Wallraff et al., Nature (24)

34 Resonant qubit-bus interaction right qubit left qubit cavit Cavit-qubit resonant interaction Vacuum Rabi splitting 2g 2g V R Flux bias on right transmon (a.u.) Background: Majer et al., Nature (27) Wallraff et al., Nature (24)

35 Dispersive qubit-qubit interactions right qubit 2 g g / ( ) Dispersive qubit-qubit swap interaction natural speed slow-down factor ~1/1 2 2 g / left qubit cavit V R Flux bias on right transmon (a.u.) Background: Majer et al., Nature (27) Wallraff et al., Nature (24)

36 Two-qubit gate: turn on interactions V R Conditional phase gate Use control lines to push qubits near a resonance cavit V R Flux bias on right transmon (a.u.)

37 Two-excitation manifold of sstem Transmon qubits have multiple levels 11 2 Two-excitation manifold Avoided crossing (16 MHz) 11 2 Flux bias on right transmon (a.u.) Strauch et al. PRL (23): proposed using interactions with higher levels for computation in phase qubits

38 Adiabatic conditional-phase gate 2 11 f + f 1 1 t f ϕa = 2 π δ fa() t dt t 1 2-excitation manifold 1-excitation manifold 1 ζ iϕ 11 e t f ϕ11 = ϕ1 + ϕ1 2 π ζ() t dt 1 1 e iϕ 1 iϕ 1 e 1 1 t Flux bias on right transmon (a.u.)

39 Implementing C-Phase with 1 fanc pulse Uˆ e i e iϕ1 e iϕ1 ϕ Adjust timing of flux pulse so that onl quantum amplitude of 11 acquires a minus sign: Uˆ π

40 Gates at the raw speed of circuit QED Q B g 1 e D2 B D2 B swap in in 1 ns e 1 f D21B gd22b D2 B D2 B c-phase in in <2 ns e Saira et al., PRL 112, 752 (214) Proposed b: G. Haack et al., PRB (21)

41 Gates at the raw speed of circuit QED Q B g 1 e D2 B D2 B swap in in 1 ns e 1 f D21B gd22b D2 B D2 B e c-phase in in <2 ns 1 1 Saira et al., PRL 112, 752 (214) Proposed b: G. Haack et al., PRB (21)

42 Gates at the raw speed of circuit QED Q B g 1 e D2 B D2 B 1 swap in in 1 ns e 1 f D21B gd22b D2 B D2 B c-phase in in <2 ns e Saira et al., PRL 112, 752 (214) Proposed b: G. Haack et al., PRB (21)

43 Generating and detecting 2-qubit entanglement Tomograph with joint readout: Filipp et al., PRL (29) Z Z ±1 ±1 ρ = σ j, k { I, X, Y, Z} 4 k σ j σ k σ j Pauli set

44 Bell inequalities z z θ x x Clauser, Horne, Shimon & Holt (1969) LHV bound: CHSH 2 CHH S = X X + X Z Z X + Z Z CHH S = X X X Z + Z X + Z Z no readout correction 1.8 ±.1 UCSB group has closed detection loophole (w/ 2.7): Ansmann et al., Nature (29)

45 Bell inequalities z z θ x x Clauser, Horne, Shimon & Holt (1969) CHH S = X X + X Z Z X + Z Z CHH S = X X X Z + Z X + Z Z with readout correction 2.57 ±.1 LHV bound: CHSH 2 not a foolproof test of hidden variables (localit & detection loopholes) With joint readout: Chow et al., PRA (21)

46 Deutsch s problem: is our coin fair? f f f f unbalanced balanced Classical Problem: You are handed a black box with one of the functions programmed in, but ou re not told which one. Determine if the function is balanced or unbalanced. f i

47 Deutsch s quantum algorithm Execute this sequence calling the quantum black box once. H x U f x H Ẑ m 1 H f( x) m = +1 m = 1 function is unbalanced function is balanced Also implemented in NMR: Chuang et al., Nature (1998) Ion traps: Guide et al., Nature (23) NV centers: Van der Sar et al., Nature (212)

48 Encoding of the functions f f f f U f1 U f2 U f3 U f x X

49 Quantum speedup in Deutsch-Jozsa algorithm Single-shot success 86% 86% 85% speedup no speedup 85% Quantum speedup This work: J. Cramer Master s thesis, TU Delft (212) First demonstration of quantum speedup in sc. circuits: Yamamoto et al., PRB (21)

50 Grover s search algorithm Consider the n-to-1 bit function: = f( x) = 1 for for x x = x x Problem: find x Case n=2: Classicall, takes on average 2.25 uses of the black box to succeed Quantum mechanicall, 1 use of the quantum black box gives right answer!

51 Grover s search algorithm Execute this sequence, which calls the quantum black box once. H H x U f x H H π H H Ẑ Ẑ m 1 m 1 H f( x) Grover s analsis inversion about mean Answer: ( m, m ) 1 ( + 1, + 1) x = ( + 1, 1) x = 1 = ( 1, + 1) x = 1 ( 1, 1) x = 11

52 Grover s search algorithm H H x U f x H H π H H Ẑ Ẑ m 1 m 1 H f( x) Quantum phase kick-back

53 Grover s search algorithm x H H H H π H H Ẑ Ẑ m 1 m ( 1 ) ( + 1 ) 1

54 Grover as a quantum game between our twins Qoogle π 1 Grover, PRL (1997)

55 Quantum searching algorithm step-b-step ψ ideal = Begin in ground state: b c 1 d π f Qoogle e g DiCarlo et al., Nature (29)

56 Quantum searching algorithm step-b-step 1 ψ ( ) ideal = Create a maximal superposition: look everwhere at once! b c 1 d π f Qoogle e g

57 Quantum searching algorithm step-b-step 1 ψ ( ) ideal = Appl Qoogle to mark the solution b c 1 d π f Qoogle e g

58 Quantum searching algorithm step-b-step ψ = 1 ( ) ideal Some more 1-qubit rotations Now we arrive in one of the four Bell states b c 1 d π f Qoogle e g

59 Quantum searching algorithm step-b-step 1 ψ ( ) ideal = Another (but known) 2-qubit operation now undoes the entanglement and makes an interference pattern that holds the answer! b c 1 d π f Qoogle e g

60 Quantum searching algorithm step-b-step ψ ideal = 1 Final 1-qubit rotations reveal the answer: Focus quantum amplitude on the answer: 1! Correct answer would found >8% of the time! b c 1 d π f Qoogle e g

61 Quantum speedup in 2-qubit Grover algorithm oracle ij π m 1 m 2

62 Quantum speedup in 2-qubit Grover algorithm oracle ij π m 1 m % 84% = P success, perfect readout 84% 87% Essentiall the same algorithmic fidelit as we did in 29: DiCarlo et al. Nature (29) (Yale)

63 Quantum speedup in 2-qubit Grover algorithm oracle 1 1 ij 11 π 87% m 1 m 2 Quantum speedup 84% = P success, perfect readout 84% 87% This work: J. Cramer Master s thesis, TU Delft (212) First demonstration of quantum speedup in sc. circuits: Dewes et al., PRL & PRB (212) (52-67% success) Single-shot success 72% 74% 71% speedup no speedup 71%

64 Grover search beond 2 qubits Grover s iteration 1 H H H x x U f f( x) H H π H H Ẑ Ẑ Grover s analsis inversion about mean m 1 m Case n>2: b computer demo.

65 Demo on computer

66 Performance of Grover s quantum algorithm O ( N) O 1 ( 2 n ) O( 2 n ) Grover s algorithm does not scale polnomiall with the number of bits n, and hence it is not efficient! The best classical approach requires, on average, N=2 n-1 calls of the classical black box. The quadratic speedup offered b Grover s is still useful.

67 Summar of Lecture #2 Superconducting quantum processors based on circuit quantum electrodnamics: Nonlinear LC oscillators as qubits; Interconnected, readout and protected using resonators. Universal gate set based on single-qubit rotations and C-Phase gates. Can controllabl create and undo entanglement. Simple quantum games (Deutsch-Josza, Grover s) with quantum speedup achieved. The power of a quantum algorithm lies in using quantum superposition to create all possible inputs at once, and then evaluating a function for all inputs in one call! The challenge in designing a quantum algorithm lies in finding an analsis step that uses quantum interference to focus quantum probabilit amplitudes toward the solution of the problem. Some computational problems can be solved more efficientl using a quantum computer. Specific example: Classical & Quantum seearch scaling N vs N

68 Tomorrow: basic quantum error correction (th and expt)