Adiabatic Quantum Computation with Noisy Qubits M.H.S. Amin D-Wave Systems Inc., Vancouver, Canada

1. Adiabatic Quantum Computation with Noisy Qubits M.H.S. Amin D-Wave Systems Inc., Vancouver, Canada

2. Collaborators: Experiment: Andrew Berkley (D-Wave) Paul Bunyk (D-Wave) Sergei Govorkov(D-Wave) Siyuan Han (Kansas) Richard Harris (D-Wave) Mark Johnson (D-Wave) Jan Johansson (D-Wave) Eric Ladizinsky (D-Wave) Sergey Uchaikin(D-Wave) Many Designers, Engineers, Technicians, etc.(D-Wave) Fabrication team(JPL) Theory: Dmitri Averin (Stony Brook) Peter Love (D-Wave, Haverford) Colin Truncik (D-Wave) Andy Wan (D-Wave) Shannon Wang (D-Wave)

3. Quantum requirements for gate model quantum computation: • Coherence • Superposition • Entanglement

4. For Adiabatic quantum computation: • Phase coherence isnot required • Ground state superposition isrequired • Ground state entanglement isrequired Superposition and entanglement can be protected by the Hamiltonian

5. Measurement basis Superposition states Energy eigenstates: Energy eigenvalues: Initializing in state “0”: Time evolution: Single Qubit Example Hamiltonian:

6. Single Qubit Example Hamiltonian: Probability of finding the qubit in state “0”: Coherent Oscillations

7. Open system density matrix: weak coupling limit Dephasing (T2) process Relaxation (T1) process Equilibrium (Boltzmann) Distribution Density Matrix Closed system density matrix: (in the energy basis)

8. g = broadening Energy gap = D Coherent Tunneling Probability of finding the qubit in state “0”: Decoherence rate Coherent oscillations well-defined gap

9. Incoherent Tunneling Probability of finding the qubit in state “0”: Decoherence rate g = broadening Energy gap = D No well-defined gap

10. Density matrix in computation basis (“0” , “1”): Signature of coherent mixture r is diagonal only if Density Matrix Density matrix in energy basis: Superposition (coherent mixture) can persist in equilibrium

11. Two-Qubit Example Hamiltonian: Ferromagnetic coupling Entangled states Lowest two energy eigenstates: Energy eigenvalues:

12. Concurrence(entanglement measure): W.K. Wootters, PRL 80, 2245 (1998) C(r) = 0, (i.e., unentangled) only if Two-Qubit Entanglement Equilibrium density matrix (J >> T , D): Entanglement can persist in equilibrium

13. Summary: • Classical limit is large T(compared to energy spacings) and not long t(compared to decoherence time) • Without a Hamiltonian, the system will be classical after the decoherence time • With a well-defined Hamiltonian (stronger than noise) system may stay quantum mechanical at small T

14. Adiabatic Quantum Computation (AQC) Energy Spectrum E. Farhi et al., Science 292, 472 (2001) System Hamiltonian: H = (1- s) Hi + s Hf Linear interpolation: s = t/tf • Ground state ofHi is easily accessible. • Ground state ofHf encodes the solution to a hard computational problem.

15. Adiabatic Quantum Computation (AQC) Energy Spectrum Effective two-state system E. Farhi et al., Science 292, 472 (2001) System Hamiltonian: H = (1- s) Hi + s Hf Linear interpolation: s = t/tf • Ground state ofHi is easily accessible. • Ground state ofHf encodes the solution to a hard computational problem. Gap = D

16. Adiabatic Theorem Error E D Success s To have small error probability: tf >> 1/D2

17. System Plus Environment D Smeared out anticrossing Environment’s energy levels e.g., Harmonic oscillator Adiabatic theorem does not apply!

18. Environment at Zero Temperature D At T=0the excitation (Landau-Zener) probability is exactly the same as that for a closed system For spin environment: A.T.S. Wan, M.H.S. Amin, S.X. Wang, cond-mat/0703085 For harmonic oscillator model: M. Wubs et al., PRL 97, 200404 (2006)

19. Incoherent tunneling rate Environment at Finite Temperature D Energy level Broadening = W If W > D, transition will be via incoherent tunneling process

20. G01 and G10 can be extracted from the initial slopes: Directional Tunneling Rates “0” “0” “1” “1” G01 G10

21. Macroscopic Resonant Tunneling (MRT) “0” “0” “1” “1” G01 G10 1st resonant peak 2nd resonant peak G10 G01 wp

22. Calculating Incoherent Tunneling Rate Two-State Model: E D e 0 System Hamiltonian: Interaction Hamiltonian: Heat bath operator

23. 1. For a Gaussian environment up to second order in D: Noise spectral density: Non-Markovian Environment M.H.S. Amin and D.V. Averin, preprint 2. If S(w) is peaked at low frequencies, one can expand e-iwt

24. Decoherence Non-Markovian Environment M.H.S. Amin and D.V. Averin, preprint Gaussian line-shape Width Shift

25. For aclassical noise, S(w) is symmetric, hence Symmetric G01 Non-Markovian Environment M.H.S. Amin and D.V. Averin, preprint Gaussian line-shape Width depends on symmetric part of S(w) depends on anti-symmetric part of S(w) Shift

26. The peak is shifted toward positive e G01(-e) < G01(+e) Quantum Noise e > 0 Let G01(-e) G01(+e) Tunneling Absorption Tunneling Emission Bose-Einstein distribution:

27. Fluctuation-Dissipation Theorem: Effective Temperature Equilibrium Environment S(w) = Ss(w) + Sa(w) depends on symmetric part Ss(w) depends on anti-symmetric part Sa(w) Can be tested experimentally

28. Magnetic flux Tunable rf-SQUID qubit: Josephson junctions: F1 F2 Double-well potential: F2 = F0/2 F2 = 0 High barrier: Low barrier: wp D ~ wp D ~ 0 wp = plasma frequency Experiment

29. Pulse Sequence High barrier Initializing Tunneling Low barrier Measurement

30. Transition Rate Measurements R. Harris et al., preprint available Tunneling to the 2nd level Tunneling to the 1st level W G10 G01 (mF0) Asymmetric tunneling Quantum Noise

31. Fit to Theory R. Harris et al., preprint available Tunneling to the 2nd level Tunneling to the 1st level W (mF0) Good agreement with theory

32. Width and Shift Measurements weakly depends on T decreases with T as ~ 1/T

33. Effective Temperature Equilibrium distribution Saturation Temperature Mixing chamber thermometer

34. Tunneling Amplitude One can extract D from data using Why T-dependent? Why increase with T? D is renormalized by high frequency environmental modes

35. At D0= 0, the two states are degenerate Environment states Degeneracy is lifted by H, with a splitting: Adiabatic Renormalization of D Consider the Hamiltonian Renormalization can only decrease D

36. Adiabatic Renormalization of D In terms of spectral density Ohmic environment: Dincreases with T ! The larger the T, the less the environment cares about the system

37. 1/f In agreement with: Astafiev et al. PRL 93, 267007 (2004) ohmic Sub-Conclusions: • The noise is coming from a quantum source, • e.g. two-state fluctuators; J. Martinis et al. PRL. 95, 210503 (2005) • Low frequency part of the noise spectrum is • peaked at zero frequency (e.g. 1/f noise). • High frequency noise is likely to be ohmic.

38. Back to AQC M.H.S. Amin and D.V. Averin, arXiv:0708.0384 Probability of success: Characteristic time scale: For a non-Markovian environment: Linear interpolation (global adiabatic evolution):

39. Normalized Closed system: (Landau-Zener probability) Not normalized Incoherent tunneling rate Width of transition region Cancel each other Computation Time Scale M.H.S. Amin and D.V. Averin, arXiv:0708.0384 Open system: Broadening (low frequency noise) does not affect the computation time

40. If ,the two regions overlap Gap Renormalization Effect High frequency modes: Low frequency modes: Gap renormalization may not happen for large n, or may happen differently from the simple one-qubit example

41. Dj = 0  Boltzmann factor: Effective temperatureTeff = T/schanges fromtoT AQC vs. Classical Annealing H = (1- s) Hi+ s Hf Optimization problem: Diagonal Non-diagonal Classical annealing

42. AQC vs. Classical Annealing H = (1- s) Hi+ s Hf Optimization problem: Diagonal Non-diagonal Dj >> T Quantum regime Dj << T Thermal regime Dj ~ T Mixed regime!

43. Single qubit noise 3. Well defined tunnel splitting: Dj > Wj 4. Accurate final Hamiltonian: hj , Jjk > Wj Noise Requirements for AQC For Gaussian distribution of the energy levels the number of the levels near the ground state will be polynomial (in n) 1. Away from classical limit: Dj > T 2. Stable final ground state: hj , Jjk > T 5. Polynomial number of energy levels within energy T from the ground state during the evolution

44. Our Multi-Qubit System Tunable rf-SQUID qubit + coupler: R. Harris et al., PRL 98, 177001 (2007) Qubits Tunable coupler Controls Jjk Controls Dj Controls Dj Controls hj

45. Double-well potential: F2 = F0/2 F2 = 0 Annealing wp Dj ~ wp Dj ~ 0 wp = plasma frequency Annealing Process Initial Hamiltonian Final Hamiltonian

46. Well-defined energy splittings Dj ~ wp > W Dj ~ wp >> T Quantum regime hj , Jjk > W,T Well-defined final Hamiltonian Annealing Regime wp ~ 400 mK 2nd resonant peak 1st resonant peak wp Effective Temperature: Teff ~ 20 mK W ~ 80mK Unlikely to be classical annealing

47. Conclusions 1. Our environment model correctly describes resonant tunneling in superconducting devices 2. Theoretical + experimental MRT investigations: dominant low frequency noise has quantum origin 3. Phase coherence (energy basis) is not required (global AQC not for local adiabatic evolution) 4. Ground state superposition and entanglement (computation basis) are required for AQC; protected by the Hamiltonian 5. Our multi-qubit system is in the right regime for AQC/Quantum Annealing

48. Additional Slides

49. Classical and Quantum Annealing = Annealing Disorder Order Slow transition Classical annealing: Disorder = Thermal mixing (introduced by entropy) Quantum annealing: Disorder = Superposition (introduced by a Hamiltonian)

50. Quantum phase transition Quantum Critical point AQC vs. Quantum Annealing (QA) H = (1- s) Hi+ s Hf Optimization problem: Problem Hamiltonian (Diagonal) Disordering Hamiltonian (Non-Diagonal) Small superposition Large superposition