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Adiabatic Quantum Computation with Noisy Qubits Mohammad Amin

Adiabatic Quantum Computation with Noisy Qubits Mohammad Amin D-Wave Systems Inc., Vancouver, Canada. Outline:. 1. Adiabatic quantum computation. 2. Density matrix approach (Markovian noise). 3. Two-state model. 4. Incoherent tunneling picture (non-Markovian).

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Adiabatic Quantum Computation with Noisy Qubits Mohammad Amin

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

  2. Outline: 1. Adiabatic quantum computation 2. Density matrix approach (Markovian noise) 3. Two-state model 4. Incoherent tunneling picture (non-Markovian)

  3. 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.

  4. 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 = gmin

  5. Adiabatic Theorem Landau-Zener transition probability: Error E gmin Success s To have small error probability: tf >> 1/gmin2

  6. System Plus Environment gmin Smeared out anticrossing Environment’s energy levels Gap is not well-defined Adiabatic theorem does not apply!

  7. Reduced density matrix: Energy basis: Instantaneous eigenstates of HS(t) Density Matrix Approach Hamiltonian: System Environment Interaction Liouville Equation: System + environment density matrix

  8. Markovian Approximation Dynamical Equation: Non-adiabatic transitions Thermal transitions For slow evolutions and small T, we can truncate the density matrix

  9. Random 16 qubit spin glass instances: • Randomly choose hi and Jij from {-1,0,1} and Di = 1 • Select small gap instances with one solution Multi-Qubit System System (Ising) Hamiltonian:

  10. Interaction Hamiltonian: Ohmic baths Multi-Qubit System Spectral density

  11. Single qubit decoherence time T2 ~ 1 ns Numerical Calculations Closed system Landau-Zener formula Probability of success T = 25 mK h = 0.5 E = 10 GHz gmin = 10 MHz Open system Evolution time Computation time can be much larger than T2

  12. Large Scale Systems Transition mainlyhappens between the first two levels and at the anticrossing A two-state model is adequate to describe such a process

  13. Matrix Elements Relaxation rate: Peak at the anticrossing Matrix elements are peaked at the anticrossing

  14. ~0 Only longitudinal coupling gives correct matrix element Effective Two-State Model Hamiltonian: Matrix element peaks:

  15. Incoherent Tunneling Regime gmin Energy level Broadening = W If W > gmin, transition will be via incoherent tunneling process

  16. Non-Markovian Environment M.H.S. Amin and D.V. Averin, arXiv:0712.0845 Assuming Gaussian low frequency noise and small gmin: Directional Tunneling Rate: Width Shift Theory agrees very well with experiment See: R. Harris et al., arXiv:0712.0838

  17. Calculating the Time Scale 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):

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

  19. Compare with Numerics Incoherent tunneling picture Probability of success T = 25 mK h = 0.5 E = 10 GHz gmin = 10 MHz Open system Evolution time Incoherent tunneling picture gives correct time scale

  20. Conclusions 1.Single qubit decoherence time does not limit computation time in AQC 2.Multi-qubit dephasing (in energy basis) does not affect performance of AQC 3. A 2-state model with longitudinal coupling to environment can describe AQC performance 4. In strong-noise/small-gap regime, AQC is equivalent to incoherent tunneling processes

  21. 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) Vicki Choi(D-Wave) Colin Truncik (D-Wave) Andy Wan (D-Wave) Shannon Wang (D-Wave)

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