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) 4. Incoherent tunneling picture (non-Markovian) 3. Two-state model

3 Adiabatic Quantum Computation (AQC) E. Farhi et al., Science 292, 472 (2001) System Hamiltonian: H = (1  s) H i + s H f Linear interpolation: s = t/t f Ground state of H i is easily accessible. Ground state of H f encodes the solution to a hard computational problem. Energy Spectrum

4 Adiabatic Quantum Computation (AQC) E. Farhi et al., Science 292, 472 (2001) System Hamiltonian: H = (1  s) H i + s H f Linear interpolation: s = t/t f Ground state of H i is easily accessible. Ground state of H f encodes the solution to a hard computational problem. Energy Spectrum Effective two-state system Gap = g min

5 Adiabatic Theorem To have small error probability: t f >>  g min  Error E s g min Landau-Zener transition probability: Success

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

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

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

9 Multi-Qubit System System (Ising) Hamiltonian: Randomly choose h i and J ij from  and  i  Select small gap instances with one solution Random 16 qubit spin glass instances:

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

11 Numerical Calculations Probability of success Evolution time Closed system Landau-Zener formula Open system T = 25 mK  = 0.5 E = 10 GHz g min = 10 MHz Single qubit decoherence time T 2 ~ 1 ns Computation time can be much larger than T 2

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

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

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

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

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

17 Calculating the Time Scale Linear interpolation (global adiabatic evolution): Probability of success: Characteristic time scale: For a non-Markovian environment: M.H.S. Amin and D.V. Averin, arXiv:

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

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

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

21 Collaborators: 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) 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)