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) 2

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

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

Adiabatic Theorem Error Landau-Zener transition probability: E gmin Success s To have small error probability: 5 tf >> 1/gmin 2

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

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

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

Multi-Qubit System (Ising) Hamiltonian: 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 9

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

Numerical Calculations Closed system Landau-Zener formula Probability of success Open system T = 25 m. K h = 0. 5 E = 10 GHz gmin = 10 MHz Evolution Single qubit decoherence time T 2 ~ 1 ns time Computation time can be much larger than T 2 11

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 12

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

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

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

Non-Markovian Environment M. H. S. Amin and D. V. Averin, ar. Xiv: 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. , ar. Xiv: 0712. 0838 16

Calculating the Time Scale M. H. S. Amin and D. V. Averin, ar. Xiv: 0708. 0384 Probability of success: Characteristic time scale: For a non-Markovian environment: Linear interpolation (global adiabatic evolution): 17

Computation Time M. H. S. Amin and D. V. Averin, ar. Xiv: 0708. 0384 Open system: Normalized Closed system: (Landau-Zener probability) Not normalized Broadening (low frequency noise) does not affect the computation time Incoherent tunneling rate Width of transition region 18 Cancel each other

Compare with Numerics Incoherent tunneling picture Probability of success Open system T = 25 m. K h = 0. 5 E = 10 GHz gmin = 10 MHz Evolution time Incoherent tunneling picture gives correct time scale 19

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 20

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) 21