Adiabatic Quantum Computation

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

• E. Farhi et al., Science 292, 472 (2001)
• 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.

Energy Spectrum
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Adiabatic Quantum Computation (AQC)

• E. Farhi et al., Science 292, 472 (2001)
• 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.

Energy Spectrum
Effective two-state system
Gap gmin
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Adiabatic Theorem
Landau-Zener transition probability
Error
E
gmin
Success
s
To have small error probability tf gtgt
1/gmin2
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System Plus Environment
gmin
Smeared out anticrossing
Environments energy levels
Gap is not well-defined
Adiabatic theorem does not apply!
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Density Matrix Approach
Hamiltonian
System Environment Interaction
Liouville Equation
System environment density matrix
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Markovian Approximation
Dynamical Equation
Non-adiabatic transitions
Thermal transitions
For slow evolutions and small T, we can
truncate the density matrix
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Multi-Qubit System
System (Ising) Hamiltonian
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Multi-Qubit System
Spectral density
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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
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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
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Matrix Elements
Relaxation rate
Peak at the anticrossing
Matrix elements are peaked at the anticrossing
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Effective Two-State Model
Hamiltonian
Matrix element peaks
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Incoherent Tunneling Regime
gmin
Energy level Broadening W
If W gt gmin, transition will be via incoherent
tunneling process
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Non-Markovian Environment
M.H.S. Amin and D.V. Averin, arXiv0712.0845
Assuming Gaussian low frequency noise and small
gmin
Directional Tunneling Rate
Width
Shift
Theory agrees very well with experiment See R.
Harris et al., arXiv0712.0838
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Calculating the Time Scale
M.H.S. Amin and D.V. Averin, arXiv0708.0384
Probability of success
Characteristic time scale
For a non-Markovian environment
Linear interpolation (global adiabatic
evolution)
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Computation Time
M.H.S. Amin and D.V. Averin, arXiv0708.0384
Open system
Broadening (low frequency noise) does not
affect the computation time
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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
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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
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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)