Quantum Adiabatic Algorithms

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CBA Lunch Seminar
Quantum Adiabatic Algorithms
Andrew LandahlHP/MIT Postdoctoral Fellow
October 28, 2003
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Adiabatic Algorithm

• Prepare system in lowest-energy state
• Slowly change applied forces
• Measure final state

Final configuration solves problem!
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Quantum Mechanics
State
Dynamics
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Adiabatic Algorithm
Path of Hamiltonians
Eigenstates
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Adiabatic Algorithm

• Prepare system in
• Evolve with for time .
• Measure energy of

Final configuration solves problem!
Farhi, Gutman, Goldstone, Sipser 2000
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Adiabatic Theorem
Given
Then
I.e., adiabatic evolution asymptotically keeps a
system in its ground state
Born, Fock 1928
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Adiabatic Approximation
Let
Error
Gap
Modulation
Perturbation theory
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Searching an unordered list
Problem Find argument minimizing
Roland, Cerf 2001
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Handling relative minima
Some problems cant be sped up.
van Dam, Mosca, Vazarani, 2001
Tunneling sometimes makes it superior to
simulated annealing
Farhi, Goldstone, Gutmann, 2002
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Robustness
Robust to

• Thermal noise
• Path control error

Not robust to

• Endpoint error

Let your path wander, but arrive at your
destination.
Childs, Farhi, Preskill 2001
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Universality
Any quantum circuit can be simulated by an
adiabatic algorithm
Proof
Based on Feynmans 1982 proof that local quantum
logic can be performed by local Hamiltonians
Aharanov, van Dam, Kempe, Landau, Lloyd, Regev,
in preparation
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Implementation
May be technologically easier
Kaminsky, Lloyd, 2002
Has been simulated in the lab
Steffen, van Dam, Hogg, Breyta, Chuang, 2003
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Hard problems
May be naturally formulated to solve hard
problems
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Experimental Progress
Cavity QED
Liquid NMR
Quantum Dot
Lattice BEC
Ion Trap