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The Limits of Adiabatic Quantum Algorithms
Alper SarikayaAdvised by Prof. Dave Bacon
Computer Science & EngineeringChemistry
University of Washington
Undergraduate Research SymposiumMay 15, 2009
Quantum Computing Theory Group: http://cs.washington.edu/homes/dabacon/qw/
Motivation
• Transistors aregetting smaller
• In 1994, Paul Shorshowed that quantum algorithms have an exponential speed-up over their classical counterpartsin factoring large prime numbers
bits
noisy bits quantum bit
cm µm nm pm
Quantum Computation• Qubit versus a classical bit .. where‘s the information stored?
• Adiabatically: take an incoming vector (input data), evolve the vector with an operator (a Hamiltonian); the answer is the smallest eigenvalue
• Think linear algebra (Math 308)!
Deterministic Probabilistic1
0.40.3
0.5
10.6
0.7
0.5
“Quantum”1
-1-1
Simulating an Adiabatic Algorithm• Benefit of Quantum Algorithms:– Infinite precision analog computation can efficiently
solve NP-complete problems
• Adiabatic theorem - A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. - Max Born, Vladimir Fock (1928)
• What is the benefit of building anadiabatic quantum computer?– Let’s compare the algorithm to a classical computer
• Testing efficiency hypothesis numerically:If the relationship between the eigenvalue gap and the number of qubits is negatively proportional, then an adiabatic quantum computer only offers a polynomial speedup over a classically-based counterpart.
• To emulate a quantum computer classically, use the Markovian matrix as the operator in this study:
where n is the number of qubits, β is varied between 0 and n, and the following two Hamiltonians are defined:
Conclusions
• There is indeed an inverse exponential relationship between the number of qubits and the smallest eigenvalue gap– Adiabatic quantum computers only offer a
polynomial speedup!
• This is only a numerical simulation of the hypothesis, not a proof
Future Directions
• Move from numerical evidence in support of the hypothesis to a formal proof to conclusively uphold the efficiency concerns
• Remember D-Wave? – Currently building an adiabatic quantum computer
and gaining lots of capital from its promise – butit probably only offers a polynomial increase in efficiency!
Acknowledgments
• Advisor: Professor Dave BaconUW Computer Science & Engineering
• Gregory CrosswhiteUW Physics, Graduate Student
• Quantum Computing Theory Grouphttp://cs.washington.edu/homes/dabacon/qw/
• This work supported in part by the National Science Foundation