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:

• Data from a sample run:

• Data from sample results:

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