The Limits of Adiabatic
Quantum Algorithms
Alper Sarikaya
Advised by Prof. Dave Bacon
Computer Science & Engineering
Chemistry
University of Washington
Undergraduate Research Symposium
May 15, 2009
Quantum Computing Theory Group:
http://cs.washington.edu/homes/dabacon/qw/
Motivation
• Transistors are
getting smaller
bits
noisy bits
• In 1994, Paul Shor
showed that quantum algorithms
have an exponential cm
speed-up
µm
nm
over their classical counterparts
in factoring large prime numbers
quantum bit
pm
Quantum Computation
• Qubit versus a classical bit .. where‘s the information
stored?
Deterministic
Probabilistic
“Quantum”
1
1
0.4
0.3
-1
0.5
-1
• 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)!
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 an
adiabatic 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 – but
it probably only offers a
polynomial increase in
efficiency!
Acknowledgments
• Advisor: Professor Dave Bacon
UW Computer Science & Engineering
• Gregory Crosswhite
UW Physics, Graduate Student
• Quantum Computing Theory Group
http://cs.washington.edu/homes/dabacon/qw/
• This work supported in part by
the National Science Foundation