1. 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:

2. Motivation Transistors are getting smaller
In 1994, Paul Shor showed that quantum algorithms have an exponential speed-up over their classical counterparts in factoring large prime numbers
bits
noisy bits
quantum bit cm µm nm pm

3. 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
Probabilistic
“Quantum” 1
0.4
0.3
0.5
0.6
0.7 1 -1 -1

4. 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

5. 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:

6. Data from a sample run:

7. Data from sample results:

8. 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

9. 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!

10. Acknowledgments
Advisor: Professor Dave Bacon UW Computer Science & Engineering
Gregory Crosswhite UW Physics, Graduate Student
Quantum Computing Theory Group
This work supported in part by the National Science Foundation