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 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

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

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