1. Efficiency of the Quantum Adiabatic Algorithm Allan Peter Young, University of California-Santa Cruz, DMR 0906366 We have investigated whether the quantum adiabatic algorithm (QAA), proposed for use on a quantum computer, could solve optimization problems efficiently. The bottlenecks are points where the gap to the first excited state is small. We investigated the median minimum gap ΔE min (averaging over about 50 samples) for the the determination of the ground state of a spin glass model by Monte Carlo (MC) simulations. Only for large sizes N (for which the MC is essential) could we see that the minimum gap decreases exponentially with N. See the main figure, which shows the median minimum gap on a log-lin scale, and fits a straight line for large sizes N. It is to be compared with the inset which is log-log so would be a straight line for a power-law gap, but this is clearly wrong for the largest size. This implies that the QAA would not be efficient for this model. We also find that there are large fluctuations between different samples for large N which implies that, for N→∞, the median minimum gap may fall off more quickly than shown in the figure. This work is in collaboration with E. Farhi (who proposed the QAA in 2001), P. Shor (whose quantum algorithm for factoring integers is famous) and others. It has just been submitted: arXiv.0908.3757. We received computing support from Google.

2. Efficiency of the Quantum Adiabatic Algorithm Allan Peter Young, University of California-Santa Cruz, DMR 0906366 A physics-inspired algorithm for optimization problems is called the Quantum Adiabatic Algorithm. To investigate whether this algorithm is would be efficient on a quantum computer for problems of large size, we have applied a method from statistical physics called Monte Carlo simulations. Recently we applied this to a problem of interest in physics, finding the ground state of a “spin glass”. Looking at small and intermediate sizes, it appears that the quantum algorithm would work very well, but studies of large sizes indicate that, in fact, it would not be more efficient than a classical algorithm. The physics technique of Monte Carlo simulations was essential to get this, unfortunately negative, result. It is known that an eventual quantum computer could solve certain specialized problems more efficiently than a classical computer. However there are great experimental difficulties in building a quantum computer which is large enough to be useful. It would an provide additional impetus for experimentalists to overcome these challenges if, in addition, quantum computers could solve a broad range of problems more efficiently than a classical computer. We have investigated whether a quantum computer could efficiently solve optimization problems, which involve finding the maximum (or minimum) of a function of many variables and are of interest in a wide range of fields in science and engineering, including physics, biology, computer science, image recognition etc. This work applies physics ideas and methods to solve a problem in a related field (quantum computing).