Bounding Run-Times of Local Adiabatic Algorithms

A common trick for designing faster quantum adiabatic algorithms is to apply the adiabaticity condition locally at every instant. However it is often difficult to determine the instantaneous gap between the lowest two eigenvalues, which is an essential ingredient in the adiabaticity condition. In this paper we present a simple linear algebraic technique for obtaining a lower bound on the instantaneous gap even in such a situation. As an illustration, we investigate the adiabatic unordered search of van Dam et al. (How powerful is adiabatic quantum computation? Proc. IEEE FOCS, pp. 279-287, 2001) and Roland and Cerf (Physical Review A 65, 042308, 2002) when the non-zero entries of the diagonal final Hamiltonian are perturbed by a polynomial (in $\log N$, where $N$ is the length of the unordered list) amount. We use our technique to derive a bound on the running time of a local adiabatic schedule in terms of the minimum gap between the lowest two eigenvalues.Comment: 11 page

A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem

A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We test one such algorithm by applying it to randomly generated, hard, instances of an NP-complete problem. For the small examples that we can simulate, the quantum adiabatic algorithm works well, and provides evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a shorter version of this article appeared in the April 20, 2001 issue of Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47