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Unformatted text preview: arXiv:quantph/0104129v1 26 Apr 2001 A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NPComplete Problem Edward Farhi, Jeffrey Goldstone * Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Sam Gutmann † Department of Mathematics, Northeastern University, Boston, MA 02115 Joshua Lapan, Andrew Lundgren, Daniel Preda ‡ Massachusetts Institute of Technology, Cambridge, MA 02139 MITCTP #3035 A shorter version of this article appeared in the April 20, 2001 issue of Science . Abstract 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 NPcomplete 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 NPcomplete problems. 1 Introduction A large quantum computer has yet to be built, but the rules for programming such a device, which are derived from the laws of quantum mechanics, are well established. It is already known that quantum computers could solve problems believed to be intractable on classical computers. (Throughout, “classical” means nonquantum.) An intractable problem is one that necessarily takes too long to solve when the input gets too big. More precisely, a classically intractable problem is one that cannot be solved using any classical algorithm whose running time grows only polynomially as a function of the length of the input. For example, all known classical factoring algorithms require more than polynomial time as a function of the number of digits in the integer to be factored. Shor’s quantum algorithm for the factoring problem [1] can factor an integer in a time that grows (roughly) as the square of the number of digits. This raises the question of whether quantum computers could solve other classically difficult problems faster than classical computers. * [email protected], [email protected] † [email protected] ‡ [email protected], [email protected], [email protected] 1 2 E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, D. Preda Beyond factoring there is a famous class of problems called NPcomplete; see, for ex ample, [2]. Hundreds of problems are known to be in this class, ranging from the practical (variants of the Traveling Salesman problem) to the whimsical (a problem derived from the game of Go). The original NPcomplete problem, Satisfiability, involves answering whether or not a Boolean formula (made up of variables connected by the operators or , and , and not ) is true for some choice of truth values for each variable. All NPcomplete problems are related in the following sense: if someone finds a polynomialtime algorithm for one NP...
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.
 Spring '11
 Aster
 Mass, Theoretical Physics

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