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

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

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arXiv:quant-ph/0104129v1 26 Apr 2001 A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete 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 MIT-CTP #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 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. 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 non-quantum.) 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
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2 E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, D. Preda Beyond factoring there is a famous class of problems called NP-complete; 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 NP-complete 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 NP-complete problems are related in the following sense: if someone finds a polynomial-time algorithm for one NP-
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