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A Solvable Model of Quantum Random Optimization Problems

A Solvable Model of Quantum Random Optimization Problems -...

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arXiv:1006.1736v2 [cond-mat.stat-mech] 25 Jun 2010 A solvable model of quantum random optimization problems Laura Foini, 1, 2 Guilhem Semerjian, 1 and Francesco Zamponi 1 1 LPTENS, CNRS UMR 8549, associ´ ee ` a l’UPMC Paris 06, 24 Rue Lhomond, 75005 Paris, France. 2 SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy We study the quantum version of a simplified model of optimization problems, where quantum fluctuations are introduced by a transverse field acting on the qubits. We find a complex low-energy spectrum of the quantum Hamiltonian, characterized by an abrupt condensation transition and a continuum of level crossings as a function of the transverse field. We expect this complex structure to have deep consequences on the behavior of quantum algorithms attempting to find solutions to these problems. PACS numbers: 75.10.Nr; 03.67.Ac; 64.70.Tg A large part of theoretical research in quantum com- puting has been devoted to the development of algo- rithms that could use quantum properties to achieve a faster velocity in performing computational tasks with respect to classical devices. A typical problem that is en- countered in almost all branches of science is that of op- timizing irregularly shaped cost functions H P : two stan- dard examples are k -SAT, H P counting the number of unsatisfied constraints on k boolean variables, and the coloring of a graph with q colors ( q -COL), H P then be- ing the number of monochromatic edges. The decision version (whether a solution, i.e. a configuration with H P = 0, exists) of both problems belong to the class of NP-complete problems. One is mostly interested in the scaling of the difficulty of these problems when the number N of variables in- volved becomes large. Besides the formal computational complexity theory which classifies the difficulty of prob- lems according to a worst-case criterion, their typical case complexity is often studied through random ensemble of instances, for instance assuming a flat probability dis- tribution over the choice of M = αN constraints on N degrees of freedom. Statistical mechanics tools have provided a very de- tailed and intricate picture of the configuration space of such typical problem Hamiltonians H P [1]. A key con- cept that emerged in this context is that of clustering of solutions . The topology of the space of solutions changes abruptly upon increasing the density of constraints α in the following way: i) at a first threshold α d it goes from a single connected cluster to a set of essentially disjoint clusters; ii) the number of clusters itself undergoes a tran- sition at α c > α d from a phase where it is exponential in N , thus defining an entropy of clusters (the complex- ity ), to a phase where the vast majority of solutions are contained in a finite number of clusters; iii) finally, the total entropy of solutions vanishes at α s > α c where the problem does not admit a solution anymore (the ground state energy becomes positive). This sequence of tran- sitions (sketched in Fig. 1) has a deep impact on the
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