Statistical mechanics of the quantumK-satisfiability problem
| Autor: | Vadim Smelyanskiy, Sergey Knysh |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Quantum Physics
Phase transition Statistical Mechanics (cond-mat.stat-mech) FOS: Physical sciences Order (ring theory) Disordered Systems and Neural Networks (cond-mat.dis-nn) General Medicine Statistical mechanics Condensed Matter - Disordered Systems and Neural Networks Classical limit Distribution (mathematics) Quantum mechanics Probability distribution Quantum Physics (quant-ph) Critical exponent Condensed Matter - Statistical Mechanics Quantum fluctuation Mathematics |
| Zdroj: | Physical Review E. 78 |
| ISSN: | 1550-2376 1539-3755 |
| Popis: | We study the quantum version of the random $K$-Satisfiability problem in the presence of the external magnetic field $\Gamma$ applied in the transverse direction. We derive the replica-symmetric free energy functional within static approximation and the saddle-point equation for the order parameter: the distribution $P[h(m)]$ of functions of magnetizations. The order parameter is interpreted as the histogram of probability distributions of individual magnetizations. In the limit of zero temperature and small transverse fields, to leading order in $\Gamma$ magnetizations $m \approx 0$ become relevant in addition to purely classical values of $m \approx \pm 1$. Self-consistency equations for the order parameter are solved numerically using Quasi Monte Carlo method for K=3. It is shown that for an arbitrarily small $\Gamma$ quantum fluctuations destroy the phase transition present in the classical limit $\Gamma=0$, replacing it with a smooth crossover transition. The implications of this result with respect to the expected performance of quantum optimization algorithms via adiabatic evolution are discussed. The replica-symmetric solution of the classical random $K$-Satisfiability problem is briefly revisited. It is shown that the phase transition at T=0 predicted by the replica-symmetric theory is of continuous type with atypical critical exponents. Comment: 35 pages, 23 figures; changed abstract, improved discussion in the introduction, added references, corrected typos |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|