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Journal ArticleDOI

Statistical mechanics of the quantum K -satisfiability problem.

24 Dec 2008-Physical Review E (Phys Rev E Stat Nonlin Soft Matter Phys)-Vol. 78, Iss: 6, pp 061128-061128
TL;DR: The replica-symmetric free-energy functional is derived within the static approximation and the saddle-point equation for the order parameter: the distribution P[h(m)] of functions of magnetizations, interpreted as the histogram of probability distributions of individual magnetizations.
Abstract: We study the quantum version of the random $K$-satisfiability problem in the presence of an external magnetic field $\ensuremath{\Gamma}$ applied in the transverse direction. We derive the replica-symmetric free-energy functional within the 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 $\ensuremath{\Gamma}$ magnetizations $m\ensuremath{\approx}0$ become relevant in addition to purely classical values of $m\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$. Self-consistency equations for the order parameter are solved numerically using a quasi\char21{}Monte Carlo method for $K=3$. It is shown that for an arbitrarily small $\ensuremath{\Gamma}$ quantum fluctuations destroy the phase transition present in the classical limit $\ensuremath{\Gamma}=0$, 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 reexamined. It is shown that the phase transition at $T=0$ predicted by the replica-symmetric theory is of continuous type with atypical critical exponents.
Citations
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Journal ArticleDOI
TL;DR: In this article, the performance of the quantum adiabatic algorithm for the solution of decision problems is investigated and the possible failure mechanisms are divided into two sets: small gaps due to quantum phase transitions and small gaps caused by avoided crossings inside a phase.
Abstract: In this review we consider the performance of the quantum adiabatic algorithm for the solution of decision problems. We divide the possible failure mechanisms into two sets: small gaps due to quantum phase transitions and small gaps due to avoided crossings inside a phase. We argue that the thermodynamic order of the phase transitions is not predictive of the scaling of the gap with the system size. On the contrary, we also argue that, if the phase surrounding the problem Hamiltonian is a Many-Body Localized (MBL) phase, the gaps are going to be typically exponentially small and that this follows naturally from the existence of local integrals of motion in the MBL phase.

60 citations


Cites background from "Statistical mechanics of the quantu..."

  • ...Thermodynamic calculations within replica theory [61,44,47] and quantum cavity theory [48] suggest random quantum first order transitions persist in at least some local models [38], although QMC data is inconclusive [64]....

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Journal ArticleDOI
TL;DR: In this paper, the Bose-Hubbard model on the Bethe lattice has been shown to be equivalent to a functional self-consistent equation in the thermodynamic limit.
Abstract: The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546 (1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.

41 citations

Posted Content
TL;DR: A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition, which might be the reason it had not been observed in the authors' numerical work.
Abstract: Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N. The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02 N) >> 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.

34 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the ability of local quantum dynamics to solve the ''energy-matching'' problem: given an instance of a classical optimization problem and a low energy state, find another macroscopically distinct low-energy state.
Abstract: We consider the ability of local quantum dynamics to solve the ``energy-matching'' problem: given an instance of a classical optimization problem and a low-energy state, find another macroscopically distinct low-energy state. Energy matching is difficult in rugged optimization landscapes, as the given state provides little information about the distant topography. Here, we show that the introduction of quantum dynamics can provide a speedup over classical algorithms in a large class of hard optimization problems. Tunneling allows the system to explore the optimization landscape while approximately conserving the classical energy, even in the presence of large barriers. Specifically, we study energy matching in the random $p$-spin model of spin-glass theory. Using perturbation theory and exact diagonalization, we show that introducing a transverse field leads to three sharp dynamical phases, only one of which solves the matching problem: (1) a small-field ``trapped'' phase, in which tunneling is too weak for the system to escape the vicinity of the initial state; (2) a large-field ``excited'' phase, in which the field excites the system into high-energy states, effectively forgetting the initial energy; and (3) the intermediate ``tunneling'' phase, in which the system succeeds at energy matching. The rate at which distant states are found in the tunneling phase, although exponentially slow in system size, is exponentially faster than classical search algorithms.

34 citations

Dissertation
01 Jan 2008
TL;DR: In this paper, the authors define a new family of constraint satisfaction problems with constant size constraints and domains and which contains problems that are NP-complete and a.s. have exponential resolution complexity.
Abstract: Despite much work over the previous decade, the Satisfiability Threshold Conjecture remains open. Random k-SAT, for constant k ≥ 3, is just one family of a large number of constraint satisfaction problems that are conjectured to have exact satisfiability thresholds, but for which the existence and location of these thresholds has yet to be proven. Of those problems for which we are able to prove an exact satisfiability threshold, each seems to be fundamentally different than random 3-SAT. This thesis defines a new family of constraint satisfaction problems with constant size constraints and domains and which contains problems that are NP-complete and a.s. have exponential resolution complexity. All four of these properties hold for k-SAT, k ≥ 3, and the exact satisfiability threshold is not known for any constraint satisfaction problem that has all of these properties. For each problem in the family defined in this thesis, we determine a value c such that c is an exact satisfiability threshold if a certain multi-variable function has a unique maximum at a given point in a bounded domain. We also give numerical evidence that this latter condition holds. In addition to studying the satisfiability threshold, this thesis finds exact thresholds for the efficient behavior of DPLL using the unit clause heuristic and a variation of the generalized unit clause heuristic, and this thesis proves an analog of a conjecture on the satisfiability of (2 + p)-SAT. Besides having similar properties as k-SAT, this new family of constraint satisfaction problems is interesting to study in its own right because it generalizes the XOR-SAT problem and it has close ties to quasigroups.

6 citations