Phase transitions in the regular random 3-SAT problem

Random quantum satisfiability

Quantum Information and Computation ◽

10.26421/qic10.1-2-1 ◽

2010 ◽

Vol 10 (1&2) ◽

pp. 1-16

Author(s):

C.R. Laumann ◽

R. Moessner ◽

A. Scarddichio ◽

S.L. Sondhi

Keyword(s):

Phase Transitions ◽

Decision Problem ◽

Quantum Phase Transitions ◽

Quantum Computers ◽

Sat Problem ◽

Graph Theoretic ◽

Theoretic Problem ◽

Np Complete ◽

Almost All ◽

Complete Quantum

Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. We study random ensembles of the QMA$_1$-complete quantum satisfiability (QSAT) problem introduced by Bravyi \cite{Bravyi:2006p4315}. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for \emph{any} hypergraph exhibit the same dimension of the satisfying manifold. This establishes the QSAT decision problem as equivalent to a, potentially new, graph theoretic problem and that the hardest typical instances are likely to be localized in a bounded range of clause density.

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Electron Microscopy of Graphite Intercalation Compounds

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100055564 ◽

1982 ◽

Vol 40 ◽

pp. 544-545

Author(s):

G. Timp ◽

L. Salamanca-Riba ◽

L.W. Hobbs ◽

G. Dresselhaus ◽

M.S. Dresselhaus

Keyword(s):

Electron Microscopy ◽

Phase Transitions ◽

Domain Wall ◽

Intercalation Compounds ◽

Lattice Plane ◽

Wall Model ◽

Graphite Intercalation Compounds ◽

Fundamental Symmetry ◽

Graphite Intercalation

Electron microscopy can be used to study structures and phase transitions occurring in graphite intercalations compounds. The fundamental symmetry in graphite intercalation compounds is the staging periodicity whereby each intercalate layer is separated by n graphite layers, n denoting the stage index. The currently accepted model for intercalation proposed by Herold and Daumas assumes that the sample contains equal amounts of intercalant between any two graphite layers and staged regions are confined to domains. Specifically, in a stage 2 compound, the Herold-Daumas domain wall model predicts a pleated lattice plane structure.

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Multiple phase transitions investigated by high-speed TEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100175880 ◽

1990 ◽

Vol 48 (4) ◽

pp. 550-551

Author(s):

Oleg Bostanjoglo ◽

Peter Thomsen-Schmidt

Keyword(s):

Phase Transitions ◽

High Speed ◽

Short Exposure ◽

Semiconductor Film ◽

Time Resolved ◽

Short Exposure Time ◽

Multiple Phase ◽

Model Substance ◽

History Of ◽

Electron Image

Thin GexTe1-x (x = 0.15-0.8) were studied as a model substance of a composite semiconductor film, in addition being of interest for optical storage material. Two complementary modes of time-resolved TEM were used to trace the phase transitions, induced by an attached Q-switched (50 ns FWHM) and frequency doubled (532 nm) Nd:YAG laser. The laser radiation was focused onto the specimen within the TEM to a 20 μm spot (FWHM). Discrete intermediate states were visualized by short-exposure time doubleframe imaging /1,2/. The full history of a transformation was gained by tracking the electron image intensity with photomultiplier and storage oscilloscopes (space/time resolution 100 nm/3 ns) /3/. In order to avoid radiation damage by the probing electron beam to detector and specimen, the beam is pulsed in this continuous mode of time-resolved TEM,too.Short events ( <2 μs) are followed by illuminating with an extended single electron pulse (fig. 1c)

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