108,731 research outputs found
Computational Difficulty of Computing the Density of States
We study the computational difficulty of computing the ground state
degeneracy and the density of states for local Hamiltonians. We show that the
difficulty of both problems is exactly captured by a class which we call #BQP,
which is the counting version of the quantum complexity class QMA. We show that
#BQP is not harder than its classical counting counterpart #P, which in turn
implies that computing the ground state degeneracy or the density of states for
classical Hamiltonians is just as hard as it is for quantum Hamiltonians.Comment: v2: Accepted version. 9 pages, 1 figur
A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates
Numerical computations of stationary states of fast-rotating Bose-Einstein
condensates require high spatial resolution due to the presence of a large
number of quantized vortices. In this paper we propose a low-order finite
element method with mesh adaptivity by metric control, as an alternative
approach to the commonly used high order (finite difference or spectral)
approximation methods. The mesh adaptivity is used with two different numerical
algorithms to compute stationary vortex states: an imaginary time propagation
method and a Sobolev gradient descent method. We first address the basic issue
of the choice of the variable used to compute new metrics for the mesh
adaptivity and show that simultaneously refinement using the real and imaginary
part of the solution is successful. Mesh refinement using only the modulus of
the solution as adaptivity variable fails for complicated test cases. Then we
suggest an optimized algorithm for adapting the mesh during the evolution of
the solution towards the equilibrium state. Considerable computational time
saving is obtained compared to uniform mesh computations. The new method is
applied to compute difficult cases relevant for physical experiments (large
nonlinear interaction constant and high rotation rates).Comment: to appear in J. Computational Physic
Computational Complexity in Electronic Structure
In quantum chemistry, the price paid by all known efficient model chemistries
is either the truncation of the Hilbert space or uncontrolled approximations.
Theoretical computer science suggests that these restrictions are not mere
shortcomings of the algorithm designers and programmers but could stem from the
inherent difficulty of simulating quantum systems. Extensions of computer
science and information processing exploiting quantum mechanics has led to new
ways of understanding the ultimate limitations of computational power.
Interestingly, this perspective helps us understand widely used model
chemistries in a new light. In this article, the fundamentals of computational
complexity will be reviewed and motivated from the vantage point of chemistry.
Then recent results from the computational complexity literature regarding
common model chemistries including Hartree-Fock and density functional theory
are discussed.Comment: 14 pages, 2 figures, 1 table. Comments welcom
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
A spectral scheme for Kohn-Sham density functional theory of clusters
Starting from the observation that one of the most successful methods for
solving the Kohn-Sham equations for periodic systems -- the plane-wave method
-- is a spectral method based on eigenfunction expansion, we formulate a
spectral method designed towards solving the Kohn-Sham equations for clusters.
This allows for efficient calculation of the electronic structure of clusters
(and molecules) with high accuracy and systematic convergence properties
without the need for any artificial periodicity. The basis functions in this
method form a complete orthonormal set and are expressible in terms of
spherical harmonics and spherical Bessel functions. Computation of the occupied
eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a
combination of preconditioned block eigensolvers and Chebyshev polynomial
filter accelerated subspace iterations. Several algorithmic and computational
aspects of the method, including computation of the electrostatics terms and
parallelization are discussed. We have implemented these methods and algorithms
into an efficient and reliable package called ClusterES (Cluster Electronic
Structure). A variety of benchmark calculations employing local and non-local
pseudopotentials are carried out using our package and the results are compared
to the literature. Convergence properties of the basis set are discussed
through numerical examples. Computations involving large systems that contain
thousands of electrons are demonstrated to highlight the efficacy of our
methodology. The use of our method to study clusters with arbitrary point group
symmetries is briefly discussed.Comment: Manuscript submitted (with revisions) to Journal of Computational
Physic
Statistical Mechanical Formulation and Simulation of Prime Factorization of Integers
We propose a new formulation of the problem of prime factorization of
integers. With replica exchange Monte Carlo simulation, the behavior which is
seemed to indicate exponential computational hardness is observed. But this
formulation is expected to give a new insight into the computational complexity
of this problem from a statistical mechanical point of view.Comment: 5 pages, 5figures, Proceedings of 4th YSM-SPIP (Sendai, 14-16
December 2012
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