84,923 research outputs found
Exploring Quantum Annealing Architectures: A Spin Glass Perspective
We study the spin-glass transition in several Ising models of relevance for
quantum annealers. We extract the spin-glass critical temperature by
extrapolating the pseudo-critical properties obtained with Replica-Exchange
Monte-Carlo for finite-size systems. We find a spin-glass phase for some random
lattices (random-regular and small-world graphs) in good agreement with
previous results. However, our results for the quasi-two-dimensional graphs
implemented in the D-Wave annealers (Chimera, Zephyr, and Pegasus) indicate
only a zero-temperature spin-glass state, as their pseudo-critical temperature
drifts towards smaller values. This implies that the asymptotic runtime to find
the low-energy configuration of those graphs is likely to be polynomial in
system size, nevertheless, this scaling may only be reached for very large
system sizes -- much larger than existing annealers -- as we observe an abrupt
increase in the computational cost of the simulations around the
pseudo-critical temperatures. Thus, two-dimensional systems with local
crossings can display enough complexity to make unfeasible the search with
classical methods of low-energy configurations.Comment: 13 page
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
Complexity, parallel computation and statistical physics
The intuition that a long history is required for the emergence of complexity
in natural systems is formalized using the notion of depth. The depth of a
system is defined in terms of the number of parallel computational steps needed
to simulate it. Depth provides an objective, irreducible measure of history
applicable to systems of the kind studied in statistical physics. It is argued
that physical complexity cannot occur in the absence of substantial depth and
that depth is a useful proxy for physical complexity. The ideas are illustrated
for a variety of systems in statistical physics.Comment: 21 pages, 7 figure
The computational complexity of density functional theory
Density functional theory is a successful branch of numerical simulations of
quantum systems. While the foundations are rigorously defined, the universal
functional must be approximated resulting in a `semi'-ab initio approach. The
search for improved functionals has resulted in hundreds of functionals and
remains an active research area. This chapter is concerned with understanding
fundamental limitations of any algorithmic approach to approximating the
universal functional. The results based on Hamiltonian complexity presented
here are largely based on \cite{Schuch09}. In this chapter, we explain the
computational complexity of DFT and any other approach to solving electronic
structure Hamiltonians. The proof relies on perturbative gadgets widely used in
Hamiltonian complexity and we provide an introduction to these techniques using
the Schrieffer-Wolff method. Since the difficulty of this problem has been well
appreciated before this formalization, practitioners have turned to a host
approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show
in DFT, although the introduction of an approximate potential leads to a
non-interacting Hamiltonian, it remains, in the worst case, an NP-complete
problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics,
Chemistry and Mathematics: A Multidisciplinary View
NMR Quantum Computation
In this article I will describe how NMR techniques may be used to build
simple quantum information processing devices, such as small quantum computers,
and show how these techniques are related to more conventional NMR experiments.Comment: Pedagogical mini review of NMR QC aimed at NMR folk. Commissioned by
Progress in NMR Spectroscopy (in press). 30 pages RevTex including 15 figures
(4 low quality postscript images
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