4,896 research outputs found
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
We analyze relationships between quantum computation and a family of
generalizations of the Jones polynomial. Extending recent work by Aharonov et
al., we give efficient quantum circuits for implementing the unitary
Jones-Wenzl representations of the braid group. We use these to provide new
quantum algorithms for approximately evaluating a family of specializations of
the HOMFLYPT two-variable polynomial of trace closures of braids. We also give
algorithms for approximating the Jones polynomial of a general class of
closures of braids at roots of unity. Next we provide a self-contained proof of
a result of Freedman et al. that any quantum computation can be replaced by an
additive approximation of the Jones polynomial, evaluated at almost any
primitive root of unity. Our proof encodes two-qubit unitaries into the
rectangular representation of the eight-strand braid group. We then give
QCMA-complete and PSPACE-complete problems which are based on braids. We
conclude with direct proofs that evaluating the Jones polynomial of the plat
closure at most primitive roots of unity is a #P-hard problem, while learning
its most significant bit is PP-hard, circumventing the usual route through the
Tutte polynomial and graph coloring.Comment: 34 pages. Substantial revision. Increased emphasis on HOMFLYPT,
greatly simplified arguments and improved organizatio
Quantum Algorithms for the Jones Polynomial
This paper gives a generalization of the AJL algorithm and unitary braid
group representation for quantum computation of the Jones polynomial to
continuous ranges of values on the unit circle of the Jones parameter. We show
that our 3-strand algorithm for the Jones polynomial is a special case of this
generalization of the AJL algorithm. The present paper uses diagrammatic
techniques to prove these results. The techniques of this paper have been used
and will be used in the future in work with R. Marx, A. Fahmy, L. H. Kauffman,
S. J. Lomonaco Jr.,A. Sporl, N. Pomplun, T. Schulte Herbruggen, J. M. Meyers,
and S. J. Glaser on NMR quantum computation of the Jones polynomial.Comment: 11 pages, 4 figures, LaTeX documen
Quantum automata, braid group and link polynomials
The spin--network quantum simulator model, which essentially encodes the
(quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable
to address problems arising in low dimensional topology and group theory. In
this combinatorial framework we implement families of finite--states and
discrete--time quantum automata capable of accepting the language generated by
the braid group, and whose transition amplitudes are colored Jones polynomials.
The automaton calculation of the polynomial of (the plat closure of) a link L
on 2N strands at any fixed root of unity is shown to be bounded from above by a
linear function of the number of crossings of the link, on the one hand, and
polynomially bounded in terms of the braid index 2N, on the other. The growth
rate of the time complexity function in terms of the integer k appearing in the
root of unity q can be estimated to be (polynomially) bounded by resorting to
the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure
STARRY: Analytic Occultation Light Curves
We derive analytic, closed form, numerically stable solutions for the total
flux received from a spherical planet, moon or star during an occultation if
the specific intensity map of the body is expressed as a sum of spherical
harmonics. Our expressions are valid to arbitrary degree and may be computed
recursively for speed. The formalism we develop here applies to the computation
of stellar transit light curves, planetary secondary eclipse light curves, and
planet-planet/planet-moon occultation light curves, as well as thermal
(rotational) phase curves. In this paper we also introduce STARRY, an
open-source package written in C++ and wrapped in Python that computes these
light curves. The algorithm in STARRY is six orders of magnitude faster than
direct numerical integration and several orders of magnitude more precise.
STARRY also computes analytic derivatives of the light curves with respect to
all input parameters for use in gradient-based optimization and inference, such
as Hamiltonian Monte Carlo (HMC), allowing users to quickly and efficiently fit
observed light curves to infer properties of a celestial body's surface map.Comment: 55 pages, 20 figures. Accepted to the Astronomical Journal. Check out
the code at https://github.com/rodluger/starr
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