2,096 research outputs found
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
Quantum geometry and quantum algorithms
Motivated by algorithmic problems arising in quantum field theories whose
dynamical variables are geometric in nature, we provide a quantum algorithm
that efficiently approximates the colored Jones polynomial. The construction is
based on the complete solution of Chern-Simons topological quantum field theory
and its connection to Wess-Zumino-Witten conformal field theory. The colored
Jones polynomial is expressed as the expectation value of the evolution of the
q-deformed spin-network quantum automaton. A quantum circuit is constructed
capable of simulating the automaton and hence of computing such expectation
value. The latter is efficiently approximated using a standard sampling
procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The
Quantum Universe'' in honor of G. C. Ghirard
The colored Jones function is q-holonomic
A function of several variables is called holonomic if, roughly speaking, it
is determined from finitely many of its values via finitely many linear
recursion relations with polynomial coefficients. Zeilberger was the first to
notice that the abstract notion of holonomicity can be applied to verify, in a
systematic and computerized way, combinatorial identities among special
functions. Using a general state sum definition of the colored Jones function
of a link in 3-space, we prove from first principles that the colored Jones
function is a multisum of a q-proper-hypergeometric function, and thus it is
q-holonomic. We demonstrate our results by computer calculations.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper29.abs.htm
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