1,384 research outputs found
Random knots in three-dimensional three-colour percolation: numerical results and conjectures
Three-dimensional three-colour percolation on a lattice made of tetrahedra is
a direct generalization of two-dimensional two-colour percolation on the
triangular lattice. The interfaces between one-colour clusters are made of
bicolour surfaces and tricolour non-intersecting and non-self-intersecting
curves. Because of the three-dimensional space, these curves describe knots and
links. The present paper presents a construction of such random knots using
particular boundary conditions and a numerical study of some invariants of the
knots. The results are sources of precise conjectures about the limit law of
the Alexander polynomial of the random knots.Comment: minor corrections in the text since v
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
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