4 research outputs found
Hyperdeterminants as integrable discrete systems
We give the basic definitions and some theoretical results about
hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability
(understood as 4d-consistency) of a nonlinear difference equation defined by
the 2x2x2-hyperdeterminant. This result gives rise to the following hypothesis:
the difference equations defined by hyperdeterminants of any size are
integrable.
We show that this hypothesis already fails in the case of the
2x2x2x2-hyperdeterminant.Comment: Standard LaTeX, 11 pages. v2: corrected a small misprint in the
abstrac
Geometric Strategy for the Optimal Quantum Search
We explore quantum search from the geometric viewpoint of a complex
projective space , a space of rays. First, we show that the optimal quantum
search can be geometrically identified with the shortest path along the
geodesic joining a target state, an element of the computational basis, and
such an initial state as overlaps equally, up to phases, with all the elements
of the computational basis. Second, we calculate the entanglement through the
algorithm for any number of qubits as the minimum Fubini-Study distance to
the submanifold formed by separable states in Segre embedding, and find that
entanglement is used almost maximally for large . The computational time
seems to be optimized by the dynamics as the geodesic, running across entangled
states away from the submanifold of separable states, rather than the amount of
entanglement itself.Comment: revtex, 10 pages, 7 eps figures, uses psfrag packag