  Quantum algorithms speeding up classical counterparts are proposed for the
problems:
  1. Recognition of eigenvalues with fixed precision. Given a quantum circuit
generating unitary mapping $U$ and a complex number the problem is to determine
is it an eigenvalue of $U$ or not.
  2. Given a molecular structure find thermodynamic functions like partitioning
function, entropy, etc. for a gas consisting of such molecules.
  3. Recognition of molecular structures. Find a molecular structure given its
spectrum.
  4. Recognition of electronic devices. Given an electronic device that can be
used only as a black box how to recognize its internal construction?
  We consider mainly structures generating sparse spectrums. These algorithms
require the time from about square root to logarithm of the time of classical
analogs and for the first three problems give exponential memory saving. Say,
the time required for distinguishing two devices with the same given spectrum
is about seventh root of the time of direct classical method, for the
recognition of eigenvalue - about sixth root. Thus microscopic quantum devices
can recognize molecular structures and physical properties of environment
faster than big classical computers.
