Quantum recognition of eigenvalues, structure of devices and thermodynamic properties

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.