Temas como introducción a la informática cuántica

Posted on 30 agosto 2008

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La Facultad de Matemática Computacional y Cibernética de la MSU, toma un examen a los estudiantes que desean hacer un post-grado en informática cuántica con los siguientes temas:

Algorithms theory

  • Models of algorithms: Turing machines, cellular automata, MArkov’s normal algorithms, their equivalence (Church thesis).
  • Computable and incomputable predicates and functions. Example of incomputable problem: an applicability of algorithm to the given input word.
  • Deterministic and nondeterministic computational models.
  • Time and space complexity of computations. P- sets, NP- sets, polynomial reductability.
  • Polynomial equivalence of all deterministic computational models. NP- complete problems.
  • Computations with oracles.

Algebra

  • Linear spaces. Dimensionality, basis, linear operators and their matrises, bilinear and qudratic forms. Changes of the basis.
  • Dot product. Hilbert space, orthonormal basises.
  • Unitary and Hermitian operators, their normal forms and relations between them.
  • Eigenvalues and eigenvectors, finding them.
  • Matrix representation of operators, matrix algebra, matrix exponential.
  • Groups, matrix representation of them.

Probability theory

  • Axioms and properties of probability.
  • Distributions, densities. Random variables, means and dispersions. Types of distributons: homogenious, normal, exponential, hi and hi-squared.
  • Central limit theorem.
  • Statistical hypothesis and criteria.
  • Random processes, Markov chains, branching processes.

Numerical methods

  • Outlook of the numerical methods for the equations of mathematical physics. Convergence and stability.

Quantum mechanics

  • Experiments showing interference of particles. Amplitudes. Quantum states. Bosons and fermions.
  • Emission and absorbtion of photons. Pauli principle. Spin 1 and 1/2. Stern-Gerlach experiment. Amplitudes transformation.
  • Dependence of amplitudes on time. Shroedinger equation. Wave function and observations. Potential energy and Hamiltonian. Molecula of ammonia. Ammonia mazer. Ion and molecula of hydrogen.
  • Spin matrix of Pauli. Photon polarization. Neutral K-mezon.
  • Systems with n states. Superfine splitting of energy levels for the atom of hydrogen.
  • Electron in 3-dimensional grid. Semiconducting. Holl effect. Spin waves.
  • Symmetry and conservation laws: impulse, energy, moment. Examples.
  • Solution of Shroedinger equation for the atom of hydrogen without spin.
  • Representation of physical values by operators.
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