We present a classical stochastic simulation technique of quantum Branching programs. This technique allows to prove the following relations among complexity classes: PrQP-BP &subuline; PP-BP and BQP-BP &subuline; PP-BP. Here BPP-BP and PP-BP stands for the classes of functions computable with bounded error and unbounded error respectively by stochastic branching program of polynomial size. BQP-BP and PrQP-BP stands the classes of functions computable with bounded error and unbounded error respectively by quantum branching program of polynomial size. Second. We present two different types, of complexity lower bounds for quantum nonuniform automata (OBDDs). We call them "metric" and "entropic" lower bounds in according to proof technique used. We present explicit Boolean functions that show that these lower bounds are tight enough. We show that when considering "almost all Boolean functions" on n variables our entropic lower bounds gives exponential (2^{c(δ)(n-log n)}) lower bound for the width of quantum OBDDs depending on the error δ allowed.

We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators, that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter (the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables), we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement.

A method of quantum tomography of arbitrary spin particle states is developed on the basis of the root approach. It is shown that the set of mutually complementary distributions of angular momentum projections can be naturally described by a set of basis functions based on the Kravchuk polynomials. The set of Kravchuk basis functions leads to a multiparametric statistical distribution that generalizes the binomial distribution. In order to analyze a statistical inverse problem of quantum mechanics, we investigated the likelihood equation and the statistical properties of the obtained estimates. The conclusions of the analytical researches are approved by the results of numerical calculations.

We investigate the procedure of Schmidt modes extraction in systems with continuous variables. An algorithm based on singular value matrix decomposition is applied to the study of entanglement in an "atom-photon" system with spontaneous radiation. Also, this algorithm is applied to the study of a bi-photon system with spontaneous parametric down conversion with type-II phase matching for broadband pump. We demonstrate that dynamic properties of entangled states in an atom-photon system with spontaneous radiation are defined by a parameter equal to the product of the fine structure constant and the atom-electron mass ratio. We then consider the evolution of the system during radiation and show that the atomic and photonic degrees of freedom are entangling for the times of the same order of magnitude as the excited state life-time. Then the degrees of freedom are de-entangling and asymptotically approach to the level of small residual entanglement that is caused by momentum dispersion of the initial atomic packet. Finally, we investigate the process of coherence loss between modes in type-II parametric down conversion that is caused by non-linear crystal properties.

Information aspects of copying quantum states via stimulated emission of an optical quantum amplifier are considered. It is shown that the measurable information very rapidly decreases after amplification of a single photon up to a level of several photons. Spontaneous emission, which leads to such behavior, is also discussed.

An algorithm and its first implementation in C# are presented for assembling arbitrary quantum circuits on the base of Hadamard and Toffoli gates and for constructing multivariate polynomial systems over the finite field Z₂ arising when applying the Feynman's sum-over-paths approach to quantum circuits. The matrix elements determined by a circuit can be computed by counting the number of common roots in Z₂ for the polynomial system associated with the circuit. To determine the number of solutions in Z₂ for the output polynomial system, one can use the Grobner bases method and the relevant algorithms for computing Grobner bases.

As a large-scale quantum register, the one-dimensional chain of the magnetic atoms with nuclear spins 1/2 in thin plate of nuclear spin-free easy-axis 3D antiferromagnet is considered. The external magnetic field is directed along the easy axis, normally to the plate surface and has a week constant gradient along the nuclear spin chain. The expression for indirect inter-spin coupling, which is due to hyperfine nuclear electron coupling in atoms and spin-wave propagation in antiferromagnet, was evaluated. It was shown that near critical point of spin-flop quantum phase transition in antiferromagnet indirect nuclear spin coupling in inhomogeneous external magnetic field might have both long-range damping and oscillating dependence from interspin distance. The external magnetic field and its gradient play the role of control parameters.

We investigate time-modulation of continuous variable (CV) quantum states of light fields in application to time-resolved quantum communications. As realizations, we consider EPR entangled states of light as well as correlated twin beams generated in nondegenerate optical parametric oscillator (NOPO) subjected by a sequence of laser pulses. We demonstrate that time-modulation of a pump field essentially improves the degrees of both cv entanglement and intensity quantum correlation in NOPO doing them beyond the standard limits.

We introduce an original model of quantum phenomena, a model that provides a picture of a "deep structure", an "underlying pattern" of quantum dynamics. We propose that the source of a particle and all of that particle's possible detectors "talk" before the particle is finally observed by just one detector. These talks do not take place in physical time. They occur in what we call "hidden time". Talks are spatially organized in such a way that the model reproduces standard quantum probability amplitudes. This is most obviously seen if one uses R. Feynman's formulation of quantum theory. We prefer the "physical level" of mathematical strictness in describing our model.

We have found that encapsulated atoms in fullerene molecules, which carry a spin, can be used for fast quantum computing. We describe the scheme for performing quantum computations, going through the preparation of the qubit state and the realization of a two-qubit quantum gate. When we apply a static magnetic field to each encased spin, we find out the ideal design for the preparation of the quantum state. Therefore, adding to our system a time dependent magnetic field, we can perform a phase-gate. The operational time related to a π-phase gate is of the order of ns. This finding shows that, during the decoherence time, which is proportional to micrometer/s, we can perform many thousands of gate operations. In addition, the two-qubit state which arises after a π-gate is characterized by a high degree of entanglement. This opens a new avenue for the implementation of fast quantum computation.

We study the problem of the most economical representation of entangled states in the classical simulations. The idea is to reduce the general form of entanglement to the bipartite entanglement which has the short representation through Schmidt expansion. The problem of such reduction is stated exactly and discussed. The example is given which shows that if we allow the linear transformation (not only unitary), the general form of entanglement cannot be described in terms of bipartite entanglement. We also study the entanglement dynamics of 2 and 3 level atoms interacting randomly and find interesting dependence of the number of its excited levels.

The method of collective behavior is based on the representation of real quantum particle by the swarm of classical particles which have all properties of the initial particle but have classical states like coordinates and impulse. Simulation with swarms can be more flexible and powerful than analytical methods because it preserves the methodology of classical description of dynamics. The method of collective behavior is illustrated on the diffusion Monte Carlo way of calculating stationary states of electrons.

Algorithmic approach to quantum theory is considered. It is based on the supposition that every evolution of many particle system can be simulated by classical algorithms of polynomial complexity. This hypothesis agrees with all known experiments but it presumes the principle cut-off of quantum formalism because it excludes a scalable quantum computer. Algorithmic approach describes quantum evolution uniformly, without separation of measurements from the unitary dynamics; it is shown how Bohrn rule for quantum probability follows from the basic principles of this approach. The radical difference of algorithmic approach from the standard and its perspectives are discussed.

Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi function in analogy with quantum mechanics. The psi function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms. A new statistical characteristic, a confidence cone, is introduced instead of a standard confidence interval. The chi-square test is considered to test the hypotheses that the estimated vector converges to the state vector of a general population and that both samples are homogeneous. The expansion coefficients are estimated by the maximum likelihood method. An iteration algorithm for solving the likelihood equation is presented. The stability and rate of convergence of the solution are studied. A special iteration parameter is introduced: its optimal value is chosen on the basis of the maximin strategy. Numerical simulation is performed using the set of the Chebyshev-Hermite functions as a basis.

Quantum cryptography (secure key distribution) systems must include procedures for correcting errors in the raw key transmitted over a quantum communication channel. Several reconciliation protocols are discussed and compared in terms of efficiency.

In this paper we prepose to enter the semiempirical relaxation operator (four-index relaxation matrix R_mjkn) in Liuville quantum equation at simulation ESR spectra of Zn-P two-spin system in a Si. Introduction of this operator allows us to expand application of Liuville equation to nonequilibrium processes. Particularly it was made to processes of absorption by Zn-P two-spin system of the superfluous micrometer-wave quantums at presence of its interaction with Si environment. The semiempirical relaxation operator R_mjkn = v_mjP_kn + P_mjv_kn contains of transitions requencies V_mn and functions overlap P_mn in spin system. Imaginary frequencies of transition ±H_mn are replaced on real V_mn, and discrete δ-function δ_mn - on overlap function P_mn as result of stochastic interaction paramagnetic Zn-P systems with Si environment and an absorbed μ-wave quantums.

This paper presents the design of a novel modified Wallace tree multiplier, using the reversible TSG gate proposed by the authors earlier. The novelty of the TSG gate is that it can also work singly as a reversible full adder. The TSG gate is also used in this paper to design various other reversible arithmetic and logical components that can be assembled to realize a primitive reversible/quantum ALU. It is also shown that these components are optimal, in terms of number of reversible gates and garbage outputs, compared to other designs existing in literature.

Experimental aspects of an application of Rydberg atoms to quantum computing are studied. A single neutral atom trapped in an antinode of the optical lattice can represent a quantum bit. Laser excitation of two atoms in neighboring antinodes allows for obtaining of quantum entanglement of the atoms via dipole-dipole interaction which is strong for high Rydberg states. A two-qubit operation could be realized in this way. The optimal values of a principal quantum number, an interatomic distance, time of a single two-qubit operation and other parameters have been estimated. The estimates were done for ²³Na and ⁸⁷Rb atoms. Also experimental results of microwave spectroscopy of a few sodium Rydberg atoms at the one-photon 37S_1/2 -> 37P_1/2 and two-photon 37S_1/2 -> 38S_1/2 transitions are presented. Microwave spectroscopy can be used to detect dipole-dipole interaction between a few Rydberg atoms. The calculations showing an influence of dipole-dipole interaction on two-atom spectra are also presented. A noticeable broadening of the five-atom spectrum was observed in the experiment due to the dipole-dipole interaction.

We study the quantum dynamics of two interacting electrons in the symmetric double-dot structure under the influence of the bichromatic resonant pulse. The state vector evolution is studied for two different pulse designs. It is shown that the laser pulse can generate the effective exchange coupling between the electron spins localized in different dots. Possible applications of this effect to the quantum information processing (entanglement generation, quantum state engineering) are discussed.

We suggest a concrete way to realize the quantum repeater protocol based on nitrogen-vacancy (NV) defects in diamond, feasible with present means of manipulating the defects.