Ia.Quantum chains of qubits

The concept of quantum information has its roots in fundamental features of quantum systems and covers measurement problems in quantum systems and creation of algorithms for quantum computing. One of the real candidates for creation of a quantum computer at the moment are long chains of atoms with spin interaction. During a long time such systems were studied in St. Petersburg Department of V.A.Steklov Institute of Mathematics of the Russian Academy of Science, in mathematical problems of physics  laboratory, lead by L.D.Faddeev. Moreover, there were studied properties of such systems, connected with integrability of important models (for example, Heisenberg model). Research in this filed led to the development of quantum groups and to understanding of their role in the theory of one-dimensional quantum systems.

In 1990 Reshetikhin and Turaev created new theory of invariants of components and three-dimensional manifolds. In 2000-s Kitaev used these results, based on conformal field theory, for development of one and two-dimensional systems theory with properties, required for quantum computing. We suggest to study the impact of topological effects on electronic properties of materials.

Ib. Optical methods of qubits manipulation

Another direction of fundamental research in the framework of the laboratory is connected with management of quantum states of qubits. In their works on optical solitons Rybin and Vadeiko with co-authors showed, that such system allows to build effective nonadiabatic gates of transparency. They demonstrated opportunity to record and read optical soliton in this system, and proved that the system is significantly sustainable in relation to typical temperature relaxation.In the papers, studying the relaxation effect on the stability of information in solitons, it was demonstrated that the system is capable of holding the state of soliton due to the swap of additional strong laser beam,performing the role of the control field.

Iс. Statistical properties of integrable quantum systems

At the current stage of development of integrable models, the task of getting and studying of exact answers for Green functions of models with broken translational invariance is relevant. This task assumes solution by combinating different methods: Bethe ansatz, quantum inverse scattering method, path integration  of spaces in commuting and anticommuting variables. Calculation of correlation functions of stochastic processes, described by non-Hermitian Hamiltonians is one of the most relevant tasks in the field of quantum computing. Group led by N.Bogolyubov is conducting research in strongly interacted boson models and has developed an approach for calculation of correlation functions. These methods are actively used both in quantum computing research and in random walk theory. The latter one is widely used in informatics, economics, biology and finance.