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Methods for complex system modeling

Priority areas of development: mathematics
The project has been carried out as part of the HSE Program of Fundamental Studies.

Goal of research

The goal in the first part of the research is to study new phenomena in essentially quantum systems that are interesting from the fundamental standpoint and perspective in applications. In particular, it was planned to perform experimental simulation of several quantum phenomena on contemporary quantum computers and to analyze the limitations of such modeling related to the errors of quantum gates and the one- and multi-qubit decoherence. It was planned to analyze the possibilities of realizing analogs of quantum electrodynamics effects in superconducting systems   and to look for new effects in systems of related qubits and resonators. The goal was to study the influence of the coordinate dependence of photon and polariton effective masses on the properties of Bose condensate of such particles. It was also planned to study the low-temperature system of bosons with long-range interparticle potential. 

The goal in the second part of this research is to study the spectral properties of connectomes and their specific features for different organisms. It is also necessary to understand which peculiarities of the neural network structure are responsible for the differences in their spectra from the spectra of all other networks and to what extent these peculiarities depend on the species differences between the organisms.

The goal in the third part of the research is, first, to study the flow of a stratified fluid with a fast oscillating interface in the case of Rayleigh-Taylor instability and its asymptotic behavior as the oscillation frequency tends to infinity under the assumption that the gravity force is much greater than the viscosity and the relative difference between the fluids is small. The second goal in this research is to obtain a general algorithm for constructing formal asymptotic solutions of the problems under study on the basis of contemporary methods for studying complex problems of mathematical physics. This problem originates from the fact that, in classical works on the three-deck structure (where only the problem of flow past a localized irregularity was considered), the solution were constructed with a great inaccuracy which lead to the Benjamin equation, where there is no relation to the problem geometry and to the functions describing the flow in a multideck structure. Obviously, such an equation does not make any sense in hydrodynamics.

The fourth goal in this research is to study quantum resonance-integrable systems. The study of tunnel effects in a hyperbolic trap with frequency resonances when the Hamiltonian is mirror symmetric consists in the construction of an effective Hamiltonian, determination of conditions for the origination of biorbital states, and representation of the tunnel splitting in terms of complexification of the phase space of the classical mechanical system. It is also necessary to study the geometric currents in a curved thin layer in the case where the cyclotron frequency coincides with the frequency of the charge transverse oscillations, to determine the influence of the layer geometry on the direction of the induced current, and consider general algebraic methods for reducing problems with resonances to Hamiltonians with an irreducible resonance.


The experimental simulation of quantum spin systems was performed remotely on quantum processor IBMqx4, and it was analyzed by analytical and numerical quantum mechanical computations. The dynamics of related systems of qubits and resonators was modeled by numerically solving the Lindblad equation in an appropriate basis of states. The Bose condensate of photons and polaritons was analyzed by numerically solving the Gross-Pitaevskii equation. The system of bosons with long-range interparticle potential was modeled numerically by the diffusion Monte Carlo method. For the regime of weak interaction, a special modification of the Bogolyubov method was developed.

The connectomes were modeled by different computer methods including the Monte Carlo probability methods and various algorithms on graphs such as algorithms for searching the communities, storage and Big Data algorithms, various algorithms for the network visualization and excitation propagation on networks. The properties of the connectome, modeled networks, and random networks were analyzed by using all standard methods of statistical physics including the method of generating functions, cluster analysis, methods developed for analyzing anomalous diffusion, and the theory of random matrices. 

The main tool underlying the investigation of hydrodynamic problems is the little-known fact that the averaging can be treated as passing to the limit in the sense of generalized functions (distributions). This immediately implies that the averaging is a local operation and the whole apparatus of the theory of generalized functions can be applied. The proposed method for constructing asymptotic solutions of flow problems is based on a combination of the method of boundary layer expansion and the averaging method (in the case of periodic irregularities) or a version of the Maslov-Whitham method (in the case of localized irregularities).

The quantum resonance-integrable systems were studied by algebraic and geometric methods such as algebraic averaging, construction of resonance algebras of symmetries, separation of the creation-annihiliation structure in them, and analysis of the topology of symplectic leaves and the classical trajectories of motion. The methods of discrete and continuous semiclassical WKB approximation were used.

Empirical base of research

As the empirical base, the results of studies in the filed of the project published in the world scientific literature were used.  

Results of research

In the first part of the research, the quantum evolution of systems of several coupled spins was modeled experimentally which demonstrated effects of quantum entanglement and excitation transmission blockade. Restrictions on the length of quantum algorithms due to the inaccuracy of quantum gate operation were obtained. A new realization of the Lamb dynamical effect in superconducting systems was proposed. An interesting effect of reinforcement of quantum phenomena in one subsystem due to the energy dissipation in the other subsystem was discovered. The influence of the coordinate dependence of the effective mass of photons and polaritons on the energy and wave functions of their Bose-Einstein condensates in optical microcavities was studied. It was shown that, using the coordinate dependence of the effective mass, one can control the particle interaction and even change their repulsion by attraction. The properties of a quantum system of bosons were studied in detail at the zero temperature with repulsion interaction potential inversely proportional to the squared distance. The phase transition between the gaseous and solid phased for the critical interaction force was discovered. The plasmon spectrum of excitations with root dispersion was discovered for a weak bounding force.

The main results obtained in the second part of this research can be formulated as follows. First, the human connectome (HC) demonstrates significant nonrandomness compared with the networks of animals. This can indicate that there is an evolution selection acting on the neural network. Second, the local clusterization mainly explains the shape of the connectome spectrum and hence plays an important role in the structure of the networks under study. Such a peculiar feature can be a consequence of the “module” hierarchical organization of the brain networks.

The result obtained in the third part of this research is, first, a system of equations regularizing the well-known Darcy equations in the sense that, in the stable case, the regularized system becomes the system of Darcy equations, and in the case of Rayleigh-Taylor instability, describes the development of the mushy region. Second, in the third part of the research, formal asymptotic solutions are obtained for nonstationary problems of flow past localized and periodic irregularities on a plate. An important result is that the Benjamin-Ono-type equation obtained by the proposed approach contains terms describing the problem geometry. Moreover, it turned out that this equation is an automatically satisfied relation, and the unknown function can itself be determined from the boundary condition for a system of Prandtl-type equations with induced pressure which describes the flow in the near-wall region.

In the fourth part of the research, the general Hamiltonian of the resonance hyperbolic trap with a pair of coinciding frequencies was investigated. It was shown that if the anharmonic part of the Hamiltonian of the hyperbolic trap is mirror symmetric, then the effective Hamiltonian has unstable equilibria and separatrixes determining separate classically admissible domains in the space of first integrals of an ideal trap. The corresponding stationary states of the confined charge can form biorbital states, i.e., such a state is localized on two distinct classical trajectories. The semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space was obtained. The semiclassial asymptotics of the tunnel splitting is represented in terms of complex periodic instantons. In the problem of a particle in a thin curved layer at the magneto-dimensional resonance, the interaction of induced currents was studied in general form, and for layers of cylindrical and parabolic shape. An example shows a layer shape for which the Hall effect is suppressed by the arising geometric currents and a layer in which the current is induced in the direction of the electric field applied to it.  

An algebraic method for reducing the problem with frequency resonance of general form to a problem with irreducible resonance was developed. The generators of the algebra of symmetries, irreducible representations, and coherent states were constructed in this case.

Level of implementation, recommendations on implementation or outcomes of the implementation of the results

The analysis of operation of 5- and 16-qubit quantum computers allowed one to develop heuristic models which permit deriving significant information even from the results of operation of a quantum computer which are distorted by the imperfection of the gate operation. The proposed realization of the Lamb dynamical effect in the connected system of a superconducting qubit-resonator can be obtained experimentally. The predicted effects of coordinate-dependent effective mass of photons and polaritons can also be verified in contemporary experiments. The considered systems of bosons with long-range repulsion can be implemented on the basis of several physical systems. 

The results obtained by investigating resonance quantum systems can be used to design and develop perspective nano-electronic devices.


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