The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), Semidefinite Programming (SDP) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of SDP is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available SDP solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized.

The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ρ^{(l)}→ϱ^{(l)}, l=1⋯M, is formulated as an optimization problem maximizing the total fidelity F=∑_{l=1}^{M}ω^{(l)}F(ϱ^{(l)},∑_{s}B_{s}ρ^{(l)}B_{s}^{†}) subject to probability preservation constraints on Kraus operators B_{s}. For F(ϱ,σ) in the form that total fidelity can be represented as a quadratic form with superoperator F=∑_{s}〈B_{s}|S|B_{s}〉 (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: (1) IN/OUT states are represented as density matrices; (2) the mapping itself is formulated as a mixed unitary quantum channel A^{OUT}=∑_{s}|w_{s}|^{2}U_{s}A^{IN}U_{s}^{†} (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states ϕ_{l}=Uψ_{l} to a quantum channel, which allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ϱ^{(l)}=Uρ^{(l)}U^{†}, in this case a quadratic on U fidelity can be constructed by considering sqrt[ρ^{(l)}]→sqrt[ϱ^{(l)}] mapping, and on a quantum channel, where quadratic on B_{s} fidelity is an approximation-a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. A software product implementing the algorithm is available from the authors.

In this work, we demonstrate experimentally that the execution flow, I = dV/dt, is the fundamental driving force of market dynamics. We develop a numerical framework to calculate execution flow from sampled moments using the Radon-Nikodym derivative. A notable feature of this approach is its ability to automatically determine thresholds that can serve as actionable triggers. The technique also determines the characteristic time scale directly from the corresponding eigenproblem. The methodology has been validated on actual market data to support these findings. Additionally, we introduce a framework based on the Christoffel function spectrum, which is invariant under arbitrary non-degenerate linear transformations of input attributes and offers an alternative to traditional principal component analysis (PCA), which is limited to unitary invariance.

We introduce Superstate Quantum Mechanics (SQM), a theory that considers states in Hilbert space subject to multiple quadratic constraints, with “energy” also expressed as a quadratic function of these states. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization with energy expressed as a quadratic form involving the Hamiltonian. When SQM represents states as unitary operators, the stationary problem becomes a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. Any stationary SQM problem is equivalent to a new algebraic problem that we address in this paper. The non-stationary SQM problem considers the evolution of the system itself, involving the same “energy” operator as in the stationary case. Two possible options for the SQM dynamic equation are considered: (1) within the framework of linear maps from higher-order quantum theory, where 2D-type quantum circuits transform one quantum system into another; and (2) in the form of a Gross-Pitaevskii-type nonlinear map. Although no known physical process currently describes such 2D dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithms. As an immediately available practical application of the theory, we consider using a quantum channel as a classical computational model; this type of computation can be performed on a classical computer.

Unusual quasi-two-dimensional crystals of a regular triangular shape, self-formed in the process of obtaining a coordination polymer based on phenazine and silver, are described and studied. X-ray diffraction studies were carried out, the interplanar distance was determined, and the spectra of Raman scattering were obtained. A mechanism is proposed that can cause the appearance of triangular crystals from nuclei of hexagonal symmetry.

A novel time-domain technique for supercapacitor characterization is developed, modeled numerically, and experimentally tested on a number of commercial supercapacitors. The method involves momentarily shorting a supercapacitor for a brief duration, denoted as $\tau$, and measuring first $\int Idt$ and second $\int I^2dt$ moments of current along with the potential before and after shorting. The effective $C(\tau)$ and $R(\tau)$ are then obtained from charge preservation and energy dissipation invariants. A linear behavior in $[R(\tau),C(\tau)]$ parametric plot is observed by several orders of $\tau$. This gives a $C/R$ characteristic slope: how much $\Delta C$ we can ``gain'' if we are ready to ``lose'' $\Delta R$ in internal resistance. The $C/R$ characteristic slope characterizes possible energy and power properties of the device in terms of materials and technology used, this is a measure of supercapacitor perfection. The technique has been proven with experimental measurements and then validated through computer modeling, analytic analysis, and impedance spectroscopy on a number of circuit types: transmission line, binary tree, etc., a new n-tree element (nTE) is introduced. The approach offers an alternative to low-frequency impedance spectroscopy and methods outlined in the IEC 62391 standard. It provides valuable insights into the performance and characteristics of supercapacitors.

Multiple instability was found on the volt-ampere characteristic of the palladium-surface-oxidized indium phosphide structure. The effect is recorded when recording the dependence of differential conductivity and differential capacitance on the applied external voltage. A mechanism for the appearance of instabilities is proposed.. Keywords: palladium, VAC, instability, impedancometry.

The problem of an optimal mapping between Hilbert spaces IN of |ψ〉 and OUT of |ϕ〉 based on a set of wavefunction measurements (within a phase) ψ_{l}→ϕ_{l}, l=1,⋯,M, is formulated as an optimization problem maximizing the total fidelity ∑_{l=1}^{M}ω^{(l)}|〈ϕ_{l}|U|ψ_{l}〉|^{2} subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT)×dim(IN) transforming operators as A^{OUT}=UA^{IN}U^{†}. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.

The application of an additional nanoparticle layer is a common practice for enhancing the optical and electrical properties of third-generation solar cells. In this study, we present the results of impedance spectroscopy (IS) for modified solar cells using Nyquist and Bode diagrams. The structure investigated consists of a conventional double junction based on crystalline silicon (c-Si) coated with thin films of inorganic perovskite nanocrystals (NC) of lead halides CsPbI3 and CsPbBr3. The latter are characterized by a significant phonon disorder, which leads to unique electron-phonon interactions and dielectric responses. The IS results indicate that, under the same conditions, the measured Nyquist plots align well with the simulated ones. An equivalent circuit model is proposed, featuring ohmic resistance, recombination resistance, and geometric capacitance. These elements arise due to charge accumulation, charge transfer resistance, and/or additional interfacial electronic states. The study finds that the introduction of a CsPbI3 layer enhances the photoresponse under bias conditions, but this photoresponse leads to a decrease in DC conductivity. In contrast, the addition of a CsPbBr3 layer obstructs the photoresponse under bias while slightly improving the photoresponse in the absence of an applied voltage. The results obtained contribute to the improvement of tandem solar cell characteristics featuring top layers of perovskite nanocrystals.

Raman scattering spectra of linear carbon chains (carbines) localized in thin gold films of variable thickness are investigated. It is shown that the integral line is inhomogeneous, and separate components are identified, the intensity of which depends in a non-trivial way on the thickness of the film. Qualitative explanations of the detected effects are proposed.

Raman scattering spectra of linear carbon chains (carbines) localized in thin gold films of variable thickness are investigated. It is shown that the integral line is inhomogeneous, and separate components are identified, the intensity of which depends in a non-trivial way on the thickness of the film. Qualitative explanations of the detected effects are proposed. Keywords: carbine, linear allotropy of carbon, localization on clusters.

An attempt to obtain market directional information from non-stationary solution of the dynamic equation: "future price tends to the value maximizing the number of shares traded per unit time" is presented. A remarkable feature of the approach is an automatic time scale selection. It is determined from the state of maximal execution flow calculated on past transactions. Both lagging and advancing prices are calculated.

Multiple instability was found on the volt-ampere characteristic of the palladium-surface-oxidized indium phosphide structure. The effect is recorded when recording the dependence of differential conductivity and differential capacitance on the applied external voltage. A mechanism for the appearance of instabilities is proposed.

The deposition of an additional layer of nanoparticles is a widely used method for improving the optical and electrical characteristics of semiconductor solar cells (SCs). In this work, films of nanocrystals (NC) of inorganic perovskites of lead halides CsPbI3 and CsPbBr3 are deposited on the surface of a solar cell based on crystalline silicon (c-Si). It is shown that the optical properties of such NC films are in good agreement with the optical properties of c-Si. It has been found that the absorption coefficient of solar cells with NC layers of inorganic perovskites is much higher in the visible region of the spectrum, which increases the photocurrent generation in the SC in the range of 370–900 nm. A significant effect of surface roughness on the photoelectric properties of solar cells has been found. CsPbI3 NC films have a textured surface and higher photocurrent than CsPbBr3 NC films, which are rougher. Enhanced photovoltaic properties of FE structures with a CsPbI3 NC layer compared to CsPbBr3 NC films due to their lower degree of roughness were observed.

A new form of ML knowledge representation with high generalization power is developed and implemented numerically. Initial 𝐼𝑁 attributes and 𝑂𝑈𝑇 class label are transformed into the corresponding Hilbert spaces by considering localized wavefunctions. A partially unitary operator optimally converting a state from 𝐼𝑁 Hilbert space into 𝑂𝑈𝑇 Hilbert space is then built from an optimization problem of transferring maximal possible probability from 𝐼𝑁 to 𝑂𝑈𝑇, this leads to the formulation of a new algebraic problem. Constructed Knowledge Generalizing Operator 𝒰 can be considered as a 𝐼𝑁 to 𝑂𝑈𝑇 quantum channel; it is a partially unitary rectangular matrix of the dimension dim(𝑂𝑈𝑇) ×dim(𝐼𝑁) transforming operators as A^𝑂𝑈𝑇=𝒰 A^𝐼𝑁𝒰^†. Whereas only operator 𝒰 projections squared are observable ⟨𝑂𝑈𝑇|𝒰|𝐼𝑁⟩^2 (probabilities), the fundamental equation is formulated for the operator 𝒰 itself. This is the reason of high generalizing power of the approach; the situation is the same as for the Schrödinger equation: we can only measure ψ^2, but the equation is written for ψ itself.

A new type of moving average is developed. Whereas a regular moving average (e.g. of price) has a built-in internal time scale (time-window, exponential weight, etc.), the moving average developed in this paper has the weight as the product of a polynomial by window factor. The polynomial is the square of a wavefunction obtained from an eigenproblem corresponding to other observable (e.g. execution flow I=dV/dt , the number of shares traded per unit time). This allows to obtain an immediate "switch" without lagging typical for regular moving average.

Unusual quasi-two-dimensional crystals of a regular triangular shape, self-formed in the process of obtaining a coordination polymer based on phenazine and silver, are described and studied. X-ray diffraction studies were carried out, the interplanar distance was determined, and the spectra of Raman scattering were obtained. A mechanism is proposed that can cause the appearance of triangular crystals from nuclei of hexagonal symmetry.

The article substantiates the need for the development of solar power plants in the energy system of various countries. Currently known solar cells have limited band gaps which makes their efficiency not very high. It has been observed that in long-term operation, degradation is the most important factor limiting the efficiency of photovoltaic cells, which is already low. Various factors caused by nature that reduce the efficiency of solar energy systems are mentioned. So, this work intends to present a method for efficient analysis of the photovoltaic cells degradation data. The necessity of developing methods for modeling the efficiency of a solar power plant for the photovoltaic cells degradation study has been proved. The calculation results for the rates of Si photovoltaic cells degradation are presented for different countries. It was noted that the main factor in the degradation process formation is manufacturing technology of photovoltaic cells.

A novel problem of quadratic form optimization with multiple constraints of the quadratic form type is studied. This type of problem arises in quantitative finance, dynamic system identification, and data science. A numerical iteration algorithm (a generalized eigenvalue problem is solved on every step) is developed and tested numerically.

A problem of timeserie data analysis is considered. Whereas standard approaches such as Kalman filter are of linear quadratic estimation (LQE) type, the technique developed in this paper views system dynamics as a sequence of unitary transformations. The approach consists of two steps: 1. Convert the sequence of vector $\mathrm{x}^{(l)}$ observations $l=1\ldots M$ to a sequence of localized at $\mathrm{x}^{(l)}$ states $\vert \psi_{\mathrm{x}(l)}\rangle$ . 2. Find unitary operator $\mathcal{U}$ ‘optimally converting $\vert \psi_{\mathrm{x}(l+1)}\}=\vert \mathcal{U}\vert \psi_{\mathrm{x}^{(l)}}\rangle;$ : the problem is reduced to finding the maximum of a quadratic form on $U$ matrix elements subject to constraints that are quadratic forms on $U$ matrix elements as well. The approach is outlier-stable and can be applied to the processes with spikes and non-Gaussian noise. The approach is gauge-invariant, e.g. the result is the same when arbitrary non-degenerate linear transform is applied to input vector $\mathrm{x}^{(l)}$ components.