arXiv subjects

Vladislav Gennadievich Malyshkin

Publications and source records attributed to Vladislav Gennadievich Malyshkin.

At least 19 recordsLinked to original sources

arXiv2026

Semidefinite Programming for Quantum Channel Learning

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.

cs.LG↗

arXiv2026

Superstate Quantum Mechanics

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.

quant-ph↗

arXiv2026

Trade Execution Flow as the Underlying Source of Market Dynamics

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 the data 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.

q-fin.CP↗

arXiv2025

Quantum Channel Learning

The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $ρ^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} ω^{(l)} F\left(\varrho^{(l)},\sum_s B_s ρ^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,σ)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (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 $A^{OUT}=\sum_s |w_s|^2 \mathcal{U}_s A^{IN} \mathcal{U}_s^{\dagger}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $ϕ_l=\mathcal{U} ψ_l$ to a quantum channel, what 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 $\varrho^{(l)}=\mathcal{U} ρ^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{ρ^{(l)}} \to \sqrt{\varrho^{(l)}}$ 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.

cs.LG↗

arXiv2024

Partially Unitary Learning

The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|ψ\right\rangle$ and $OUT$ of $\left|ϕ\right\rangle$ based on a set of wavefunction measurements (within a phase) $ψ_l \to ϕ_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\sum_{l=1}^{M} ω^{(l)} \left|\langleϕ_l|\mathcal{U}|ψ_l\rangle\right|^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). The constructed operator $\mathcal{U}$ can be considered as an $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension $\dim(OUT) \times \dim(IN)$ transforming operators as $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. 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.

cs.LG↗

arXiv2024

On Supercapacitors Time-Domain Spectroscopy. C/R Characteristic Slope

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 $τ$, and measuring first $\int Idt$ and second $\int I^2dt$ moments of current along with the potential before and after shorting. The effective $C(τ)$ and $R(τ)$ are then obtained from charge preservation and energy dissipation invariants. A linear behavior in $[R(τ),C(τ)]$ parametric plot is observed by several orders of $τ$. This gives a $C/R$ characteristic slope: how much $ΔC$ we can ``gain'' if we are ready to ``lose'' $ΔR$ 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.

physics.chem-ph↗

arXiv2022

On Machine Learning Knowledge Representation In The Form Of Partially Unitary Operator. Knowledge Generalizing Operator

A new form of ML knowledge representation with high generalization power is developed and implemented numerically. Initial $\mathit{IN}$ attributes and $\mathit{OUT}$ class label are transformed into the corresponding Hilbert spaces by considering localized wavefunctions. A partially unitary operator optimally converting a state from $\mathit{IN}$ Hilbert space into $\mathit{OUT}$ Hilbert space is then built from an optimization problem of transferring maximal possible probability from $\mathit{IN}$ to $\mathit{OUT}$, this leads to the formulation of a new algebraic problem. Constructed Knowledge Generalizing Operator $\mathcal{U}$ can be considered as a $\mathit{IN}$ to $\mathit{OUT}$ quantum channel; it is a partially unitary rectangular matrix of the dimension $\mathrm{dim}(\mathit{OUT}) \times \mathrm{dim}(\mathit{IN})$ transforming operators as $A^{\mathit{OUT}}=\mathcal{U} A^{\mathit{IN}} \mathcal{U}^{\dagger}$. Whereas only operator $\mathcal{U}$ projections squared are observable $\left\langle\mathit{OUT}|\mathcal{U}|\mathit{IN}\right\rangle^2$ (probabilities), the fundamental equation is formulated for the operator $\mathcal{U}$ 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.

cs.LG↗

arXiv2022

Market Directional Information Derived From (Time, Execution Price, Shares Traded) Sequence of Transactions. On The Impact From The Future

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.

q-fin.CP↗

arXiv2021

On The Radon-Nikodym Spectral Approach With Optimal Clustering

Problems of interpolation, classification, and clustering are considered. In the tenets of Radon--Nikodym approach $\langle f(\mathbf{x})ψ^2 \rangle / \langleψ^2\rangle$, where the $ψ(\mathbf{x})$ is a linear function on input attributes, all the answers are obtained from a generalized eigenproblem $|f|ψ^{[i]}\rangle = λ^{[i]} |ψ^{[i]}\rangle$. The solution to the interpolation problem is a regular Radon-Nikodym derivative. The solution to the classification problem requires prior and posterior probabilities that are obtained using the Lebesgue quadrature[1] technique. Whereas in a Bayesian approach new observations change only outcome probabilities, in the Radon-Nikodym approach not only outcome probabilities but also the probability space $|ψ^{[i]}\rangle$ change with new observations. This is a remarkable feature of the approach: both the probabilities and the probability space are constructed from the data. The Lebesgue quadrature technique can be also applied to the optimal clustering problem. The problem is solved by constructing a Gaussian quadrature on the Lebesgue measure. A distinguishing feature of the Radon-Nikodym approach is the knowledge of the invariant group: all the answers are invariant relatively any non-degenerated linear transform of input vector $\mathbf{x}$ components. A software product implementing the algorithms of interpolation, classification, and optimal clustering is available from the authors.

cs.LG↗

arXiv2020

On The Inverse Relaxation Approach To Supercapacitors Characterization

A novel inverse relaxation technique for supercapacitor characterization is developed, modeled numerically, and experimentally tested on a number of commercial supercapacitors. It consists in shorting a supercapacitor for a short time $τ$, then switching to the open circuit regime and measuring an initial rebound and long-time relaxation. The results obtained are: the ratio of "easy" and "hard" to access capacitance and the dependence $C(τ)$, that determines what the capacitance the system responds at time-scale $τ$; it can be viewed as an alternative to used by some manufacturers approach to characterize a supercapacitor by fixed capacitance and time-scale dependent internal resistance. Among the advantages of proposed technique is that it does not require a source of fixed current, what simplifies the setup and allows a high discharge current regime. The approach can be used as a replacement of low-frequency impedance measurements and the ones of IEC 62391 type, it can be effectively applied to characterization of supercapacitors and other relaxation type systems with porous internal structure. The technique can be completely automated by a microcontroller to measure, analyze, and output the results.

physics.app-ph↗

arXiv2020

On Numerical Estimation of Joint Probability Distribution from Lebesgue Integral Quadratures

An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$- and $g$- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density-matrix correlation. Examples of the density-matrix correlation can be a value-correlation $V_{f^{[i]};g^{[j]}}$, similar to a regular correlation concept, and a new one, a probability-correlation $P_{f^{[i]};g^{[j]}}$. If Christoffel function average is used instead of regular average the approach can be extended to an estimation of joint probability of three and more random processes. The theory is implemented numerically; the software is available under the GPLv3 license.

math.NA↗

arXiv2020

On Lebesgue Integral Quadrature

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$-point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$-point discrete distribution, producing the Lebesgue integral. Mathematically, the problem is reduced to a generalized eigenvalue problem: Lebesgue quadrature value-nodes are the eigenvalues and the corresponding weights are the square of the averaged eigenvectors. A numerical estimation of an integral as the Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value-nodes) and the probability of the outcome (weight). For this reason, it is especially well-suited for the study of non-Gaussian processes. The software implementing the theory is available from the authors.

math.NA↗

arXiv2019

Market Dynamics: On Directional Information Derived From (Time, Execution Price, Shares Traded) Transaction Sequences

A new approach to obtaining market--directional information, based on a non-stationary solution to the dynamic equation "future price tends to the value that maximizes the number of shares traded per unit time" [1] is presented. In our previous work[2], we established that it is the share execution flow ($I=dV/dt$) and not the share trading volume ($V$) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow $I$ (the dynamic impact) than to the traded volume $V$ (the regular impact). In this paper, an important advancement is achieved: we define the "scalp-price" ${\cal P}$ as the sum of only those price moves that are relevant to market dynamics; the criterion of relevance is a high $I$. Thus, only "follow the market" (and not "little bounce") events are included in ${\cal P}$. Changes in the scalp-price defined this way indicate a market trend change - not a bear market rally or a bull market sell-off; the approach can be further extended to non-local price change. The software calculating the scalp--price given market observations triples (time, execution price, shares traded) is available from the authors.

q-fin.TR↗

arXiv2019

Market Dynamics. On A Muse Of Cash Flow And Liquidity Deficit

A first attempt at obtaining market--directional information from a non--stationary solution of the dynamic equation "future price tends to the value that maximizes the number of shares traded per unit time" [1] is presented. We demonstrate that the concept of price impact is poorly applicable to market dynamics. Instead, we consider the execution flow $I=dV/dt$ operator with the "impact from the future" term providing information about not--yet--executed trades. The "impact from the future" on $I$ can be directly estimated from the already--executed trades, the directional information on price is then obtained from the experimentally observed fact that the $I$ and $p$ operators have the same eigenfunctions (the exact result in the dynamic impact approximation $p=p(I)$). The condition for "no information about the future" is found and directional prediction quality is discussed. This work makes a substantial contribution toward solving the ultimate market dynamics problem: find evidence of existence (or proof of non--existence) of an automated trading machine which consistently makes positive P\&L on a free market as an autonomous agent (aka the existence of the market dynamics equation). The software with a reference implementation of the theory is provided.

q-fin.TR↗

arXiv2018

Generalized Radon--Nikodym Spectral Approach. Application to Relaxation Dynamics Study

Radon--Nikodym approach to relaxation dynamics, where probability density is built first and then used to calculate observable dynamic characteristic is developed and applied to relaxation type signals study. In contrast with $L^2$ norm approaches, such as Fourier or least squares, this new approach does not use a norm, the problem is reduced to finding the spectrum of an operator (virtual Hamiltonian), which is built in a way that eigenvalues represent the dynamic characteristic of interest and eigenvectors represent probability density. The problems of interpolation (numerical estimation of Radon--Nikodym derivatives is developed) and obtaining the distribution of relaxation rates from sampled timeserie are considered. Application of the theory is demonstrated on a number of model and experimentally measured timeserie signals of degradation and relaxation processes. Software product, implementing the theory is developed.

math.NA↗

arXiv2016

Mathematical Foundations of Realtime Equity Trading. Liquidity Deficit and Market Dynamics. Automated Trading Machines

We postulates, and then show experimentally, that liquidity deficit is the driving force of the markets. In the first part of the paper a kinematic of liquidity deficit is developed. The calculus-like approach, which is based on Radon--Nikodym derivatives and their generalization, allows us to calculate important characteristics of observable market dynamics. In the second part of the paper this calculus is used in an attempt to build a dynamic equation in the form: future price tend to the value maximizing the number of shares traded per unit time. To build a practical automated trading machine P&L dynamics instead of price dynamics is considered. This allows a trading automate resilient to catastrophic P&L drains to be built. The results are very promising, yet when all the fees and trading commissions are taken into account, are close to breakeven. In the end of the paper important criteria for automated trading systems are presented. We list the system types that can and cannot make money on the market. These criteria can be successfully applied not only by automated trading machines, but also by a human trader.

q-fin.CP↗

arXiv2016

Market Dynamics vs. Statistics: Limit Order Book Example

Commonly used limit order book attributes are empirically considered based on NASDAQ ITCH data. It is shown that some of them have the properties drastically different from the ones assumed in many market dynamics study. Because of this difference we propose to make a transition from "Statistical" type of order book study (typical for academics) to "Dynamical" type of study (typical for market practitioners). Based on market data analysis we conclude, that most of market dynamics information is contained in attributes with spikes (e.g. executed trades flow $I=dv/dt$), there is no any "stationary case" on the market and typical market dynamics is a "fast excitation and then slow relaxation" type of behavior with a wide distribution of excitation frequencies and relaxation times. A computer code, providing full depth order book information and recently executed trades is available from authors [1].

q-fin.TR↗

arXiv2016

Market Dynamics. On Supply and Demand Concepts

The disbalance of Supply and Demand is typically considered as the driving force of the markets. However, the measurement or estimation of Supply and Demand at price different from the execution price is not possible even after the transaction. An approach in which Supply and Demand are always matched, but the rate $I=dv/dt$ (number of units traded per unit time) of their matching varies, is proposed. The state of the system is determined not by a price $p$, but by a probability distribution defined as the square of a wavefunction $ψ(p)$. The equilibrium state $ψ^{[H]}$ is postulated to be the one giving maximal $I$ and obtained from maximizing the matching rate functional $ /<ψ^2(p)>$, i.e. solving the dynamic equation of the form "future price tend to the value maximizing the number of shares traded per unit time". An application of the theory in a quasi--stationary case is demonstrated. This transition from Supply and Demand concept to Liquidity Deficit concept, described by the matching rate $I$, allows to operate only with observable variables, and have a theory applicable to practical problems.

econ.GN↗