Searcharxiv⌕ Search

arXiv subjects

Alexander Pechen

Publications and source records attributed to Alexander Pechen.

At least 19 recordsLinked to original sources
arXiv2026

Diverse efficiency of observable optimization for four-level quantum systems with higher-order traps

In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a $V-V$ system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.

quant-ph↗
arXiv2026

Learning Hidden Structures in Open Quantum Dynamics

We introduce a machine-learning approach for identifying hidden structural features of open quantum dynamics under restricted experimental access. Unlike most existing data-driven methods which focus on detection or prediction of dynamical behavior, our framework targets the inference of invariant algebraic structures underlying the effective Markovian evolution. Measurement limitations, symmetries, and superselection rules are incorporated through a *-algebraic description of accessible observables. The learning problem is formulated as maximum-likelihood estimation from multi-time measurement sequences, where the algebraic type of an invariant subalgebra, particularly a decoherence-free subalgebra, is treated as a discrete structural hypothesis. The feasibility of the approach is illustrated on multiple synthetic models and a waveguide quantum electrodynamics system, where nontrivial intermediate algebraic structures are identified directly from measurement data.

quant-ph↗
arXiv2025

On the occasion of Dr. Ivan Dmitrievich Remizov's 40th birthday

This article celebrates the 40th anniversary of Dr. Ivan Dmitrievich Remizov, a mathematician who made a number of important contributions to the theory of one-parameter operator semigroups -- a branch of functional analysis which has applications to differential equations, mathematical physics, random processes, control theory, and quantum mechanics. Born on December 7, 1984, Dr. Remizov obtained his Ph.D. in 2018 from Moscow State University and has made substantial contributions, particularly in the study of Chernoff approximations of one-parameter semigroups of operators. This article reviews his academic background, research achievements, and his impact on the mathematical community.

math.HO↗
arXiv2025

Weak coupling limit for quantum systems with unbounded weakly commuting system operators

This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles in the dipole approximation. The free system Hamiltonian and the system part of the Hamiltonian describing interaction with the reservoir are considered as unbounded operators with continuous spectrum which are commuting in a weak sense. We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir which are non-zero in the WCL. Then we prove that the resulting reduced system dynamics converges to unitary dynamics (such behavior sometimes called as Quantum Cheshire Cat effect) with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian. We obtain exact form of the modified Hamiltonian and estimate the rate of convergence to the limiting dynamics. For Fermi reservoir, we prove the convergence of the full Dyson series. For Bose case the convergence is understood term by term.

math-ph↗
arXiv2024

Gradient projection method for constrained quantum control

In this work, we adopt the Gradient Projection Method (GPM) to problems of quantum control. For general $N$-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals, and provide the projection versions of the Pontryagin maximum principle and the GPM, all directly in terms of quantum objects such as evolution operator, Hamiltonians, density matrices, etc. Various forms of the GPM, including one- and two-step, are provided and compared. We formulate the GPM both for closed and open quantum systems, latter for the general case with simultaneous coherent and incoherent controls. The GPM is designed to perform local gradient based optimization in the case when bounds are imposed on the controls. The main advantage of the method is that it allows to exactly satisfy the bounds, in difference to other approaches such as adding constraints as weight to objective. We apply the GPM to several examples including generation of one- and two-qubit gates and two-qubit Bell and Werner states for models of superconducting qubits under the constraint when controls are zero at the initial and final times, and steering an open quantum system state to a target density matrix for simulating action of the Werner-Holevo channel, etc.

quant-ph↗
arXiv2024

Quantum channels, complex Stiefel manifolds, and optimization

Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known representations of quantum channels are described by Choi matrices and by Kraus operator-sum representation (OSR). As was shown before, one can use Kraus OSR to parameterize quantum channels by points of a suitable quotient of some complex Stiefel manifold by the action of the unitary group. In this work, we establish a homeomorphism between the topological space of quantum channels and the quotient of the complex Stiefel manifold. This homeomorphism can be applied to various quantum optimization problems. As an example, we apply it to the analysis of extrema points for a wide variety of quantum control objective functionals defined on the complex Stiefel manifolds, including mean value, generation of quantum gates, thermodynamic quantities involving entropy, etc. Finally, a metric on the space of quantum channels induced by the Riemannian metric on the Stiefel manifold is defined, and we show that it is a generalization of the Bures angle between density matrices.

quant-ph↗
arXiv2024

Some Aspects of Remote State Restoring in State Transfer Governed by XXZ-Hamiltonian

We consider the remote state restoring and perfect transfer of the zero-order coherence matrix (PTZ) in a spin system governed by the XXZ-Hamiltonian conserving the excitation number. The restoring tool is represented by several nonzero Larmor frequencies in the Hamiltonian. To simplify the analysis we use two approximating models including either step-wise or pulse-type time-dependence of the Larmor frequencies. Restoring in spin chains with up to 20 nodes is studied. Studying PTZ, we consider the zigzag and rectangular configurations and optimize the transfer of the 0-order coherence matrix using geometrical parameters of the communication line as well as the special unitary transformation of the extended receiver. Overall observation is that XXZ-chains require longer time for state transfer than XX-chains, which is confirmed by the analytical study of the evolution under the nearest-neighbor approximation. We demonstrate the exponential increase of the state-transfer time with the spin chain length.

quant-ph↗
arXiv2024

Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space

We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.

quant-ph↗
arXiv2024

Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives

This article is devoted to developing an approach for manipulating the von Neumann entropy $S(\rho(t))$ of an open two-qubit system with coherent control and incoherent control inducing time-dependent decoherence rates. The following goals are considered: (a) minimizing or maximizing the final entropy $S(\rho(T))$; (b) steering $S(\rho(T))$ to a given target value; (c) steering $S(\rho(T))$ to a target value and satisfying the pointwise state constraint $S(\rho(t)) \leq \overline{S}$ for a given $\overline{S}$; (d) keeping $S(\rho(t))$ constant at a given time interval. Under the Markovian dynamics determined by a Gorini--Kossakowski--Sudarshan--Lindblad type master equation, which contains coherent and incoherent controls, one- and two-step gradient projection methods and genetic algorithm have been adapted, taking into account the specifics of the objective functionals. The corresponding numerical results are provided and discussed.

quant-ph↗
arXiv2024

Phenomenon of a stronger trapping behaviour in $\Lambda$-type quantum systems with symmetry

$\Lambda$, $V$, $\Xi$ (ladder), and other three-level quantum systems with one forbidden transition (referred here as $\Lambda$-type systems) play an important role in quantum physics. Various applications require manipulation of such systems using as control shaped laser field. In this work, we study how degeneracy of energy states or of Bohr frequencies in these systems affects the efficiency or difficulty of finding optimal shape of the control field. For this, we adopt the notion of higher order traps, which was introduced in [A.N. Pechen and D.J. Tannor, Are there traps in quantum control landscapes? Phys. Rev. Lett. {\bf 106}, 120402 (2011)], where second/third order traps were discovered for $\Lambda$-atom with one forbidden transition and with non-degenerate energy levels. We theoretically study control of such systems with and without degeneracy in their eigenstates and Bohr frequencies, and investigate numerically using GRAPE and l-BFGS algorithms how this degeneracy influences on the efficiency of optimizing the control laser field. We find that the degeneracy of the Bohr frequencies in the $\Xi$ system, which makes the system energy levels symmetrically distributed, leads to the appearance of a seventh order trap with a more significant attracting domain resulting in a more difficult optimization, while the degeneracy of energy states in generic $\Lambda$-type systems does not lead to an increase of the order of the zero control trap compared to the non-degenerate case. We also find that when not only the Bohr frequencies are degenerate in the system $\Xi$, but also the dipole moments for the two allowed transitions coincide (in this case $\Xi$ system is not controllable), then true traps arise in the quantum control landscape. In particular, the constant zero control becomes a trap.

quant-ph↗
arXiv2024

Incoherent GRAPE (inGRAPE) for optimization of quantum systems with environmentally assisted control

In this work, we review several results on development and application of incoherent version of GRAPE (Gradient Ascent Pulse Engineering) approach, inGRAPE, to optimization for open quantum systems driven by both coherent and incoherent controls. In the incoherent control approach, the environment serves as a control together with coherent field, and decoherence rates become generally time-dependent. For a qubit, explicit analytic expressions for evolution of the density matrix were obtained by solving a cubic equation via Cardano method. We discuss applications of incoherent GRAPE method to high fidelity gate generation for open one- and two-qubit systems and surprising properties of the underlying control landscapes, forming two groups - smooth single peak landscapes for Hadamard, C-NOT and C-Z gates, and more complicated with two peaks for T (or $\pi/8$) gate. For a qutrit, a formulation of the environment-assisted incoherent control with time-dependent decoherence rates is provided.

quant-ph↗
arXiv2024

Quantum control by the environment: Turing uncomputability, Optimization over Stiefel manifolds, Reachable sets, and Incoherent GRAPE

The ability to control quantum systems is necessary for many applications of quantum technologies ranging from gate generation in quantum computation to NMR and laser control of chemical reactions. In many practical situations, the controlled quantum systems are open, i.e., interacting with the environment. While often influence of the environment is considered as an obstacle for controlling the systems, in some cases it can be exploited as a useful resource. In this note, we briefly review some results on control of open quantum systems using environment as a resource, including control by engineered environments and by non-selective measurements, Turing uncomputability of discrete quantum control, parametrization of Kraus maps by points of the Stiefel manifolds and corresponding Riemanninan optimization, control by dissipation and time-dependent decoherence rates, reachable sets, and incoherent GRAPE (Gradient Ascent Pulse Engineering) -- inGRAPE -- for gradient-based optimization.

quant-ph↗
arXiv2023

Amplification of quantum transfer and quantum ratchet

Amplification of quantum transfer and ratchet--type processes are important for quantum technologies. We also expect that quantum ratchet works in quantum photosynthesis, where possible role of quantum effects is now widely discussed but the underlying dynamical processes are still not clearly known. In this work, we study a model of amplification of quantum transfer and making it directed which we call the quantum ratchet model. The model is based on a special quantum control master equation with dynamics induced by a feedback-type process. The ratchet effect is achieved in the quantum control model with dissipation and sink, where the Hamiltonian depends on vibrations in the energy difference synchronized with transitions between energy levels. A similarity between this model and the model of coherent transport in quantum photosynthesis, where the time dependence of the Hamiltonian arises due to vibrons, is studied. Amplitude and frequency of the oscillating vibron together with the dephasing rate are the parameters of the quantum ratchet which determine its efficiency. We study with which parameters the quantum ratchet minimizes the exction recombination time and show that the experimentally known values of the parameters of the photosynthetic reaction center correspond to values of the parameters of the quantum ratchet which realize a local minimum of the exciton recombination time. We also find another values of the parameters of the quantum ratchet minimizing the exciton recombination time, which corresponds to a twice smaller frequency of the vibron compared to that observed in experiments.

quant-ph↗
arXiv2023

Generation of C-NOT, SWAP, and C-Z Gates for Two Qubits Using Coherent and Incoherent Controls and Stochastic Optimization

In this work, we consider a general form of the dynamics of open quantum systems determined by the Gorini-Kossakowsky-Sudarchhan-Lindblad type master equation with simultaneous coherent and incoherent controls with three particular forms of the two-qubit Hamiltonians. Coherent control enters in the Hamiltonian and incoherent control enters in both the Hamiltonian and the superoperator of dissipation. For these systems, we analyze the control problems of generating two-qubit C-NOT, SWAP, and C-Z gates using with piecewise constant controls and stochastic optimization in the form of an adapted version of the dual annealing algorithm. In the numerical experiment, we analyze the minimal infidelity obtained by the dual annealing for various values of strength of the interaction between the system and the environment.

quant-ph↗
arXiv2023

On reconstruction of states from evolution induced by quantum dynamical semigroups perturbed by covariant measures

In this work, we show the ability to restore states of quantum systems from evolution induced by quantum dynamical semigroups perturbed by covariant measures. Our procedure describes reconstruction of quantum states transmitted via quantum channels and as a particular example can be applied to reconstruction of photonic states transmitted via optical fibers. For this, the concept of perturbation by covariant operator-valued measure in a Banach space is introduced and integral representation of the perturbed semigroup is explicitly constructed. Various physically meaningful examples are provided. In particular, a model of the perturbed dynamics in the symmetric (boson) Fock space is developed as covariant measure for a semiflow of shifts and its perturbation in the symmetric Fock space, and its properties are investigated. Another example may correspond to the Koopman-von Neumann description of a classical oscillator with bounded phase space.

quant-ph↗
arXiv2023

Quantum control landscape for generation of $H$ and $T$ gates in an open qubit with both coherent and environmental drive

An important problem in quantum computation is generation of single-qubit quantum gates such as Hadamard ($H$) and $\pi/8$ ($T$) gates which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to decrease of the gate fidelity, in some cases it can be used as a resource. Here we consider the problem of optimal generation of $H$ and $T$ gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we study quantum control landscape which represents the behaviour of the infidelity as a functional of the controls. We consider three landscapes, with infidelities defined by steering between two, three (via Goerz-Reich-Koch approach), and four matrices in the qubit Hilbert space. We observe that for the $H$ gate, which is Clifford gate, for all three infidelities the distributions of minimal values obtained with gradient search have a simple form with just one peak. However, for $T$ gate which is a non-Clifford gate, the situation is surprisingly different - this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, that might indicate possible existence of two isolated minima in the control landscape. Important is that among these three infidelities only those defined with three and four matrices guarantee closeness of generated gate to a target and can be used as a good measure of closeness. We study sets of optimized solutions for this most general and not treated before case of coherent and incoherent controls acting together, and discover that they form submanifolds in the control space, and unexpected, in some cases two isolated submanifolds.

quant-ph↗
arXiv2023

Krotov Type Optimization of Coherent and Incoherent Controls for Open Two-Qubit Systems

This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini-Kossakowski-Sudarshan-Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert-Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert-Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous constrained controls and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap's values close to their upper estimates. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.

quant-ph↗
arXiv2023

Optimization of Time-Dependent Decoherence Rates and Coherent Control for a Qutrit System

The work considers an open qutrit system whose density matrix $\rho(t)$ evolution is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the superoperator of dissipation) controls. Incoherent control makes the decoherence rates depending on time in a specific controlled manner and within clear physical mechanics. We consider the problem of maximizing the Hilbert-Schmidt overlap between the system's final state $\rho(T)$ and a given target state $\rho_{\rm target}$ and the problem of minimizing the squared Hilbert-Schmidt distance between these states. For the both problems, we perform their realifications, derive the corresponding Pontryagin function, adjount system (with the two cases of transversality conditions in view of the two terminal objectives), and gradients of the objectives, adapt the one-, two-, three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze, first, the methods' operation and, second, the obtained control processes, in respect to considering the environment as a resource via incoherent control.

quant-ph↗