arXiv subjects

Sergey Denisov

Publications and source records attributed to Sergey Denisov.

At least 19 recordsLinked to original sources

arXiv2026

Error accumulation in dissipative quantum circuits

Present-day quantum processors are open systems in which dissipation and decoherence degrade the information carried by a quantum state as it propagates through a circuit. We study this degradation in dissipative random quantum circuits, modeling each two-qubit gate as a diluted unitary that interpolates between the intended unitary operation and a random dissipative quantum channel. Using the fidelity between the ideal and noisy output states as a diagnostic, we show that sufficiently random unitary gates or dissipative Kraus operators lead to a universal decay of the average fidelity. This decay is determined only by the dissipation strength, system size, and circuit depth, while microscopic details of the gates and noise, including the Kraus rank, appear only in subleading corrections to higher moments. We derive a closed-form expression for the average fidelity in terms of three physically meaningful error parameters, dissipation strength, coherent two-qubit gate-error strength, and connectivity-error probability. Finally, we identify the cases when dissipative effects can be reproduced by an unitary-noise model and when dissipation-induced decoherence remains distinguishable from coherent noise at the level of the average fidelity.

quant-ph↗

arXiv2026

What We Talk About When We Talk About Dissipative Quantum Chaos

Dissipative quantum chaos is an emerging theory that is expected to extend the ideas, concepts, and methodology of conventional Hamiltonian quantum chaos from coherent evolution to open quantum dynamics. The new theory should provide a set of tools to distinguish chaotic open quantum systems from integrable ones, as well as quantitative measures of their chaoticity (or, conversely, integrability). The foundations of this theory were laid in the late 1980s, and from the very start it was clear that, like its Hamiltonian predecessor, it had to be based on the spectral properties of the operators governing open quantum evolution. After these first steps, the field remained relatively quiet for many years and it is only over the last decade that the development of dissipative quantum chaos has received a strong boost, as confirmed by a large number of publications on this topic and, very recently, the first experiments performed to test its theoretical predictions. In this chapter, we review these recent developments and outline the basic foundations of dissipative quantum chaos.

quant-ph↗

arXiv2025

$k$-Positive Maps: New Characterizations and a Generation Method

We study $k$-positive linear maps on matrix algebras and address two problems, (i) characterizations of $k$-positivity and (ii) generation of non-decomposable $k$-positive maps. On the characterization side, we derive optimization-based conditions equivalent to $k$-positivity that (a) reduce to a simple check when $k=d$, (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for $k<d$), and (c) recast $k$-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single $k$-positive map, produces one-parameter families that remain $k$-positive and non-decomposable for small enough times. We illustrate this by generating such families for $d=3$ and $d=4$. We also formulate a semi-definite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying $k$-positivity and a systematic way to sample $k$-positive non-decomposable maps.

quant-ph↗

arXiv2025

Spectra of noisy parameterized quantum circuits: Single-Ring universality

Random unitaries are an important resource for quantum information processing. While their universal properties have been thoroughly analyzed, it is not known what happens to these properties when the unitaries are sampled on the present-day noisy intermediate-scale quantum (NISQ) computers. We implement parameterized circuits, which have been proposed as a means to generate random unitaries, on an IBM Quantum processor and model these implementations as quantum maps. To retrieve the maps, a machine-learning assisted tomography is used. We find the spectrum of a map to be either an annulus or a disk depending on the circuit depth and detect an annulus-disk transition. By their spectral properties, the retrieved maps appear to be very similar to a recently introduced ensemble of random maps, for which spectral densities can be analytically evaluated. Our results establish, via Dissipative Quantum Chaos theory, a connection between intrinsic properties of present-day noisy intermediate-scale quantum (NISQ) computing platforms and non-Hermitian random matrix theory.

quant-ph↗

arXiv2025

Experimental Detection of Dissipative Quantum Chaos

More than four decades of research on chaos in isolated quantum systems have led to the identification of universal signatures -- such as level repulsion and eigenstate thermalization -- that serve as cornerstones in our understanding of complex quantum dynamics. The emerging field of dissipative quantum chaos explores how these properties manifest in open quantum systems, where interactions with the environment play an essential role. We report the first experimental detection of dissipative quantum chaos and integrability by measuring the complex spacing ratios (CSRs) of open many-body quantum systems implemented on a high-fidelity superconducting quantum processor. Employing gradient-based tomography, we retrieve a ``donut-shaped'' CSR distribution for chaotic dissipative circuits, a hallmark of level repulsion in open quantum systems. For an integrable circuit, spectral correlations vanish, evidenced by a sharp peak at the origin in the CSR distribution. As we increase the depth of the integrable dissipative circuit, the CSR distribution undergoes an integrability-to-chaos crossover, demonstrating that intrinsic noise in the quantum processor is a dissipative chaotic process. Our results reveal the universal spectral features of dissipative many-body systems and establish present-day quantum computation platforms, which are predominantly used to run unitary simulations, as testbeds to explore dissipative many-body phenomena.

quant-ph↗

arXiv2025

The dynamical law behind eye movements: distinguishing between Lévy and intermittent strategies

Foraging is a complex spatio-temporal process which is often described with stochastic models. Two particular ones, Lévy walks (LWs) and intermittent search (IS), became popular in this context. Researchers from the two communities, each advocating for either Lévy or intermittent approach, independently analyzed foraging patterns and reported agreement between empirical data and the model they used. We resolve this Lévy-intermittent dichotomy for eye-gaze trajectories collected in a series of experiments designed to stimulate free foraging for visual information. By combining analytical results, statistical quantifiers, and basic machine learning techniques, we devise a method to score the performance of the models when they are used to approximate an individual gaze trajectory. Our analysis indicates that the intermittent search model consistently yields higher scores and thus approximates the majority of the eye-gaze trajectories better.

cond-mat.stat-mech↗

arXiv2025

Dissipation-Induced Threshold on Integrability Footprints

The presence of a dissipative environment substantially disrupts unitary evolution. Nevertheless, key features of the evolution, such as its integrable or chaotic nature, are not immediately erased by dissipation. To understand this, we model environment-induced dissipation as a convex combination of unitary evolution and a random Kraus map, and examine how signatures of integrability gradually fade with increasing dissipation strength. Our analysis shows that in the weakly dissipative regime, the complex eigenvalue spectrum organizes into well-defined, high-density clusters. We estimate the critical dissipation threshold beyond which these clusters disappear, rendering the dynamics indistinguishable from chaotic evolution. This threshold depends only on the number of spectral clusters and the rank of the Kraus map. To characterize this transition, we introduce the eigenvalue angular velocity as a diagnostic of integrability loss. We illustrate our findings using several integrable quantum circuits, including the dissipative quantum Fourier transform. Our results provide a quantitative picture of how noise gradually erases the footprints of unitary integrability in open quantum systems.

quant-ph↗

arXiv2025

Quantum Liouvillian Tomography

Characterization of near-term quantum computing platforms requires the ability to capture and quantify dissipative effects. This is an inherently challenging task, as these effects are multifaceted, spanning a broad spectrum from Markovian to strongly non-Markovian dynamics. We introduce Quantum Liouvillian Tomography (QLT), a protocol to capture and quantify non-Markovian effects in time-continuous quantum dynamics. The protocol leverages gradient-based quantum process tomography to reconstruct dynamical maps and utilizes regression over the derivatives of Pauli string probability distributions to extract the Liouvillian governing the dynamics. We benchmark the protocol using synthetic data and quantify its accuracy in recovering Hamiltonians, jump operators, and dissipation rates for two-qubit systems. Finally, we apply QLT to analyze the evolution of an idling two-qubit system implemented on a superconducting quantum platform to extract characteristics of Hamiltonian and dissipative components and, as a result, detect inherently non-Markovian dynamics. Our work introduces the first protocol capable of retrieving generators of generic open quantum evolution from experimental data, thus enabling more precise characterization of many-body non-Markovian effects in near-term quantum computing platforms.

quant-ph↗

arXiv2025

Spectral delineation of Markov Generators: Classical vs Quantum

The celebrated theorem of Perron and Frobenius implies that spectra of classical Markov operators, represented by stochastic matrices, are restricted to the unit disk. This property holds also for spectra of quantum stochastic maps (quantum channels), which describe quantum Markovian evolution in discrete time. Moreover, the spectra of stochastic $N \times N$ matrices are additionally restricted to a subset of the unit disk, called Karpeleviuc region, the shape of which depends on $N$. We address the question of whether the spectra of generators, which induce Markovian evolution in continuous time, can be bound in a similar way. We propose a rescaling that allows us to answer this question affirmatively. The eigenvalues of the rescaled classical generators are confined to the modified Karpeleviuc regions, whereas the eigenvalues of the rescaled quantum generators fill the entire unit disk.

cond-mat.stat-mech↗

arXiv2023

Random Lindblad operators obeying detailed balance

We introduce different ensembles of random Lindblad operators $\cal L$, which satisfy quantum detailed balance condition with respect to the given stationary state $σ$ of size $N$, and investigate their spectral properties. Such operators are known as `Davies generators' and their eigenvalues are real; however, their spectral densities depend on $σ$. We propose different structured ensembles of random matrices, which allow us to tackle the problem analytically in the extreme cases of Davies generators corresponding to random $σ$ with a non-degenerate spectrum for the maximally mixed stationary state, $σ= \mathbf{1} /N$. Interestingly, in the latter case the density can be reasonably well approximated by integrating out the imaginary component of the spectral density characteristic to the ensemble of random unconstrained Lindblad operators. The case of asymptotic states with partially degenerated spectra is also addressed. Finally, we demonstrate that similar universal properties hold for the detailed balance-obeying Kolmogorov generators obtained by applying superdecoherence to an ensemble of random Davies generators. In this way we construct an ensemble of random classical generators with imposed detailed balance condition.

cond-mat.stat-mech↗

arXiv2023

Random sparse generators of Markovian evolution and their spectral properties

The evolution of a complex multi-state system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter $φ$, which is the number of non-zero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with $D$ vertices (number of system states) and $2 φD$ edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of non-sparse random generators. We show that the first moment of the eigenvalue distribution scales as $\sim φ$, while its variance is $\sim \sqrtφ$. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions, and clarify the behavior of the spectral gap as a function of $D$. Finally, we analyze complex spacing ratio statistics of ultra-sparse generators, $φ= \mathrm{const}$, and find that starting already at $φ\geqslant 2$, spectra of the generators exhibit universal properties typical of Ginibre's Orthogonal Ensemble.

cond-mat.stat-mech↗

arXiv2023

Improved Differential Privacy for SGD via Optimal Private Linear Operators on Adaptive Streams

Motivated by recent applications requiring differential privacy over adaptive streams, we investigate the question of optimal instantiations of the matrix mechanism in this setting. We prove fundamental theoretical results on the applicability of matrix factorizations to adaptive streams, and provide a parameter-free fixed-point algorithm for computing optimal factorizations. We instantiate this framework with respect to concrete matrices which arise naturally in machine learning, and train user-level differentially private models with the resulting optimal mechanisms, yielding significant improvements in a notable problem in federated learning with user-level differential privacy.

cs.LG↗

arXiv2021

High-frequency expansions for time-periodic Lindblad generators

Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be non-trivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet generator (describing the stroboscopic evolution) is either again Markovian and of Lindblad type or not. On the other hand, even though in the high-frequency regime a Lindbladian Floquet generator (Floquet Linbladian) is numerically found to exist, this behaviour is, curiously, not correctly reproduced within analytical high-frequency expansions. Here, we demonstrate that a proper Floquet Lindbladian can still be obtained from a high-frequency expansion, when treating the problem in a suitably chosen rotating frame. Within this approach, we can then also describe the transition to a phase at lower driving frequencies, where no Floquet Lindbladian exists, and show that the emerging non-Markovianity of the Floquet generator can entirely be attributed to the micromotion of the open driven system.

quant-ph↗

arXiv2021

Asymptotic densities of planar Lévy walks: a non-isotropic case

Lévy walks are a particular type of continuous-time random walks which results in a super-diffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a new unidirectional motion event either in the positive or in the negative direction. We consider two-dimensional generalization of Lévy walks in the form of the so-called XY-model. It describes a particle moving with a constant velocity along one of the four basic directions and randomly switching between them when starting a new motion event. We address the ballistic regime and derive solutions for the asymptotic density profiles. The solutions have a form of first-order integrals which can be evaluated numerically. For specific values of parameters we derive an exact expression. The analytic results are in perfect agreement with the results of finite-time numerical samplings.

cond-mat.stat-mech↗

arXiv2020

Quasi-stationary states of game-driven systems: a dynamical approach

Evolutionary game theory is a framework to formalize the evolution of collectives ("populations") of competing agents that are playing a game and, after every round, update their strategies to maximize individual payoffs. There are two complementary approaches to modeling evolution of player populations. The first addresses essentially finite populations by implementing the apparatus of Markov chains. The second assumes that the populations are infinite and operates with a system of mean-field deterministic differential equations. By using a model of two antagonistic populations, which are playing a game with stationary or periodically varying payoffs, we demonstrate that it exhibits metastable dynamics that is reducible neither to an immediate transition to a fixation (extinction of all but one strategy in a finite-size population) nor to the mean-field picture. In the case of stationary payoffs, this dynamics can be captured with a system of stochastic differential equations and interpreted as a stochastic Hopf bifurcation. In the case of varying payoffs, the metastable dynamics is much more complex than the dynamics of the means.

nlin.AO↗

arXiv2020

Transforming the Lindblad Equation into a System of Linear Equations: Performance Optimization and Parallelization of an Algorithm

With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper we address master equations of the Lindblad form, which are a popular theoretical tool in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell-Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. This allows us to use standard high-performance parallel algorithms to integrate the equations and thus to emulate open quantum dynamics in a computationally efficient way. Recently we presented an implementation of the transform with the computational complexity scaling as $O(N^5 log N)$ for dense Lindbaldians and $O(N^3 log N)$ for sparse ones. However, infeasible memory costs remain a serious obstacle on the way to large models. Here we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension $N=2000$ and a dense random Lindbladian model of the dimension $N=200$ by using $25$ nodes with $64$ GB RAM per node.

physics.comp-ph↗

arXiv2020

Continuity of weighted operators, Muckenhoupt $A_p$ weights, and Steklov problem for orthogonal polynomials

We consider weighted operators acting on $L^p(\mathbb{R}^d)$ and show that they depend continuously on the weight $w\in A_p(\mathbb{R}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w(\mathbb{T})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2(\mathbb{T})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.

math.CA↗

arXiv2019

Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Let $μ$ be a measure from Szegő class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $μ$. In this paper, we prove a version of the classical Szegő's formula which controls the oscillation of $f_n$ on $\mathbb T$ for all $n \ge 0$. Then, we focus on an analog of Lusin's conjecture for polynomials $\{φ_n\}$ orthogonal with respect to measure $μ$ and prove that pointwise convergence of $\{|φ_n|\}$ almost everywhere on $\mathbb T$ is equivalent to a certain condition on zeroes of $φ_n$.

math.CV↗