arXiv subjects

A. V. Lebedev

Publications and source records attributed to A. V. Lebedev.

At least 19 recordsLinked to original sources

arXiv2026

Noise structuring in fixed-depth Trotter simulation: stationary channels and observable-level depolarization

We analyze fixed-depth Trotter simulation as a method for structuring hardware noise in digital many-body dynamics. The number of layers is chosen using the largest endpoint time and is then kept fixed throughout the time scan, making the total noise dose approximately independent of the endpoint time. For local stochastic faults, we show that, once propagated faults lose memory of their insertion layer, the noisy circuit factorizes into ideal evolution followed by a stationary finite-depth binomial channel. In the dilute-layer limit, this channel reduces to a Poissonian exponential. The memory time of a single fault is related to a Loschmidt echo. An important consequence is observable-level depolarization: for selected macroscopic observables at low to moderate noise levels, the stationary channel can act as an almost time-independent affine contrast correction, even though the full channel need not be depolarizing, which is crusial for error mitigation purposes. At short times, the same protocol produces a digital Zeno-like transient, in which a fixed number of noise opportunities competes with a vanishing coherent angle per layer. Our results also reveal limitations of naive zero-noise extrapolatin strategies based on oversimplified functions.

quant-ph↗

arXiv2026

Spectral theory of energy-selective quantum search with Ising Hamiltonian phase oracles

We develop an exact spectral-response theory for the Grover-type iterate $W_T=D_\xi\exp(-\ii T H)$, in which the evolution generated by a diagonal Ising Hamiltonian is used directly as a continuous phase oracle. An energy-grouped recurrence and its generating-function solution show how the empirical characteristic function determines the position, width, height, and saturation time of an energy-selective resonance. For an annealed Gaussian density of states, a high-density-tail resonance containing $M$ configurations is reached after $\Theta(\sqrt{2^n/M})$ oracle calls with success probability $\Theta(1)$, giving a quadratic query improvement over independent uniform sampling with classical energy evaluation. For correlated random Ising spectra, overlap-dependent covariances lead to a realization-dependent resonance shift with root-mean-square scale $O(n^{-3})$, parametrically larger than the resonance width, and can also reduce the peak height. The shift is both an algorithmic detuning and a coherent probe of sample-specific spectral fluctuations whose ensemble statistics reflect the Ising overlap structure. Spectral symmetrization and iterative calibration can remove or compensate the resonance-center displacement for prescribed-energy targeting. We also clarify the relation to designed spectral filters and the precision and coherence requirements of this asymptotic primitive.

quant-ph↗

arXiv2024

Dynamical quantum Ansatz tree approach for the heat equation

Quantum computers can be used for the solution of various problems of mathematical physics. In the present paper, we consider a discretized version of the heat equation and address its solution on quantum computer using variational Anzats tree approach (ATA). We extend this method originally proposed for the system of linear equations to tackle full time dependent heat equation. The key ingredients of our method are (i) special probabilistic quantum circuit in order to add heat sources to temperature distribution, (ii) limiting auxiliary register in the preparation of quantum state, (iii) utilizing a robust cluster of repetitive nodes in the anzats tree structure. We suggest that our procedure provides an exponential speedup compared to the classical algorithms in the case of time dependent heat equation.

quant-ph↗

arXiv2023

On $C^*$-algebras associated to transfer operators for countable-to-one maps

Our initial data is a transfer operator $L$ for a continuous, countable-to-one map $φ:Δ\to X$ defined on an open subset of a locally compact Hausdorff space $X$. Then $L$ may be identified with a `potential', i.e. a map $\varrho:Δ\to X$ that need not be continuous unless $φ$ is a local homeomorphism. We define the crossed product $C_0(X)\rtimes L$ as a universal $C^*$-algebra with explicit generators and relations, and give an explicit faithful representation of $C_0(X)\rtimes L$ under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver $C^*$-algebras of Muhly and Tomforde, $C^*$-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid $C^*$-algebras associated to Deaconu-Renault groupoids. We describe spectra of core subalgebras of $C_0(X)\rtimes L$ and use it to characterise simplicity of $C_0(X)\rtimes L$ and prove the uniqueness theorem for $C_0(X)\rtimes L$. We give efficient criteria for $C_0(X)\rtimes L$ to be a Kirchberg algebra, and we discuss relationship between KMS states on the core subalgebra of $C_0(X)\rtimes L$ and conformal measures for $φ$.

math.OA↗

arXiv2023

Depth analysis of variational quantum algorithms for heat equation

Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution are finite difference schemes, which can be reduced to the linear algebra problem. We consider three approaches to solve the heat equation on a quantum computer. Using the direct variational method we minimize the expectation value of a Hamiltonian with its ground state being the solution of the problem under study. Typically, an exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speed up to be achieved. The Hadamard test based approach solves this problem, however, the performed simulations do not evidently prove that the ansatz circuit has a polynomial depth with respect to the number of qubits. The ansatz tree approach exploits an explicit form of the matrix what makes it possible to achieve an advantage over classical algorithms. In our numerical simulations with up to $n=11$ qubits, this method reveals the exponential speed up.

quant-ph↗

arXiv2023

Hadamard and Vandermonde determinants and Bernoulli-Euler-Lagrange-Aitken-Nikiporets type numerical method for roots of polynomials

In the article we develop Euler-Lagrange method and calculate all the roots of an arbitrary complex polynomial $P(z)$ on the base of calculation (similar to the Bernoulli-Aitken-Nikiporets methods) of the limits of ratios of Hadamard determinants built by means of coefficients of expansions into Taylor and Laurent series of the function~$\frac{P'(z)}{P(z)}$.

math.CA↗

arXiv2023

Renormalization flow of a weak extended backscattering Hamiltonian in a non-chiral Tomonaga-Luttinger liquid

We consider a non-chiral Luttinger liquid in the presence of a backscattering Hamiltonian which has an extended range. Right/left moving fermions at a given location can thus be converted as left/right moving fermions at a different location, within a specific range. We perform a momentum shell renormalization group treatment which gives the evolution of the relative degrees of freedom of this Hamiltonian contribution under the renormalization flow, and we study a few realistic examples of this extended backscattering Hamiltonian. We find that, for repulsive Coulomb interaction in the Luttinger liquid, any such Hamiltonian contribution evolves into a delta-like scalar potential upon renormalization to a zero temperature cutoff. On the opposite, for attractive couplings, the amplitude of this kinetic Hamiltonian is suppressed, rendering the junction fully transparent. As the renormalization procedure may have to be stopped because of experimental constraints such as finite temperature, we predict the actual spatial shape of the kinetic Hamiltonian at different stages of the renormalization procedure, as a function of the position and the Luttinger interaction parameter, and show that it undergoes structural changes. This renormalized kinetic Hamiltonian has thus to be used as an input for the perturbative calculation of the current, for which we provide analytic expressions in imaginary time. We discuss the experimental relevance of this work by looking at one-dimensional systems consisting of carbon nanotubes or semiconductor nanowires.

cond-mat.str-el↗

arXiv2022

Critical phase boundary and finite-size fluctuations in Su-Schrieffer-Heeger model with random inter-cell couplings

A dimerized fermion chain, described by Su-Schrieffer-Heeger (SSH) model, is a well-known example of 1D system with a non-trivial band topology. An interplay of disorder and topological ordering in the SSH model is of a great interest owing to experimental advancements in synthesized quantum simulators. In this work, we investigate a special sort of a disorder when inter-cell hopping amplitudes are random. Using a definition for $\mathbb{Z}_2$-topological invariant $ν\in \{ 0; 1\}$ in terms of a non-Hermitian part of the total Hamiltonian, we calculate $\langleν\rangle$ averaged by random realizations. This allows to find (i) an analytical form of the critical surface that separates phases of distinct topological orders and (ii) finite size fluctuations of $ν$ for arbitrary disorder strength. Numerical simulations of the edge modes formation and gap suppression at the transition are provided for finite-size system. In the end, we discuss a band-touching condition derived within the averaged Green function method for a thermodynamic limit.

quant-ph↗

arXiv2021

Analysis of relationships between spectral potential of transfer operators, $t$-entropy, entropy and topological pressure

The paper is devoted to the analysis of relationships between principal objects of the spectral theory of dynamical systems (transfer and weighted shift operators) and basic characteristics of information theory and thermodynamic formalism (entropy and topological pressure). We present explicit formulae linking these objects with $t$-entropy and spectral potential. Herewith we uncover the role of inverse rami-rate, forward entropy along with essential set and the property of non-contractibility of a dynamical system.

math.FA↗

arXiv2021

Linear Ascending Metrological Algorithm

The ubiquitous presence of shot noise sets a fundamental limit to the measurement precision in classical metrology. Recent advances in quantum devices and novel quantum algorithms utilizing interference effects are opening new routes for overcoming the detrimental noise tyranny. However, further progress is limited by the restricted capability of existing algorithms to account for the decoherence pervading experimental implementations. Here, adopting a systematic approach to the evaluation of effectiveness of metrological procedures, we devise the Linear Ascending Metrological Algorithm (LAMA), which offers a remarkable increase in precision in the demanding situation where a decohering quantum system is used to measure a continuously distributed variable. We introduce our protocol in the context of magnetic field measurements, assuming superconducting transmon devices as sensors operated in a qudit mode. Our findings demonstrate a quantum-metrological procedure capable of mitigating detrimental dephasing and relaxation effects.

quant-ph↗

arXiv2020

Heat generation due to the Anderson catastrophe in mesoscopic devices

Anderson's orthogonality catastrophe (AOC) theorem establishes that the ground state of the many-body fermion system is asymptotically orthogonal to the ground state of the same system perturbed by a scattering potential, so that the overlap between the original and new ground states decays to zero with the system size. We adopt the AOC for a description of heat production in a complementary metal-oxide-semiconductor (CMOS) transistor. We find that the heat released in the transistor comprises two distinct components, contribution from the dissipation accompanying electron transmission under the applied voltage and purely quantum-mechanical AOC part due to the change in scattering matrix for electrons upon switching between high and low conductance regimes. We calculate the AOC-induced heat production, which we call switching heat.

cond-mat.mes-hall↗

arXiv2019

Time-reversal of an unknown quantum state

For decades, researchers have sought to understand how the irreversibility of the surrounding world emerges from the seemingly time symmetric, fundamental laws of physics. Quantum mechanics conjectured a clue that final irreversibility is set by the measurement procedure and that the time reversal requires complex conjugation of the wave function, which is overly complex to spontaneously appear in nature. Building on this Landau-Wigner conjecture, it became possible to demonstrate that time reversal is exponentially improbable in a virgin nature and to design an algorithm artificially reversing a time arrow for a given quantum state on the IBM quantum computer. However, the implemented arrow-of-time reversal embraced only the known states initially disentangled from the thermodynamic reservoir. Here we develop a procedure for reversing the temporal evolution of an arbitrary unknown quantum state. This opens the route for general universal algorithms sending temporal evolution of an arbitrary system backwards in time.

quant-ph↗

arXiv2019

H-theorem for Systems with an Interaction Invariant Distribution Function

H-theorem gives necessary conditions for a system to evolve in time with a non-diminishing entropy. In a quantum case the role of H-theorem plays the unitality criteria of a quantum channel transformation describing the evolution of the system's density matrix under the presence of the interaction with an environment. Here, we show that if diagonal elements of the system's density matrix are robust to the presence of interaction the corresponding quantum channel is unital.

quant-ph↗

arXiv2019

Variational principles for spectral radius of weighted endomorphisms of $C(X,D)$

We give formulas for the spectral radius of weighted endomorphisms $aα: C(X,D)\to C(X,D)$, $a\in C(X,D)$, where $X$ is a compact Hausdorff space and $D$ is a unital Banach algebra. Under the assumption that $α$ generates a partial dynamical system $(X,φ)$, we establish two kinds of variational principles for $r(aα)$: using linear extensions of $(X,φ)$ and using Lyapunov exponents associated with ergodic measures for $(X,φ)$. This requires considering (twisted) cocycles over $(X,φ)$ with values in an arbitrary Banach algebra $D$, and thus our analysis can not be reduced to any of mutliplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with $α: C(X,D)\to C(X,D)$. In particular, they are far reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin and others. They are most efficient when $D=\mathcal{B}(F)$, for a Banach space $F$, and endomorphisms of $\mathcal{B}(F)$ induced by $α$ are inner isometric. As a by product we obtain a dynamical variational principle for an arbitrary operator $b\in \mathcal{B}(F)$ and that it's spectral radius is always a Lyapunov exponent in some direction $v\in F$, when $F$ is reflexive.

math.FA↗

arXiv2019

Sup-sums principles for F-divergence, Kullback--Leibler divergence, and new definition for t-entropy

The article presents new sup-sums principles for integral F-divergence for arbitrary convex function F and arbitrary (not necessarily positive and absolutely continuous) measures. As applications of these results we derive the corresponding sup-sums principle for Kullback--Leibler divergence and work out new `integral' definition for t-entropy explicitly establishing its relation to Kullback--Leibler divergence.

math.ST↗

arXiv2019

Realization of the Werner-Holevo and Landau-Streater quantum channels for qutrits on quantum computers

We realize Landau-Streater (LS) and Werner-Holevo (WH) quantum channels for qutrits on the IBM quantum computers. These channels correspond to interaction between the qutrit and its environment that result in the globally unitarily covariant qutrit transformation violating multiplicativity of the maximal $p$-norm. Our realization of LS and WH channels is based on embedding qutrit states into states of two qubits and using single-qubit and two-qubit CNOT gates to implement the specific interaction. We employ the standard quantum gates hence the developed algorithm suits any quantum computer. We run our algorithm on a 5-qubit and a 20-qubit computer as well as on a simulator. We quantify the quality of the implemented channels comparing their action on different input states with theoretical predictions. The overall efficiency is quantified by fidelity between the theoretical and experimental Choi states implemented on the 20-qubit computer.

quant-ph↗

arXiv2018

Extended quantum Maxwell demon acting over macroscopic distances

A quantum Maxwell demon is a device that can lower the entropy of a quantum system by providing it with purity. The functionality of such a quantum demon is rooted in a quantum mechanical SWAP operation exchanging mixed and pure states. We describe the setup and performance of a quantum Maxwell demon that purifies an energy-isolated system from a distance. Our cQED-based design involves two transmon qubits, where the mixed-state target qubit is purified by a pure-state demon qubit connected via an off-resonant transmission line; this configuration naturally generates an iSWAP gate. Although less powerful than a full SWAP gate, we show that assuming present-day performance characteristics of a cQED implementation, such an extended quantum Maxwell demon can purify the target qubit over macroscopic distances on the order of meters and tolerates elevated temperatures of the order of a few Kelvin in the transmission line.

quant-ph↗

arXiv2018

Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom

Phase estimation algorithms are key protocols in quantum information processing. Besides applications in quantum computing, they can also be employed in metrology as they allow for fast extraction of information stored in the quantum state of a system. Here, we implement two suitably modified phase estimation procedures, the Kitaev- and the semiclassical Fourier-transform algorithms, using an artificial atom realized with a superconducting transmon circuit. We demonstrate that both algorithms yield a flux sensitivity exceeding the classical shot-noise limit of the device, allowing one to approach the Heisenberg limit. Our experiment paves the way for the use of superconducting qubits as metrological devices which are potentially able to outperform the best existing flux sensors with a sensitivity enhanced by few orders of magnitude.

quant-ph↗