arXiv subjects

A. Petukhov

Publications and source records attributed to A. Petukhov.

11 recordsLinked to original sources

arXiv2025

Low rank matrix completion and realization of graphs: results and problems

The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).

math.HO↗

arXiv2021

Accurately computing electronic properties of a quantum ring

A promising approach to study condensed-matter systems is to simulate them on an engineered quantum platform. However, achieving the accuracy needed to outperform classical methods has been an outstanding challenge. Here, using eighteen superconducting qubits, we provide an experimental blueprint for an accurate condensed-matter simulator and demonstrate how to probe fundamental electronic properties. We benchmark the underlying method by reconstructing the single-particle band-structure of a one-dimensional wire. We demonstrate nearly complete mitigation of decoherence and readout errors and arrive at an accuracy in measuring energy eigenvalues of this wire with an error of ~0.01 rad, whereas typical energy scales are of order 1 rad. Insight into this unprecedented algorithm fidelity is gained by highlighting robust properties of a Fourier transform, including the ability to resolve eigenenergies with a statistical uncertainty of 1e-4 rad. Furthermore, we synthesize magnetic flux and disordered local potentials, two key tenets of a condensed-matter system. When sweeping the magnetic flux, we observe avoided level crossings in the spectrum, a detailed fingerprint of the spatial distribution of local disorder. Combining these methods, we reconstruct electronic properties of the eigenstates where we observe persistent currents and a strong suppression of conductance with added disorder. Our work describes an accurate method for quantum simulation and paves the way to study novel quantum materials with superconducting qubits.

quant-ph↗

arXiv2021

Realizing topologically ordered states on a quantum processor

The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of $\ln2$, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results demonstrate the potential for quantum processors to provide key insights into topological quantum matter and quantum error correction.

quant-ph↗

arXiv2020

Direct measurement of non-local interactions in the many-body localized phase

The interplay of interactions and strong disorder can lead to an exotic quantum many-body localized (MBL) phase. Beyond the absence of transport, the MBL phase has distinctive signatures, such as slow dephasing and logarithmic entanglement growth; they commonly result in slow and subtle modification of the dynamics, making their measurement challenging. Here, we experimentally characterize these properties of the MBL phase in a system of coupled superconducting qubits. By implementing phase sensitive techniques, we map out the structure of local integrals of motion in the MBL phase. Tomographic reconstruction of single and two qubit density matrices allowed us to determine the spatial and temporal entanglement growth between the localized sites. In addition, we study the preservation of entanglement in the MBL phase. The interferometric protocols implemented here measure affirmative correlations and allow us to exclude artifacts due to the imperfect isolation of the system. By measuring elusive MBL quantities, our work highlights the advantages of phase sensitive measurements in studying novel phases of matter.

cond-mat.dis-nn↗

arXiv2020

Demonstrating a Continuous Set of Two-qubit Gates for Near-term Quantum Algorithms

Quantum algorithms offer a dramatic speedup for computational problems in machine learning, material science, and chemistry. However, any near-term realizations of these algorithms will need to be heavily optimized to fit within the finite resources offered by existing noisy quantum hardware. Here, taking advantage of the strong adjustable coupling of gmon qubits, we demonstrate a continuous two-qubit gate set that can provide a 3x reduction in circuit depth as compared to a standard decomposition. We implement two gate families: an iSWAP-like gate to attain an arbitrary swap angle, $θ$, and a CPHASE gate that generates an arbitrary conditional phase, $ϕ$. Using one of each of these gates, we can perform an arbitrary two-qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic Simulation, or fSim, gate set. We benchmark the fidelity of the iSWAP-like and CPHASE gate families as well as 525 other fSim gates spread evenly across the entire fSim($θ$, $ϕ$) parameter space achieving purity-limited average two-qubit Pauli error of $3.8 \times 10^{-3}$ per fSim gate.

quant-ph↗

arXiv2015

Frequency shifts and relaxation rates for spin 1/2 particles moving in electromagnetic fields

We discuss the behaviour of the Larmor frequency shift and the longitudinal relaxation rate due to non-uniform electromagnetic fields on an assembly of spin 1/2 particles, in adiabatic and nonadiabatic regimes. We also show some general relations between the various frequency shifts and between the frequency shifts and relaxation rates. The remarkable feature of all our results is that they were obtained without any specific assumptions on the explicit form of the correlation functions of the fields. Hence, we expect that our results are valid both for diffusive and ballistic regime of motion and arbitrary cell shapes and surface scattering. These results can then be applied to a wide variety of realistic systems.

physics.atom-ph↗

arXiv2015

Homogeneous projective varieties with semi-continuous rank function

Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this paper, we classify all equivariantly embedded homogeneous projective varieties $\mathbb X\subset\mathbb P(V)$ with tame secant varieties. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, $\mathbb X$ is the orbit in $\mathbb P(V)$ of a highest weight line in an irreducible representation $V$ of a reductive algebraic group $G$. Thus, our result is a list of all irreducible representations of reductive groups, where the resulting $\mathbb X$ has tame secant varieties.

math.AG↗

arXiv2014

Annihilators of highest weight $\frak{sl}(\infty)$-modules

We give a criterion for the annihilator in U$(\frak{sl}(\infty))$ of a simple highest weight $\frak{sl}(\infty)$-module to be nonzero. As a consequence we show that, in contrast with the case of $\frak{sl}(n)$, the annihilator in U$(\frak{sl}(\infty))$ of any simple highest weight $\frak{sl}(\infty)$-module is integrable, i.e., coincides with the annihilator of an integrable $\frak{sl}(\infty)$-module. Furthermore, we define the class of ideal Borel subalgebras of $\frak{sl}(\infty)$, and prove that any prime integrable ideal in U$(\frak{sl}(\infty))$ is the annihilator of a simple $\frak b^0$-highest weight module, where $\frak b^0$ is any fixed ideal Borel subalgebra of $\frak{sl}(\infty)$. This latter result is an analogue of the celebrated Duflo Theorem for primitive ideals.

math.RT↗

arXiv2011

Comment on "Pressure dependence of wall relaxation in polarized He^3 gaseous cells." by W. Zheng, H. Gao, Q. Ye, and Y.Zhang

The authors have demonstrated a strong linear pressure dependence of the longitudinal relaxation time for He^3 at room and cryogenic temperatures in a given experimental setup. They offer a theoretical explanation of the effect based on diffusion theory in the bulk and an unusual boundary condition. We question the physical basis of the boundary condition and suggest some alternate explanations of the observations.

physics.atom-ph↗

arXiv2011

Spectral properties of correlation functions of fields with arbitrary position dependence in restricted geometries from the ballistic to the diffusive regimes

The transition between ballistic and diffusive motion poses difficult problems in several fields of physics. In this work we show how to calculate the spectra of the correlation functions between fields of arbitrary spatial dependence as seen by particles moving through the fields in regions bounded by specularly reflecting walls valid for diffusive and ballistic motion as well as the transition region in between for motion in 2 and 3 dimensions. Applications to relaxation in nmr are discussed.

physics.data-an↗

arXiv2008

Metal-insulator transition and giant anisotropic magnetoresistance in ultra thin (Ga,Mn)As

MBE-grown, 5 nm-thick annealed Ga0.95Mn0.05As films with Tc~90K demonstrate transition from metallic to insulating state below To~10K, where sheet resistances Rsh~h/e2 and both longitudinal Rxx and transverse Rxy components become comparable. Below metal-insulator transition we found giant anisotropic magnetoresistance (GAMR), which depends on orientation of magnetization to crystallographic axes and manifests itself in positive magnetoresistance near 50% for Rxx at T=1.7K, H//[110] crystallographic direction and parallel to current in contrast to smaller and negative magnetoresistance for H// direction. We connect GAMR with anisotropic spin-orbit interaction resulting in formation of high- and low- resistance states with different localization along non-equivalent easy axes.

cond-mat.mtrl-sci↗