Number Of Necessary Measurements Research Articles

Adaptive Quantum State Tomography with Active Learning

Recently, tremendous progress has been made in the field of quantum science and technologies: different platforms for quantum simulation as well as quantum computing, ranging from superconducting qubits to neutral atoms, are starting to reach unprecedentedly large systems. In order to benchmark these systems and gain physical insights, the need for efficient tools to characterize quantum states arises. The exponential growth of the Hilbert space with system size renders a full reconstruction of the quantum state prohibitively demanding in terms of the number of necessary measurements. Here we propose and implement an efficient scheme for quantum state tomography using active learning. Based on a few initial measurements, the active learning protocol proposes the next measurement basis, designed to yield the maximum information gain. We apply the active learning quantum state tomography scheme to reconstruct different multi-qubit states with varying degree of entanglement as well as to ground states of the XXZ model in 1D and a kinetically constrained spin chain. In all cases, we obtain a significantly improved reconstruction as compared to a reconstruction based on the exact same number of measurements and measurement configurations, but with randomly chosen basis configurations. Our scheme is highly relevant to gain physical insights in quantum many-body systems as well as for benchmarking and characterizing quantum devices, e.g. for quantum simulation, and paves the way for scalable adaptive protocols to probe, prepare, and manipulate quantum systems.

Problems of informatization of rural schools within the framework of the National Project “Education”

The education system is a significant component of the socio-economic development of any state, since it directly affects the development and formation of human capital. The special role of the school in rural areas is not one of just an educational organization, it is primarily an “anchor” for the preservation of rural settlements, a symbol that the village has prospects and a future. The purpose of the presented study is to assess the state of information and communication technologies (ICT) in rural schools, taking into account the impact of the National Project (NP) “Education”. The main research methods were methods of comparative analysis, data comparison, exponential smoothing. As a result of the conducted research, a forecast was made of the number of students in the general education system in rural areas and school institutions in rural areas at the beginning of the 2024/2025 academic year. The forecast revealed the risk of not achieving the targets of NP “Education” in rural areas, including in the field of ICT. The practical significance of the research results lies in the fact that the obtained forecast results allow us to take a number of necessary measures to prevent possible risks of not achieving the indicators. It is necessary to review the mechanisms for assessing the effectiveness and achievability of the indicators laid down in the NP, increase funding and the pace of creating new school places in rural areas, increase the availability of modern information resources and equipment.

Антикризове управління як протидія банкрутству: досвід українських підприємств мікрокредитування

The proposed scientific article is devoted to the topical issue of the need for crisis management in the microcrediting market. The article considers the crisis (resulting from the full-scale invasion) as a factor influencing the smooth functioning of the company. The importance and relevance of crisis management of the company in conditions of uncertainty during the wartime is emphasized. Analyzing and summarizing scientific works of domestic and foreign scholars, it was proposed to study crisis management through the prism of the stages of crisis development, i. e.: crisis prevention, crisis management, overcoming the consequences of the crisis. Each of the stages requires a number of necessary measures that should be applied by the company for effective crisis management. Among the main ones of these measures are: early diagnosis of the crisis; rapid response to crises; implementation of preventive measures; creation of an anti-crisis program; implementation of post-crisis strategies. The article analyzes crisis management during the wartime on the example of a representative of the microcrediting market – LLC «MoneyVEO Quick Financial Assistance». The major anti-crisis measures of the company include: improvement of IT infrastructure; setting up automated and well-coordinated work of top management; attraction of external financial resources; compliance with legal requirements; cost reduction; improvement of the system for assessing the creditworthiness of counter-parties. An analysis of the main financial indicators of the company was carried out, on the basis of which a conclusion was made about the effectiveness of crisis management during the wartime. The probability of bankruptcy remains high, but there are opportunities to bypass it and bring the company out of the crisis. Prospects for further research in this direction are the need to apply preventive crisis management instruments and improve the mechanism of strategic crisis management.

Assessment of Complex Signal Parameters Effect on the Energy Concealment Degree

The article deals with the problem of obtaining a qualitative assessment of the energy concealment degree of radio signals. Energy concealment is regarded as the most significant form of signal concealment, since a designer of radio transmitting devices has the most influence and the greatest freedom in defining these properties. The purpose of the article is to systematize the available knowledge about the methods of assessment of energy concealment in order to conduct a mutual comparison of the characteristics of various signals. The authors describe and analyze a number of methods for assessment of energy concealment of signals based on the number of necessary measurements of the signal level range of the signal reconnaissance, as well as probabilistic assessment of signal disclosure. The results of the analysis showed that described methods are not suitable for obtaining a comparative assessment. Based on the conclusions made in the study energy concealment assessment methods, the author's method for assessment the degree of energy concealment of signals in terms of transmitted symbol energy is proposed. The article clarifies the application of the method to assess the parameters of chirp signals, narrowband and wideband FM, BPSK, QPSK, as well as quadrature signals. A comparative assessment of the chirp, FM, BPSK, QPSK and QAM-16 signal parameters for the decimeter VHF band is obtained, taking into account the transmission speed and power. Assessment of energy concealment of signals were obtained at typical information transfer rates. On the basis of the results, some regularities were described that make it possible to evaluate the effect of changing individual parameters of a radio transmitting device on final energy concealment.

URBAN PLANNING METHODS FOR PROTECTING TERRITORIES FROM NOISE

The purpose of the article is to study the features of the formation of noise loads depending on the nature of development along the road network. Consideration of different typologies of development depending on the characteristics of the passage of the street and road network and the impact on the environmental load on the urban area.
 The assessment of the acoustic load of the noise level of inter-main areas is an important step in the work of city planners, it determines the functionality of the use of the territory in the future, the number of necessary measures to protect the territory from noise. All this increases the role of assessment methods used in the analysis and selection of the most effective method of noise protection, the importance of urban planning aspects of the impact on the quality of the living environment of the population in the city.

ПРОБЛЕМИ РОЗВИТКУ СТРАХУВАННЯ ЖИТТЯ В УКРАЇНІ ТА ШЛЯХИ ЇХ ВИРІШЕННЯ

Today, insurance is a source of enrichment for the state's economy, not just an instrument of financial protection. The article examines the essence of life insurance and the relevance of life insurance in Ukraine. Life insurance is insurance that protects the property interests of the insured person related to his life and death. In order to insure the life of any person, you need the presence of insurance interest in it from the insured. The analysis of the insurance market is investigated. The insurance market showed a moderate increase in the volume of eligible assets and insurance reserves. At the same time, the assets of life insurers are constantly growing, despite the corona crisis. As for the assets of Non-life insurers, there is a slight reduction. The volume of total assets of insurers, the structure of accepted assets, insurance premiums and insurance payments of insurance companies, the amount of paid insurance payments are analyzed. The structure of eligible assets consists of more than 50% of bank deposits. This is due to both profitability and a detailed study of the supervision of the encumbrance and availability of such deposits and not including them in eligible assets. Stably high share in eligible assets belongs to real estate, which is partly represented by land and residential real estate, which are regularly revalued. The life insurance market is not large and has a high concentration, so a list of the largest risk insurers by gross premiums, life insurers by premiums, life insurers by benefits. At the same time, at the moment there are a number of problems in the Ukrainian insurance market that do not allow this type of activity to develop at the same level as in developed countries. The solution of the selected problems is possible only by combining the efforts of the state and insurance companies. A number of necessary measures have been proposed to increase the efficiency of life insurance in Ukraine. Namely: to enable insurance companies to participate in solving social problems, the state should increase the requirements for insurance companies and the necessary interesting and attractive insurance programs that would meet the needs of the population, would respond to new social problems.

Improving the accuracy and efficiency of quantum connected moments expansions * *This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce

The still-maturing noisy intermediate-scale quantum technology faces strict limitations on the algorithms that can be implemented efficiently. In the realm of quantum chemistry, the variational quantum eigensolver (VQE) algorithm has become ubiquitous, with many variations. Alternatively, a promising new avenue has been unraveled by the quantum variants of techniques grounded on expansions of the moments of the Hamiltonian, notably the connected moments expansion (CMX) and the Peeters–Devreese–Soldatov (PDS) energy functional. Common to those approaches is that, upon preparing an approximate ground state used to compute the necessary moments, the accuracy of the estimated ground state energy depends on the degree of overlap between the prepared state and the true ground state. Thus, we use the ADAPT-VQE algorithm to test shallow circuit construction strategies for the purpose of increasing the overlap with the exact ground state, validated by the sizable accuracy improvement herein reported in the PDS and CMX ground state energies. We also show that we can take advantage of the fact that the terms to be measured are highly recurring in different moments, incurring a substantial reduction in the number of necessary measurements. By coupling this measurement caching with a threshold that determines whether a given term is to be measured based on its associated scalar coefficient, we observe a further reduction in the number of circuit implementations while allowing for tunable accuracy.

Number of independent measurements required to obtain reliable mean scattering properties of irregular particles having a small size parameter, using microwave analogy measurements

Laboratory measurements of light scattered by a cloud of randomly oriented levitating particles are often used to interpret remote sensing measurements of dust in space and in Earth's atmosphere. It is necessary to know how many particles or how many different orientations of the same particles must be considered to retrieve the mean scattering function of brightness and polarization. New laboratory measurements were conducted using the microwave analogy method between frequencies of 3 to 18 GHz, where an "analog" particle with a small size parameter in a range of 0.5-12 will have a size of several cm. Twelve such "analog" particles from compact shapes to aggregates with small fractal dimensions were fabricated by additive manufacturing (3D printing) and were studied. The number of necessary measurements to reach the mean scattering properties of a particle with an accuracy of about 5% is obtained for less than 20 different orientations. To reach a 1.5% (1-σ) error in brightness and a 0.5% (1-σ) error in polarization, the number of necessary measurements is in a range of 20 to 70, depending on the shape, fluffiness, deviation from a perfect sphere, and surface irregularities of the particle. These results show that several tens of randomly oriented particles of the same size are sufficient to retrieve the mean light scattering properties. Also, several tens of orientations of the same particles provide mean scattering properties, compared to modelling calculations using the Finite Element Method, for an aggregate composed of identical monomers.

THEORETICAL ASPECT OF STRUCTURIZATION OF NATIONAL ECONOMY IN TRANSFORMATION CONDITIONS

The article investigates the theoretical aspects of structuring the national economy in transformational conditions. It is substantiated that the process of transformation is a continuous qualitative expansion of the boundaries of changes in the system of development of the national economy. It is proved that the indicator of adaptability of state regulation of the national economy reflects the scale of adaptation, as well as the mechanism itself, which adjusts to changing segments of the economy at the macro and meso levels. This indicator reveals the possibility of the process of large-scale regulation of the national environment model to get used to the new conditions. It was found that economic adaptation is the adaptation of the mechanism of regulation of the components of the national economy to the dynamic conditions of the external environment. Segmentation of structural changes provides an opportunity to conclude that as a result of adaptation the mechanism of regulation of the macro-level structure forms a qualitatively new system that meets modern requirements and trends of globalization. It is proved that the mechanism of state influence by means of methods of transformation and adaptation of market levers provides long-term and stable development of the national macroeconomic environment. The formation of a balanced system of state regulation is based on solving tasks aimed at achieving strategic goals of sustainable socio-economic development and improving the welfare of the population, taking into account globalization trends.The methodological basis of the article is a set of methods, including methods of scientific cognition, analysis and synthesis, systematization and scientific abstraction. The scientific basis of the conducted researches is the scientific work of domestic and foreign scientists in the field of state institutional regulation of open economy in modern transformational conditions.The processes of structuring the national economy in transformational conditions take place in a permanent turbulent environment, which requires a comprehensive study of complex, multilateral and multilevel interdependencies of structural segments of the system of state regulation. It should be noted that a number of important reforms have been carried out recently and a significant number of necessary measures have been implemented, which reflects the adaptive and stabilizing state of the macroeconomic environment. Under such transformational conditions, the potential of the investment climate can increase. In addition, the solution of certain issues of budget, tax and customs regulations allows for a steady increase in the level of economic security of the state, which in turn has a significant impact on the state of the national economy in the context of globalization.The principles aimed at ensuring effective state regulation of the national economy in the context of globalization are defined.

Human capital as a fundamental determinant of rural development

The Krasnoyarsk territory is a vast area. According to the Federal State Service of the Statistic the territory population density is 1.2 people per 1 square km. Thus it makes some special demands on the quality of human capital, and it is human capital which makes it possible to develop both the rural territory and the entire economy as a whole. The problem is that human capital has suffered a lack of attention for many years. Among the ways to overcome this situation we see a number of necessary measures aimed at increasing wages for agricultural workers, improving the demographic situation, diversifying the areas of employment, improving labor conditions, and improving the quality of life. In this study, we used a sociological research method by applying a questionnaire we had compiled for the survey of rural residents. Here the author defines the term “human capital of the rural population” in her own opinion and lines up a system of indicators characterizing the basic components of the human capital of the rural population, such as health capital, education capital, labor capital, cultural and behavioral capital, as well as intellectual one. To overcome the negative situation taking place in rural areas of the Krasnoyarsk territory, it is necessary to assess human capital on the previously mentioned 5 basic components, and on base of the results to propose a socially attractive model for rural development, which in its turn will attract additional human capital. The proposed measures make it possible to increase employment in rural areas by creating agricultural cooperatives, small businesses and farms, and expand the ways of use of maternity benefit in the development of rural households and small businesses, to reduce the income tax rate for individual entrepreneurs who are registered in rural areas, as well as to increase the standards for the total area for housing construction for young families, and to apply digital forms of cultural and information services to the rural population.

Information-Theoretic Lower Bounds for Compressive Sensing With Generative Models

It has recently been shown that for compressive sensing, significantly fewer measurements may be required if the sparsity assumption is replaced by the assumption the unknown vector lies near the range of a suitably-chosen generative model. In particular, in (Bora et al., 2017) it was shown roughly O(k log L) random Gaussian measurements suffice for accurate recovery when the generative model is an L-Lipschitz function with bounded k-dimensional inputs, and O(kdlogw) measurements suffice when the generative model is a k-input ReLU network with depth d and width w. In this paper, we establish corresponding algorithm-independent lower bounds on the sample complexity using tools from minimax statistical analysis. In accordance with the above upper bounds, our results are summarized as follows: (i) We construct an L-Lipschitz generative model capable of generating group-sparse signals, and show that the resulting necessary number of measurements is Ω(k logL); (ii) Using similar ideas, we construct ReLU networks with high depth and/or high depth for which the necessary number of measurements scales as Ω(kd logw/logn) (with output dimension n), and in some cases Ω(kd logw). As a result, we establish that the scaling laws derived in (Bora et al., 2017) are optimal or near-optimal in the absence of further assumptions.

Optimal Multiplexing of Spatially Encoded Information across Custom-Tailored Configurations of a Metasurface-Tunable Chaotic Cavity

Combining tunable metasurfaces with chaotic cavities opens new avenues for finely tailored dynamic control of microwaves with programmable coding metacavities (PCMs). There is currently a strong interest in utilizing PCMs to overcome the notorious difficulty of coherent measurements across large apertures at radiofrequencies, with important applications in imaging and sensing for security screening, medical diagnosis and human-computer interaction. Such approaches rely on multiplexing spatially encoded information across a random sequence of PCM coding patterns for single-port single-frequency acquisition. Here, it is shown that a judiciously tailored rather than random coding sequence is necessary to unlock the full potential of PCMs for analog multiplexing. Specifically, the singular value spectrum of the multiplexing channel matrix is tailored to be perfectly flat - as opposed to downward sloping. In-situ experiments show that thereby the number of necessary measurements to achieve a given reconstruction quality is lowered by a factor of 2.5. Computational imaging and other microwave metrology applications are expected to benefit from the resulting reduction in acquisition time, processing burden and latency. The proposed approach and platform also set the stage for future studies in the emerging field of wave control through engineered wave chaos.

Метод определения мест утечек в водопроводе путем контроля градиента давления

The objective was to develop a method for determining leaks in a branchless water pipeline section, when a pressure drop in the pipe under the action of friction forces is observed. The problem of detecting leaks can be solved by developing new methods for monitoring the operation of water supply and distribution systems. A well-known method for detecting leaks in branchless pipelines consists in determination of the coordinates of the intersection point of lines that characterize the pressure change in the pipe before and after the leak incident. On this basis, a method is proposed for determining several leak locations in the water supply system with fewer necessary measurements. A well-known method for determining leaking sites in a horizontal water supply section with a constant pipe diameter was analysed. One disadvantage of this method is that determination of a leaking site may require numerous measurements to establish the ratio of the pipeline length to the most likely minimum length of the interval between nearby leak points. The developed method for detecting water leaks by monitoring the pressure gradient eliminates this drawback. A method of equivalent points was proposed, which allows the number of necessary measurements to be significantly reduced. Formulas were obtained for estimating the multiplicity of reducing the number of necessary measurements in the proposed method. It was determined that, upon an excessive increase in the number of leak points, the specified multiplicity decreases to zero. However, this indicates a completely obsolete water supply system, which requires replacement

P1644Personalized screening intervals for measurement of n-terminal pro-b-type natriuretic peptide improve efficiency of prognostication in patients with chronic heart failure

Abstract Background Predefined screening intervals and target levels do not account for variations in temporal patterns of biomarkers between individuals, which may hamper their potential use for therapy guidance. Conversely, a personalized screening approach with screening intervals and target levels based on the evolution of biomarkers in individual patients may further improve risk assessment and therapy guidance. Purpose We hypothesize that personalized screening intervals for N-terminal pro-B-type natriuretic peptide (NT-proBNP) measurements in patients with chronic heart failure (CHF) maximize information gain on the individual patient's disease progression, while minimizing the number of necessary measurements. We aim to compare such personalized scheduling of NT-proBNP measurements to a predefined fixed scheduling approach. Methods In 263 CHF patients from the Bio-SHiFT study, NT-proBNP was measured trimonthly according to a prespecified, fixed schedule [median: 9 (IQR: 5–10) measurements per patient].The primary composite endpoint (PE) comprised cardiac death, cardiac transplantation, left ventricular assist device implantation or heart failure hospitalization, and occurred in 70 patients (26.6%). Using joint models for time-to-event and longitudinal data, we modelled the association between repeated NT-proBNP measurements and the PE. Using the fitted joint model, for each patient at each follow-up visit, we determined the optimal time point of the next NT-proBNP measurement based on the patient's individual risk profile and the maximum information gain on the patient's prognosis as assessed by the Kullback-Leibler divergence. Personalized scheduling was compared to fixed (trimonthly) scheduling by means of a realistic simulation study, based on a replica of the study population included in the Bio-SHiFT study. In this simulation study, we stopped monitoring NT-proBNP to potentially enable appropriate timely intervention if the cumulative risk of PE exceeded an arbitrary risk threshold of 7.5% within 3-months. We compared personalized scheduling with fixed scheduling in terms of capability of identification of high-risk intervals (whether timely intervention was enabled before occurrence of PE), number of measurements needed, and costs. Results Compared to fixed scheduling, personalized scheduling saved on average 2 measurements [personalized; median: 7 (IQR: 7–8) vs. fixed; 9 (IQR: 8–10) measurements], while the start of the time-window identified for therapeutic intervention to avoid the occurrence of PE was similar in both approaches [personalized; median: 6.6 (IQR: 4.5–11.3) vs. fixed; 6.3 (IQR: 4.2–10.3) months before occurrence of PE]. Costs saved were €165 per patient per year. Figure 1 Conclusion Personalized scheduling of NT-proBNP measurements in CHF patients shows similar prognostic performance as fixed scheduling, but requires fewer NT-proBNP measurements. This may improve efficiency of natriuretic guided therapy, if the latter were to be installed. Acknowledgement/Funding Funding for this study was provided by the Jaap Schouten Foundation and Erasmus MC Efficiency Research grant

Reduced-Code Static Linearity Test of Split-Capacitor SAR ADCs Using an Embedded Incremental <inline-formula> <tex-math notation="LaTeX">$\Sigma\Delta$ </tex-math> </inline-formula> Converter

Reduced-code techniques for an analog-to-digital converter (ADC) static linearity test have the potential to drastically reduce the number of necessary measurements for a complete static linearity characterization. These techniques take advantage of the repetitive operation of certain families of converters such as pipelines, successive-approximation registers (SARs), cyclic, etc. In this paper, we present a novel reduced-code technique for the static linearity test of split-capacitor SAR ADCs based on the on-chip generation and measurement of the major carrier transitions of the input digital-to-analog converter of the converter. The proposed test method does not require a test stimulus, and we show that the necessary measurements can be easily extracted by reconfiguring portions of the SAR into a low-resolution incremental ${\Sigma \Delta }$ converter. The proposed technique is validated with both behavioral and electrical simulations of a 10-bit SAR ADC in a 65-nm CMOS technology.

The Curse of Class Imbalance and Conflicting Metrics with Machine Learning for Side-channel Evaluations

We concentrate on machine learning techniques used for profiled sidechannel analysis in the presence of imbalanced data. Such scenarios are realistic and often occurring, for instance in the Hamming weight or Hamming distance leakage models. In order to deal with the imbalanced data, we use various balancing techniques and we show that most of them help in mounting successful attacks when the data is highly imbalanced. Especially, the results with the SMOTE technique are encouraging, since we observe some scenarios where it reduces the number of necessary measurements more than 8 times. Next, we provide extensive results on comparison of machine learning and side-channel metrics, where we show that machine learning metrics (and especially accuracy as the most often used one) can be extremely deceptive. This finding opens a need to revisit the previous works and their results in order to properly assess the performance of machine learning in side-channel analysis.

Compressed sensing ECG using restricted Boltzmann machines

Abstract Recently, it has been shown that compressed sensing (CS) has the potential to lower energy consumption in wireless electrocardiogram (ECG) systems. By reducing the number of acquired measurements, the communication burden is decreased and energy is saved. In this paper, we aim at further reducing the number of necessary measurements to achieve faithful reconstruction by exploiting the representational power of restricted Boltzmann machines (RBMs) to model the probability distribution of the sparsity pattern of ECG signals. The motivation for using this approach is to capture the higher-order statistical dependencies between the coefficients of the ECG sparse representation, which in turn, leads to superior reconstruction accuracy and reduction in the number of measurements, as it is shown via experiments.

The Reliabilities of Several Measurement Methods of Cervical Sagittal Alignment in Cases with Cervical Spine Rotation Using X-ray Findings in Cervical Spine Disorders.

Introduction Several measurement methods designed to provide an understanding of cervical sagittal alignment have been reported, but few studies have compared the reliabilities of these measurement methods. The purpose of the present study was to investigate the intraexaminer and interexaminer reliabilities of several cervical sagittal alignment measurement methods and of the rotated cervical spine using plain lateral cervical spine X-rays of patients with cervical spine disorders. Methods Five different measurement methods (Borden's method; Ishihara index method (Ishihara method); C2-7 Cobb method (C2-7 Cobb); posterior tangent method: absolute rotation angle C2-7 (ARA); and classification of cervical spine alignment (CCSA)) were applied by seven examiners to plain lateral cervical spine X-rays of 20 patients (10 randomly extracted cases from a rotated cervical spine group and 10 from a nonrotated group) with cervical spine disorders. Case 1 and Case 2 intraclass correlation coefficients (ICCs) were used to analyze intraexaminer and interexaminer reliabilities. The necessary number of measurements and the necessary number of examiners were also determined. The target coefficient of correlation was set at ≥0.81 (almost perfect ICC). Results In both groups, an ICC(1, 1) ≥ 0.81 was obtained with Borden's method, the Ishihara method, C2-7 Cobb, and ARA by all examiners. The necessary number of measurements was 1. With CCSA, a kappa coefficient of at least 0.9 was obtained. In both groups, with Borden's method, the Ishihara method, C2-7 Cobb, and ARA, the ICC(2, 1) was ≥0.9, indicating that the necessary number of examiners was 1. The standard error of measurement (SEM) was lowest with Borden's method, and the Ishihara method and C2-7 Cobb had almost the same values. Conclusions Among cervical sagittal alignment measurement methods for cervical spine disorders, regardless of cervical spine rotation, Borden's method, Ishihara method, and C2-7 Cobb offer stronger reliability in terms of the ICC and SEM.

Sparsity and incoherence in orthogonal matching pursuit

Recovery of sparse signals via approximation methods has been extensively studied in recently years. We consider the nonuniform recovery of orthogonal matching pursuit (OMP) from fewer noisy random measurements. Rows of sensing matrices are assumed to be drawn independently from a probability distribution obeying the isotropy property and the incoherence property. Our models not only include the standard sensing matrices in compressed sensing context, but also cover other new sensing matrices such as random convolutions, subsampled tight or continuous frames. Given m admissible random measurements of a fixed s-sparse signal $$\varvec{x}\in \mathbb {R}^n$$ , we show that OMP can recover the support of $$\varvec{x}$$ exactly after s iterations with overwhelming probability provided that $$\begin{aligned} m=O( s(s+\log (n-s))). \end{aligned}$$ It follows that the approximation order of OMP is $$\begin{aligned} \Vert \varvec{x}- \varvec{x}_j\Vert =O(\eta ^j) \end{aligned}$$ where $$0<\eta <1$$ and $$\varvec{x}_j$$ denotes the recovered signal at j-th iteration. As a byproduct of the proof, the necessary number of measurements to ensure sparse recovery by $$l_1$$ -minimization with random partial circulant or Toeplitz matrices is proved to be optimal.

Hybrid reconstruction of quantum density matrix: when low-rank meets sparsity

Both the mathematical theory and experiments have verified that the quantum state tomography based on compressive sensing is an efficient framework for the reconstruction of quantum density states. In recent physical experiments, we found that many unknown density matrices in which people are interested in are low-rank as well as sparse. Bearing this information in mind, in this paper we propose a reconstruction algorithm that combines the low-rank and the sparsity property of density matrices and further theoretically prove that the solution of the optimization function can be, and only be, the true density matrix satisfying the model with overwhelming probability, as long as a necessary number of measurements are allowed. The solver leverages the fixed-point equation technique in which a step-by-step strategy is developed by utilizing an extended soft threshold operator that copes with complex values. Numerical experiments of the density matrix estimation for real nuclear magnetic resonance devices reveal that the proposed method achieves a better accuracy compared to some existing methods. We believe that the proposed method could be leveraged as a generalized approach and widely implemented in the quantum state estimation.