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We develop a new method for high-precision tomography of ion qubit registers under conditions of limited distinguishability of its logical states. It is not always possible to achieve low error rates during the readout of the quantum states of ion qubits due to the finite lifetime of excited levels, photon scattering, dark noise, low numerical aperture, etc. However, the model of fuzzy quantum measurements makes it possible to ensure precise tomography of quantum states. To do this, we developed a fuzzy measurement model based on counting the number of fluorescent photons. A statistically adequate algorithm for the reconstruction of quantum states of ion qubit registers based on fuzzy measurement operators is proposed. The algorithm uses the complete information available in the experiment and makes it possible to account for systematic measurement errors associated with the limited distinguishability of the logical states of ion qubits. We show that the developed model, although computationally more complex, contains significantly more information about the state of the qubit and provides a higher accuracy of state reconstruction compared to the model based on the threshold algorithm., Comment: 9 pages, 4 figures

The wave plate is a basic device for transforming and measuring the polarization states of light. It is known that the transformation of light by means of two wave plates makes it possible to measure the state of polarization in an arbitrary basis. The finite spectral width of the light, however, leads to a chromatic aberration of the polarization quantum transformation caused by the parasitic dispersion of the birefringence of the plate material. This causes systematic errors in the tomography of quantum polarization states and significantly reduces its accuracy. This study is a development of our work1, in which an adequate model for quantum measurements of polarization qubits under chromatic aberration was first formulated. This work includes a generalization of the results obtained earlier for the cases of two-qubit states. Along with examples of random states those uniformly distributed over the Haar measure are considered. Using a matrix of complete information, it is quantitatively traced how the presence of chromatic aberrations under conditions of a finite spectral width of light leads to the loss of information in quantum measurements. It is shown that the use of the developed model of fuzzy measurements instead of the model of standard projection measurements makes it possible to suppress systematic errors of quantum tomography even when using high-order wave plates. It turns out that the fuzzy measurement model can give a significant increase in the reconstruction accuracy compared to the standard measurement model., Comment: 8 pages, 7 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021

The method of quantum tomography, which allows us to track with high accuracy the evolution of multilevel quantum systems (qudits) in Hilbert spaces of various dimensions is presented. The developed algorithms for quantum control are based on the use of the spinor representation of the Lorentz transformation group. In the simplest case of one-qubit states, it turns out that, in addition to three-dimensional rotations on the Bloch sphere, one can introduce four-dimensional Lorentz pseudorotations, similar to the transformations of the special theory of relativity. We show that feedback through weakly perturbing adaptive quantum measurements turns out to be capable of providing high-precision control of the quantum system, while introducing only weak perturbations into the initial quantum state. It turns out that, together with the control of a quantum system through its weak perturbation, the developed algorithms for controlling the evolution of the state of a quantum system can be super-efficient, providing a higher measurement accuracy than any standard POVM (Positive-Operator Valued Measure) protocols. The results of the study are important for the development of optimal adaptive methods for quantum states and operations controlling. The results obtained are essential for the development of high-precision control methods for quantum information technologies., Comment: 8 pages, 3 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021

Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state - an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems., Comment: 10 pages, 4 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021

Quantum tomography is an important tool for obtaining information about the quantum state from experimental data. In this study, we conduct a comparative analysis of various quantum tomography protocols, including protocols based on highly symmetric sets of state vectors, on mutually unbiased bases (MUB), and two-level transformations. Using the universal distribution for the fidelity of quantum state tomography, we estimate the fidelity and efficiency of quantum measurements for qudits of various dimensions as applied to a set of random states uniformly distributed over the Haar measure., Comment: 8 pages, 6 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021

The study of the effect of quantum noise on the accuracy of modeling quantum systems on a quantum computer using the Zalka-Wiesner method is carried out. The efficiency of the developed methods and algorithms is demonstrated by the example of solving the nonstationary Schrodinger equation with allowance for quantum noise for a particle moving in the Poschl-Teller potential. The analysis and prediction of accuracy for the Zalka-Wiesner method are carried out taking into account the complexity of the quantum system and the strength of quantum noise. In our opinion, the considered problem provides a useful test for assessing the quality and efficiency of quantum computing devices on various physical platforms currently being developed. The obtained results are essential for the development of high-precision methods for controlling the technologies of quantum computations., Comment: 10 pages, 4 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021

The present work is devoted to the development of a method for high-precision tomography of ion qubit registers under conditions of limited distinguishability of the states of a logical value 0 and a logical value 1. In the considered ion qubits, the identification of the quantum state is achieved by measuring the fluorescence of the ion by repeated excitation of the cyclic transition, which includes only the lower energy state that sets a logical value 0 and becomes bright, but does not include the upper metastable state that remains dark and sets logical value 1. It is important to note that it is not always possible to achieve low levels of registration errors due to the finite lifetime of excited levels, photon scattering, dark noise, low numerical aperture values, etc. However, even under such conditions, with use of the model of fuzzy quantum measurements, it is possible to provide precise control of quantum states. We show that a model that is characterized by relatively high levels of errors under conditions, where we have a reliable statistical model of their occurrence, is more accurate than the case when the considered errors are small, but we do not have an adequate statistical model for the occurrence of these errors. In the given illustrative examples, we show that the factor of reducing the loss of accuracy with the use of the model of fuzzy measurements can reach values of the order of 1000 or more in comparison with standard measurements. The obtained results are essential for the development of high-precision methods for controlling the technology of quantum computing on the ion platform., Comment: 9 pages, 4 figures, The 14th International Conference 'Micro- and Nanoelectronics - 2021' (ICMNE-2021) with the Extended Session 'Quantum Informatics', Zvenigorod, October 4-8, 2021

Thermal states of light are widely used in quantum optics for various quantum phenomena testing. Particularly, they can be utilized for characterization of photon creation and photon annihilation operations. During the last decade the problem of photon subtraction from multimode quantum states become of much significance. Therefore, in this work we present a technique for statistical parameter estimation of multimode multiphoton subtracted thermal states of light, which can be used for multimode photon annihilation test., Comment: 9 pages, 9 figures

In this work we present an algorithm of building an adequate model of polarizing quantum state measurement. This model takes into account chromatic aberration of the basis change transformation caused by the parasitic dispersion of the wave plates crystal and finite radiation spectral bandwidth. We show that the chromatic aberration reduces the amount of information in the measurements results. Using the information matrix approach we estimate the impact of this effect on the qubit state reconstruction fidelity for different values of sample size and spectral bandwidth. We also demonstrate that our model outperforms the standard model of projective measurements as it could suppress systematic errors of quantum tomography even when one performs the measurements using wave plates of high order.

Thermal states of light are widely used in quantum optics due to their correlation properties. As is well known, their correlation properties and the photon number distribution as a whole are strongly dependent on the mode number selected by the detection scheme. The same changes can be caused by photon subtraction. Therefore, we describe the general case of the multimode thermal state after a multiphoton subtraction, when the photon number statistics is registered by the detector selecting a part of the initial modes. We present an analytical form of the obtained photon number distribution and its general properties and check them in the experiment.

In this report we present a general approach for estimating quantum circuits by means of measurements. We apply the developed general approach for estimating the quality of superconducting and optical quantum chips. Using the methods of quantum states and processes tomography developed in our previous works, we have defined the adequate models of the states and processes under consideration., Comment: 9 pages, 7 figures

We present a general approach to the classical dynamical systems simulation. This approach is based on classical systems extension to quantum states. The proposed theory can be applied to analysis of multiple (including non-Hamiltonian) dissipative dynamical systems. As examples, we consider the logistic model, the Van der Pol oscillator, dynamical systems of Lorenz, R\"ossler (including R\"ossler hyperchaos) and Rabinovich-Fabrikant. Developed methods and algorithms integrated in quantum simulators will allow us to solve a wide range of problems with scientific and practical significance., Comment: 12 pages, 5 figures

The quantum measurement procedure based on the Lorentz transformation formalism and weak perturbation of the system is considered. In the simple case of a single-qubit it turns out that one can perform 4-dimension pseudo-rotation along with ordinary 3-dimension rotations on the Bloch sphere. These pseudo-rotations are similar to the Lorentz transformation in special relativity theory. The extension of the Lorentz transformation for many-qubit systems is also considered. The quantum measurement protocols based on the Lorentz transformation are proposed. It has been shown that these protocols cease to form the decomposition of unity and could be superefficient providing the fidelity higher than any POVM-measurement protocol. However, one can perform the complement of the Lorentz protocol to POVM-protocol by an additional measurement operator. If the initial mixed state is close to the pure one this operator corresponds to weak perturbation of the state while the original Lorentz protocol sets the strong perturbations. As the result, the feedback provides an effective control of a quantum system introducing weak perturbations to the quantum state. The results of this research are essential for the development of methods for the control of quantum information technologies., Comment: 12 pages, 3 figures

The work is devoted to the theoretical and experimental study of quantum states of light conditionally prepared by subtraction of a random number of photons from the initial multimode thermal state. A fixed number of photons is subtracted from a multimode quantum state, but only a subsystem of a lower number of modes is registered, in which the number of subtracted photons turns out to be a non-fixed random variable. It is shown that the investigation of multiphoton subtracted multimode thermal states provides a direct study of the fundamental quantum-statistical properties of bosons using a simple experimental implementation. The developed experimental setup plays a role of a specific boson lototron, which is based on the fundamental link between the statistics of boson systems and the Polya distribution. It is shown that the calculation of the photon number distribution based on the Polya urn scheme is equivalent to a calculation using statistical weights for boson systems. A mathematical model based on the composition of the Polya distribution and thermal state is developed and verified. The experimental results are in a good agreement with the developed theory., Comment: 10 pages, 3 figures

The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal form of Boolean function, and quantum logic circuits is described. It is shown that quantum information approach provides simple algorithm to construct Zhegalkin polynomial using truth table. Developed methods and algorithms have arbitrary Boolean function generalization with multibit input and multibit output. Such generalization allows us to use many-valued logic (k-valued logic, where k is a prime number). Developed methods and algorithms can significantly improve quantum technology realization. The presented approach is the baseline for transition from classical machine logic to quantum hardware., Comment: 19 pages, 4 figures

The polarizing multi-photon quantum states tomography with non-unit quantum efficiency of detectors is considered. A new quantum tomography protocol is proposed. This protocol considers events of losing photons of multi-photon quantum state in one or more channels among with n-fold coincidence events. The advantage of the proposed protocol compared with the standard n-fold coincidence protocol is demonstrated using the methods of statistical analysis., Comment: 8 pages; 3 figures; submitted to Laser Physics (international journal)

The estimation of high order correlation function values is an important problem in the field of quantum computation. We show that the problem can be reduced to preparation and measurement of optical quantum states resulting after annihilation of a set number of quanta from the original beam. We apply this approach to explore various photon bunching regimes in optical states with gamma-compounded Poisson photon number statistics. We prepare and perform measurement of the thermal quantum state as well as states produced by subtracting one to ten photons from it. Maximum likelihood estimation is employed for parameter estimation. The goal of this research is the development of highly accurate procedures for generation and quality control of optical quantum states., Comment: 8 pages, 1 figure. International Conference on Micro- and Nano-Electronics 2016. Zvenigorod, Russian Federation, October 3-7, 2016

The new method of multivariate data analysis based on the complements of classical probability distribution to quantum state and Schmidt decomposition is presented. We considered Schmidt formalism application to problems of statistical correlation analysis. Correlation of photons in the beam splitter output channels, when input photons statistics is given by compound Poisson distribution is examined. The developed formalism allows us to analyze multidimensional systems and we have obtained analytical formulas for Schmidt decomposition of multivariate Gaussian states. It is shown that mathematical tools of quantum mechanics can significantly improve the classical statistical analysis. The presented formalism is the natural approach for the analysis of both classical and quantum multivariate systems and can be applied in various tasks associated with research of dependences., Comment: 10 pages, 6 figures. International Conference on Micro- and Nano-Electronics 2016. Zvenigorod, Russian Federation, October 3-7, 2016

We consider realistic measurement systems, where measurements are accompanied by decoherence processes. The aim of this work is the construction of methods and algorithms for precise quantum measurements with fidelity close to the fundamental limit. In the present work the notions of ideal and non-ideal quantum measurements are strictly formalized. It is shown that non-ideal quantum measurements could be represented as a mixture of ideal measurements. Based on root approach the quantum state reconstruction method is developed. Informational accuracy theory of non-ideal quantum measurements is proposed. The monitoring of the amount of information about the quantum state parameters is examined, including the analysis of the information degradation under the noise influence. The study of achievable fidelity in non-ideal quantum measurements is performed. The results of simulation of fidelity characteristics of a wide class of quantum protocols based on polyhedrons geometry with high level of symmetry are presented. The impact of different decoherence mechanisms, including qubit amplitude and phase relaxation, bit-flip and phase-flip, is considered., Comment: 10 pages, 4 figures. International Conference on Micro- and Nano-Electronics 2016. Zvenigorod, Russian Federation, October 3-7, 2016

We present a study of optical quantum states generated by subtraction of photons from the thermal state. Some aspects of their photon number and quadrature distributions are discussed and checked experimentally. We demonstrate an original method of up to ten photon subtracted state preparation with use of just one single-photon detector. All the states where measured with use of balanced homodyne technique, and the corresponding density matrices where reconstructed. The fidelity between desired and reconstructed states exceeds 99%

We have developed a method for complementing an arbitrary classical dynamical system to a quantum system using the Lorenz and R\"ossler systems as examples. The Schr\"odinger equation for the corresponding quantum statistical ensemble is described in terms of the Hamilton-Jacobi formalism. We consider both the original dynamical system in the position space and the conjugate dynamical system corresponding to the momentum space. Such simultaneous consideration of mutually complementary position and momentum frameworks provides a deeper understanding of the nature of chaotic behavior in dynamical systems. We have shown that the new formalism provides a significant simplification of the Lyapunov exponents calculations. From the point of view of quantum optics, the Lorenz and R\"ossler systems correspond to three modes of a quantized electromagnetic field in a medium with cubic nonlinearity. From the computational point of view, the new formalism provides a basis for the analysis of complex dynamical systems using quantum computers., Comment: Submitted to Laser Physics

Application of root density estimator to problems of statistical data analysis is demonstrated. Four sets of basis functions based on Chebyshev-Hermite, Laguerre, Kravchuk and Charlier polynomials are considered. The sets may be used for numerical analysis in problems of reconstructing statistical distributions by experimental data. Based on the root approach to reconstruction of statistical distributions and quantum states, we study a family of statistical distributions in which the probability density is the product of a Gaussian distribution and an even-degree polynomial. Examples of numerical modeling are given. The results of present paper are of interest for the development of tomography of quantum states and processes., Comment: 9 pages, 4 figures, report for the International Symposium "Quantum Informatics-2014" (QI-2014), Zvenigorod, Moscow region, October 06-10, 2014

Application of root density estimator to problems of statistical data analysis is demonstrated. Four sets of basis functions based on Chebyshev-Hermite, Laguerre, Kravchuk and Charlier polynomials are considered. The sets may be used for numerical analysis in problems of reconstructing statistical distributions by experimental data. Examples of numerical modeling are given.