Recent advances in quantum computers and simulators are steadily leading us towards full-scale quantum computing devices. Due to the fact that debugging is necessary to create any computing device, quantum tomography (QT) is a critical milestone on t
his path. In practice, the choice between different QT methods faces the lack of comparison methodology. Modern research provides a wide range of QT methods, which differ in their application areas, as well as experimental and computational complexity. Testing such methods is also being made under different conditions, and various efficiency measures are being applied. Moreover, many methods have complex programming implementations; thus, comparison becomes extremely difficult. In this study, we have developed a general methodology for comparing quantum state tomography methods. The methodology is based on an estimate of the resources needed to achieve the required accuracy. We have developed a software library (in MATLAB and Python) that makes it easy to analyze any QT method implementation through a series of numerical experiments. The conditions for such a simulation are set by the number of tests corresponding to real physical experiments. As a validation of the proposed methodology and software, we analyzed and compared a set of QT methods. The analysis revealed some method-specific features and provided estimates of the relative efficiency of the methods.
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 d
ynamical systems. As examples, we consider the logistic model, the Van der Pol oscillator, dynamical systems of Lorenz, Rossler (including Rossler 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.
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.
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 Boole
an 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.
The method of noisy multiqubit quantum circuits modeling is proposed. The analytical formulas for the dependence of quantum algorithms accuracy on qubits count and noise level are obtained for Grover algorithm and quantum Fourier transform. It is sho
wn that the proposed approach is very much in line with results obtained by Monte Carlo statistical modeling method. The developed theory makes it possible to predict the influence of quantum noise on the accuracy of the forward-looking multiqubit quantum systems under development.
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 s
et 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.
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.
Three different levels of noisy quantum schemes modeling are considered: vectors, density matrices and Choi-Jamiolkowski related states. The implementations for personal computers and supercomputers are described, and the corresponding results are sh
own. For the level of density matrices, we present the technique of the fixed rank approximation and show some analytical estimates of the fidelity level.
The influence of amplitude and phase relaxation on evolution of quantum states within the formalism of quantum operations is considered. The model of polarizing qubits where noises are determined by the existence of spectral degree of freedom that sh
ows up during the light propagation inside anisotropic mediums with dispersion is studied. Approximate analytic model for calculation of phase plate impact on polarizing state with dispersion influence taken into consideration is suggested.
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 an
alysis 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.
