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We propose a protocol for studying the purely imaginary Fisher zeros of the Ising model on a quantum computer. Our protocol is based on the direct relation between the partition function for purely imaginary temperature and the evolution operator of the Ising model. In this case, the inverse temperature is equal to the time of evolution. However, this protocol does not allow for measuring the zeros in the complex plane, only those localized on the imaginary axes. We test this protocol on the ibm-lagos quantum computer in the cases of a 3-spin chain and a triangle cluster in a purely imaginary magnetic field, as well as a 7-spin cluster in which the interaction between spins reproduces the architecture of the quantum computer. We also obtain the equations which determine the Lee-Yang zeros of a 3-spin chain and a triangle cluster., Comment: 7 pages, 7 figures

In the paper a review of results for recovering of the weak equivalence principle in a space with deformed commutation relations for operators of coordinates and momenta is presented. Different types of deformed algebras leading to a space quantization are considered among them noncommutative algebra of canonical type, algebra of Lie type, nonlinear deformed algebra with arbitrary function of deformation depending on momenta. A motion of a particle and a composite system in gravitational field is examined and the implementation of the weak equivalence principle is studied. The principle is preserved in quantized space if we consider parameters of deformed algebras to be dependent on mass. It is also shown that dependencies of parameters of deformed algebras on mass lead to preserving of the properties of the kinetic energy in quantized spaces and solving of the problem of significant effect of space quantization on the motion of macroscopic bodies (the problem is known as the soccer-ball problem).

We consider simulation of the high spins on a quantum computer. The protocols which allow one to measure the mean value of spin and correlations between spins are proposed. As a result, we determine the time dependence of the mean values of spin-1 in the magnetic field prepared on the ibmq-santiago quantum computer. In addition, we study the evolution of two interacting spins on the ibmq-lima quantum computer. The time-dependencies of the mean value of spin-1 and correlations between these spins are detected. Finally, we generalize these protocols for the spins of arbitrary values., Comment: 8 pages, 7 figures

Spin-1 tunneling and splitting of energy levels as a result of tunneling are observed explicitly on IBM's quantum computer, ibmq-bogota. The spin-1 is realized with two spins-1/2. We detect oscillations of spin-1 between the states $|1\rangle$, $|-1\rangle$ in the result of tunneling on the basis of studies of the time dependence of the mean value of z-component of spin-1 on the quantum device. The energy level splitting is observed quantifying on the IBM's quantum computer the eigenvalues of Hamiltonian which describes the spin tunneling.

We propose the definition of the geometric measure of entanglement for continuous variable states. On the basis of this definition we examine entanglement of the graph states obtained as a result of action of a unitary operator on the ground state of a system of $N$ noninteracting harmonic oscillators. We find that the entanglement of a harmonic oscillator with other ones is defined by the value of its vertex degree.

We consider the Dirac $\delta$-function potential problem in general case of deformed Heisenberg algebra leading to the minimal length. Exact bound and scattering solutions of the problem in quasiposition representation are presented. We obtain that for some resonance energy the incident wave is completely reflected. We conclude that this effect is very sensitive to the choice of the deformation function.

We propose a method for detection of energy levels of arbitrary spin system on a quantum computer based on studies of evolution of only one probe spin. On the basis of the proposed method energy levels of spin systems are found on IBM's quantum computer ibmq-bogota, among them are spin chain in magnetic field, triangle spin cluster, Ising model on squared lattice in magnetic field. The results of quantum calculations are in agreement with the theoretical ones. The method is efficient for estimation of the energy levels of many-spin systems and opens a possibility to achieve quantum supremacy in solving eigenvalue problem with development of multi-qubit quantum computers.

Geometric properties of evolutionary graph states of spin systems generated by the operator of evolution with Ising Hamiltonian are examined, using their relationship with fluctuations of energy. We find that the geometric characteristics of the graph states depend on properties of the corresponding graphs. Namely, it is obtained that the fluctuations of energy in graph states and therefore the velocity of quantum evolution, the curvature and the torsion of the states are related with the total number of edges, triangles and squares in the corresponding graphs. The obtained results give a possibility to quantify the number of edges, triangles and squares in a graph on a quantum devise and achieve quantum supremacy in solving this problem with the development of a multi-qubit quantum computer. Geometric characteristics of graph states corresponding to a chain, a triangle, and a square are detected on the basis of calculations on IBM's quantum computer ibmq-manila.

In general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the $\delta'(x)$ potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for $\delta'(x)$ and $\delta(x)-\delta'(x)$ potentials in deformed space with arbitrary function of deformation. The energy spectrum for different partial cases of deformation function is analysed.

We show that the time dependence of mean value of a physical quantity is related with the transition energies of a quantum system. In the case when the operator of a physical quantity anticommutes with the Hamiltonian of a system, studies of the evolution of its mean value allow determining the energy levels of the system. On the basis of the result, we propose a method for determining energy levels of physical systems on a quantum computer. The method opens a possibility to achieve quantum supremacy in solving the problem of finding minimal or maximal energy of Ising model with spatially anisotropic interaction using multi-qubit quantum computers. We apply the method for spin systems (spin in magnetic field, spin chain, Ising model on squared lattice) and realize it on IBM's quantum computers.

We study the entanglement between a certain qubit and the remaining system in the Schr\"odinger cat state prepared on the ibmq-melbourne quantum computer. The protocol, which we use for this purpose, is based on the determination of the mean value of spin corresponding to a certain qubit. We explore the dependence of the entanglement on a parameter of the Schr\"odinger cat state which consists of different numbers of qubits. In addition, we explore the entanglement of each qubit with the remaining system in the maximum entangled Schr\"odinger cat state., Comment: 8 pages, 5 figures, 1 table

We consider graph states generated by operator of evolution with Ising Hamiltonian. The geometric measure of entanglement of a spin with other spins in the graph state is obtained analytically and quantified on IBM's quantum computer, IBM Q Valencia. The results of quantum calculations are in good agreement with the theoretical ones. We conclude that the geometric measure of entanglement of a spin with other spins in the graph state is related with degree of vertex representing the spin in the corresponding graph.

We study the entanglement between a certain qubit and the remaining system in rank- 2 mixed states prepared on the quantum computer. The protocol, which we propose for this purpose, is based on the relation of geometric measure of entanglement with correlations between qubits. As a special case, we consider a two-qubit rank-2 mixed state and find the relation of concurrence with the geometric measure of entanglement. On the ibmq-melbourne quantum computer we measure the geometric measure of entanglement in the cases of 2- and 4-qubit mixed quantum states which consist of Schr\"odinger cat states. We study the dependence of the value of entanglement on the parameter which defines the weight of pure states. Finally, we determine the concurrence of 2-qubit mixed state., Comment: 13 pages, 6 figures

We derive continuity equation and exact expression for flow probability density in a space with arbitrary deformed algebra leading to minimal length. In coordinate representation the flow probability density is presented as infinite series over parameter of deformation which in momentum representation can be casted into exact closed form determined by deformed kinetic energy. The flow probability density is calculated explicitly for plane wave and for superposition of two plane waves.

We implement a protocol to determine the degree of entanglement between a qubit and the rest of the system on a quantum computer. The protocol is based on results obtained in paper [Frydryszak et al. (2017)]. This protocol is tested on a 5-qubit superconducting quantum processor called ibmq-ourense provided by the IBM company. We determine the values of entanglement of the Schr\"odinger cat and the Werner states prepared on this device and compare them with the theoretical ones. In addition, a protocol for determining the entanglement of rank-2 mixed states is proposed. We apply this protocol to the mixed state which consists of two Bell states prepared on the ibmq-ourense quantum device., Comment: 6 pages, 9 figures

We study entanglement of spin degrees of freedom with continuous one in supersymmetric (SUSY) quantum mechanics. Concurrence is determined by mean value of spin and is calculated explicitly for SUSY states. We show that eigenstates of supercharges are maximally entangled. As an example the entanglement of atom state with photon state and SUSY in Jaynes-Cummings model are considered.

A space with deformed commutation relations for coordinates and momenta leading to generalized uncertainty principle (GUP) is studied. We show that GUP causes great violation of the weak equivalence principle for macroscopic bodies, violation of additivity property of the kinetic energy, dependence of the kinetic energy on composition, great corrections to the kinetic energy of macroscopic bodies. We find that all these problems can be solved in the case of arbitrary deformation function depending on momentum if parameter of deformation is proportional inversely to squared mass.

We find stringent upper bound on the momentum scale in noncommutative phase space of canonical type on the basis of studies of perihelion shift of the Mercury planet with taking into account features of description of motion of macroscopic body in the space with noncommutativity of coordinates and noncommutativity of momenta. Using results for precession of perihelion of the Mercury planet from ranging to the MESSENGER spacecraft we obtain upper bound for parameter of momentum noncommutativity $10^{-80}\textrm{kg}^2\textrm{m}^2/\textrm{s}^2$ which is many orders less than known in the literature., Comment: to appear in EPL

A space with deformed Poisson brackets for coordinates and momenta leading to the minimal length is considered. Features of description of motion of a body in the space are examined. We propose conditions on the parameters of deformation on which Poisson brackets for coordinates and momenta of the center-of-mass reproduce relations of deformed algebra, kinetic energy of a body is independent of its composition, and the weak equivalence principle is preserved in the deformed space. Influence of minimal length on the motion of the Sun-Earth-Moon system is studied. We find that deformation of the Poisson brackets leads to corrections to the accelerations of the Earth and the Moon toward the Sun, as a result the Eotvos-parameter does not vanish even if we consider equality of gravitational and inertial masses. The upper bound for the minimal length is estimated using results of the Lunar laser ranging experiment., Comment: to appear in International Journal of Modern Physics D

We study the Lee-Yang zeros of the ferromagnetic Ising bath via the interaction with the two probe spins. Similarly as in paper [Bo-Bo Wei, Ren-Bao Liu, Phys. Rev. Lett. 109, 185701 (2012)] the problem of detecting the zeros is reduced to the exploration of time evolution of probe spins. As a result, the relation between the Lee-Yang zeros of the bath and correlation functions of the probe system is obtained. Also we obtain relation between the Lee-Yang zeros and values of the entanglement of probe spins. We apply these results to the 1D Ising spin model with nearest-neighbor interaction which can be prepared on trapped atoms., Comment: 18 pages, 3 figures

We study a problem of description of macroscopic body motion in the frame of nonrelativistic Snyder model. It is found that the motion of the center-of-mass of a body is described by an effective parameter which depends on the parameters of Snyder algebra for coordinates and momenta of particles forming the body and their masses. We also show that there is reduction of the effective parameter with respect to parameters of Snyder algebra for coordinates and momenta of individual particles. As a result the problem of extremely small result for the minimal length obtained on the basis of studies of the Mercury motion in the Snyder space is solved. In addition we find that relation of parameter of Snyder algebra with mass opens possibility to preserve the property of independence of the kinetic energy on composition, to recover the weak equivalence principle, to consider coordinates as kinematic variables, to recover proportionality of momenta to mass and to consider Snyder algebra for coordinates and momenta of the center-of-mass of a body defined in the traditional way.

Considering increasing requirements to the coordinates measurement precision by the end of XX century International Astronomical Union commenced implementation of the new astrometric system ICRF (International Celestial Reference Frame). This quasi-inertial reference frame system centered in the barycenter of the Solar System and has axes defined by the positions of distant extragalactic sources - frames. Unlike equatorial system ICRF has no shortcomings of the coordinates identification due to the Earth axis precession, stellar proper motions and other factors. Extragalactic frames of the ICRF system are mostly quasars, radio galaxies, blazars and Seyfert galaxies i.e. different types of the active galaxy nuclei (AGN). Active galaxy nuclei are characterized by processes with significant. Such processes quite often are followed by X-ray emission generation. The purpose of this work is to consider X-ray emission of ICRF sources and features of their possible proper motions. Among 295 selected reference frames of the system we identified 54 X-ray sources which were observed by space observatory XMM Newton and noticed rapid variability of the blazars 2E 2673 (W Com) and 2E 1802 which enables to conclude that they have some very active processes in the sources centers. With regards to the future more detailed analysis we beleive that evidences of the objects proper motion could be found in their spectra. Based on the constructed luminosity and spectral graphs we could conclude that apart from above mentioned AGNs rest 52 objects do not show veriability and special attention should be paid to blazars within ICRF system development and use. Major part of the X-ray sources between the reference frames are stable., Comment: 5 pages, 6 figures, 18th Gamow Conference-School: 12-18 August, 2018, Odessa, Ukraine

In general case of deformed Heisenberg algebra leading to the minimal length we present a definition of the square inverse position operator. Our proposal is based on the functional analysis of the square position operator. Using this definition a particle in the field of the square inverse position potential is studied. We have obtained analytical and numerical solutions for the energy spectrum of the considerable problem in different cases of deformation function. We find that the energy spectrum slightly depends on the choice of deformation function., Comment: This work was partly supported by the grant of the President of Ukraine for support of scientific researches of young scientists (F-75)

Time reversal symmetry is studied in a space with noncommutativity of coordinates and noncommutativity of momenta of canonical type. The circular motion is examined as an apparent example of time reversal symmetry breaking in the space. On the basis of exact solution of the problem we show that because of noncommutativity the period of the circular motion depends on its direction. We propose the way to recover the time reversal and rotational symmetries in noncommutative phase space of canonical type. Namely, on the basis of idea of generalization of parameters of noncommutativity to tensors we construct noncommutative algebra which is rotationally-invariant, invariant under time reversal, besides it is equivalent to noncommutative algebra of canonical type.

Recently in paper [Peng et al., Phys. Rev. Lett. 114, 010601 (2015)] the experimental observation of the Lee-Yang zeros of an Ising-type spin-1/2 bath, by measuring the coherence of a probe spin, was reported. We generalize this problem to the case of an arbitrary high-spin bath. Namely, we consider the evolution of a probe arbitrary spin which interacts with bath composed by N arbitrary spins. As a result, the connection between the observed values of the probe spin, such as magnetization and susceptibility, and the Lee-Yang zeros is found. We apply these results to some models, namely, a triangle spin cluster, the Ising model with a long-range interaction and the 1D Ising model with nearest-neighbor interaction. Also we propose the implementation of these models on real physical systems., Comment: 13 pages, 6 figures

We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is interesting that this expression corresponds to the squared Euclidian distance between the mixed state and the pure one in space of eigenvalues of the density matrix. As an example, geometric measure of mixing for spin-1/2 states is calculated., Comment: 4 pages

Features of motion of macroscopic body in gravitational field in a space with noncommutativity of coordinates and noncommutativity of momenta are considered in general case when coordinates and momenta of different particles satisfy noncommutative algebra with different parameters of noncommutativity. Influence of noncommutativity on the motion of three-body Sun-Earth-Moon system is examined. We show that because of noncommutativity the free fall accelerations of the Moon and the Earth toward the Sun in the case when the Moon and the Earth are at the same distance to the source of gravity are not the same even if gravitational and inertial masses of the bodies are equal. Therefore, the Eotvos-parameter is not equal to zero and the weak equivalence principle is violated in noncommutative phase space. We estimate the corrections to the Eotvos-parameter caused by noncommutativity on the basis of Lunar laser ranging experiment results. We obtain that with high precision the ratio of parameter of momentum noncommutativity to mass is the same for different particles., Comment: accepted for publication in International Journal of Theoretical Physics

Influence of noncommutativity on the motion of composite system is studied in noncommutative phase space of canonical type. A system composed by $N$ free particles is examined. We show that because of momentum noncommutativity free particles of different masses with the same velocities at the initial moment of time do not move together. The trajectory and the velocity of free particle in noncommutative phase space depend on its mass. So, a system of the free particles flies away. Also, it is shown that the total momentum defined in the traditional way is not integral of motion in a space with noncommutativity of coordinates and noncommutativity of momenta. We find that in the case when parameters of noncommutativity corresponding to a particle are determined by its mass the trajectory and velocity of free particle are independent of the mass, also the total momenta as integrals of motion can be introduced in noncommutative phase space.

We study generalized (1+1)-dimensional Dirac oscillator in nonuniform electric field. It is shown that in the case of specially chosen electric field the eigenvalue equation can be casted in the form of supersymmetric quantum mechanics. It gives a possibility to find exact solution for the energy spectrum of the generalized Dirac oscillator in nonuniform electric field. Explicit examples of exact solutions are presented. We show that sufficiently large electric field destroys the bounded eigenstates.

The two-time correlation function for probe spin interacting with spin system (bath) is studied. We show that zeros of this function correspond to zeros of partition function of spin system in complex magnetic field. The obtained relation gives new possibility to observe the Lee-Yang zeros experimentally. Namely, we show that measuring of the time dependence of correlation function allows direct experimental observation of the Lee-Yang zeros.

The time-dependent correlation functions of q-deformed Bose gas are studied. We find relation of zeros of the correlation functions with the Fisher zeros of partition function of the system. Complex temperature appears as a result of q-deformation and evolution of correlation function. A particular case of q-deformed Bose particles on two levels is examined and zeros of correlation functions and Fisher zeros of partition function are analyzed.

Composite system is studied in noncommutative phase space with preserved rotational symmetry. We find conditions on the parameters of noncommutativity on which commutation relations for coordinates and momenta of the center-of-mass of composite system reproduce noncommutative algebra for coordinates and momenta of individual particles. Also, on the conditions the coordinates and the momenta of the center-of-mass satisfy noncommutative algebra with effective parameters of noncommutativity which depend on the total mass of the system and do not depend on its composition. Besides, it is shown that on these conditions the coordinates in noncommutative space do not depend on mass and can be considered as kinematic variables, the momenta are proportional to mass as it has to be. A two-particle system with Coulomb interaction is studied and the corrections to the energy levels of the system are found in rotationally invariant noncommutative phase space. On the basis of this result the effect of noncommutativity on the spectrum of exotic atoms is analyzed.

Two-time correlation functions of a system of Bose particles are studied. We find relation of zeros of the correlation functions with the Lee-Yang zeros of partition function of the system. Obtained relation gives the possibility to observe the Lee-Yang zeros experimentally. A particular case of Bose particles on two levels is examined and zeros of two-time correlation functions and Lee-Yang zeros of partition function of the system are analyzed.

We reduce two-body problem to the one-body problem in general case of deformed Heisenberg algebra leading to minimal length.Two-body problems with delta and Coulomb-like interactions are solved exactly. We obtain analytical expression for the energy spectrum for partial cases of deformation function. The dependence of the energy spectrum on the center-of-mass momentum is found. For special case of deformation function, which correspondes to cutoff procedure in momentum space it is shown that this dependence is more likely to observe for identical particles.

We construct algebra with noncommutativity of coordinates and noncommutativity of momenta which is rotationally invariant and equivalent to noncommutative algebra of canonical type. Influence of noncommutativity on the energy levels of hydrogen atom is studied in rotationally invariant noncommutative phase space. We find corrections to the levels up to the second order in the parameters of noncommutativity and estimate the upper bounds of these parameters.

The weak equivalence principle is studied in a space with noncommutativity of coordinates and noncommutativity of momenta. We find conditions on the parameters of noncommutativity which give the possibility to recover the equivalence principle in noncommutative phase space. It is also shown that in the case when these conditions are satisfied the motion of the center-of-mass of a composite system in noncommutative phase space and the relative motion are independent, the kinetic energy of composite system has additivity property and is independent on the systems composition. So, we propose conditions on the parameters of noncommutativity which give the possibility to solve the list of problems in noncommutative phase space.