arXiv subjects
Publications and source records attributed to V. M. Tkachuk.
The supersymmetric properties of the generalized Dirac oscillator allow us to determine the entanglement entropy between the spin and the continuous variable in a purely algebraic manner. The entanglement has a relativistic origin and disappears in the nonrelativistic limit. The entanglement entropy attains its maximal value in the limit of infinite energy.
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. This protocol allows one to measure the zeros 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 consider the Dirac $δ$-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 present a method of a quantum simulation of a quantum harmonic oscillator in a special case of the deformed commutation relation, which corresponds to the so-called q-deformed oscillator on an IBM quantum computer. Using the method of detection of energy levels of a spin system on a quantum computer by probe spin evolution we obtain the energy levels of both the q-deformed quantum harmonic and anharmonic oscillators.
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.
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).
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.
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 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.
In general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the $δ'(x)$ potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for $δ'(x)$ and $δ(x)-δ'(x)$ 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 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 Schrödinger 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.
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 the Schrödinger 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 Schrödinger 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 Schrödinger cat state.
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 Schrödinger 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.
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.
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.