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Visnyk of the Lviv University Series Physics

Quantum calculations of the speed of evolution of quantum states of a two-spin system with \\ XY interaction

Markian Kachmar, Kh. P. Gnatenko

Published Jan 1, 2025• 0 citations
Publisher

Abstract

In this paper the speed of evolution of a two-spin system with XY interaction is studied.The analysis is carried out both analytically and using quantum computing.The dependence of the speed on the coupling constants and the parameters of the initial state is investigated.The evolution speed is calculated through quantum calculations on AerSimulator, based on the study of quantum correlators and the time dependence of the mean value of the evolution operator.The results of the quantum calculations are in close agreement with the theoretical predictions.

DOI
10.30970/vph.62.2025.67

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In this paper the speed of evolution of a two-spin system with XY interaction is studied.The analysis is carried out both analytically and using quantum computing.The dependence of the speed on the coupling constants and the parameters of the initial state is investigated.The evolution speed is calculated through quantum calculations on AerSimulator, based on the study of quantum correlators and the time dependence of the mean value of the evolution operator.The results of the quantum calculations are in close agreement with the theoretical predictions.
The entanglement distance of evolutionary quantum states of a two-spin system with the XXZ model has been studied. The analysis has been conducted both analytically and using quantum computing. An analytical dependence of the entanglement distance on the values of the model coupling constants and the parameters of the initial states has been obtained. The speed of evolution of a two-spin system has been investigated. The analysis has been performed analytically and using quantum computing. An explicit dependence of the speed of evolution on the coupling constants and on the parameters of the initial state has been obtained. The results of quantum computations are in good agreement with the theoretical predictions.
The dynamics of two 1∕21∕21∕2-spin qubits under the influence of a quantum Heisenberg XYXYXY-type spin-bath is studied. After the Holstein-Primakoff transformation, numerical polynomial scheme is used to give the time-evolution calculation of the center qubits initially prepared in a product state or a Bell state. Then the concurrence of the two qubits, the zzz-component moment of either of the subsystem spins, and the fidelity of the subsystem are shown; they exhibit sensitive dependence on the anisotropic parameter, the temperature, the coupling strength, and the initial state. It is found that (i) the larger the anisotropic parameter $\ensuremath{\gamma}$, the bigger the probability of maintaining the initial state of the two qubits; (ii) with increasing temperature TTT, the bath plays a stronger destroy effect on the dynamics of the subsystem, so does the interaction g0{g}_{0}g0​ between the subsystem and the bath; and (iii) the time evolution of the subsystem is dependent on the initial state. The revival of the concurrence does not always mean the restoration of the state. Further, the dynamical properties of the subsystem should be judged by the combination of concurrence and fidelity.
Exact analytical calculations of eigenvalues and eigenstates are presented for quantum coupled double-well (DW) systems with Razavy's hyperbolic potential. With the use of four kinds of initial wavepackets, we have calculated the tunneling period TTT and the orthogonality time τττ which signifies a time interval for an initial state to evolve to its orthogonal state. We discuss the coupling dependence of TTT and τττ, and the relation between τττ and the concurrence CCC which is a typical measure of the entanglement in two qubits. Our calculations have shown that it is not clear whether the speed of quantum evolution may be measured by TTT or τττ and that the evolution speed measured by τττ (or TTT) is not necessarily increased with increasing CCC. This is in contrast with the earlier study [V. Giovannetti, S. Lloyd and L. Maccone, Europhys. Lett. {\bf 62} (2003) 615] which pointed out that the evolution speed measured by τττ is enhanced by the entanglement in the two-level model.
Abstract 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.
We study the time evolution of a single spin coupled by exchange interaction to an environment of interacting spin bath modeled by the XY Hamiltonian. By evaluating the spin correlator of the single spin, we observed that the decay rate of the spin oscillations strongly depends on the relative magnitude of the exchange coupling between the single spin and its nearest neighbor J(') and coupling among the spins in the environment J. The decoherence time varies significantly based on the relative coupling magnitudes of J and J('). The decay rate law has a Gaussian profile when the two exchange couplings are of the same order J(') approximately J but converts to exponential and then a power law as we move to the regimes of J(')>J and J(')<J. We also show that the spin oscillations propagate from the single spin to the environmental spins with a certain speed. The effect of varying the anisotropic parameter, external magnetic field, and temperature on the decaying rate of the spin state is also discussed.