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IEEE International Conference on Quantum Computing and Engineering (QCE)

Detecting Energy Levels of Spin Systems on IBM’s Quantum Computer by Evolution of Mean Value of Physical Quantity

Kh. P. Gnatenko, H. P. Laba, V. M. Tkachuk

Published Oct 1, 2021• 0 citations
Publisher

Abstract

Finding energy levels of a quantum system is an important problem that can be solved using quantum computers. We showed that transition energies of a quantum system are related with the evolution of mean value of a physical quantity. In particular case when the operator of a physical quantity anticommutes with the Hamiltonian of a system we found that studies of the evolution of the mean value of this physical quantity give a possibility to determine the energy levels of the system and their degeneracy. On the basis of the results the energy levels of spin systems were detected on IBMs quantum computers ibmq manila and ibmq melbourne. It is worth noting that spin systems are one of the most suitable systems for modeling them on a quantum computer. We considered a spin in magnetic field, spin chain with Ising interaction, spin square-lattice with isotropic and spatially anisotropic Ising interactions. The results obtained on the quantum devices correspond to the analytical one. It is important to note that finding the maximal or the minimal energy of the Ising model on square lattice with spatially anisotropic interaction is not a trivial problem. Therefore, the proposed algorithm for obtaining the energy levels opens a possibility to achieve the quantum supremacy in solving this problem on many qubit quantum computers.

DOI
10.1109/qce52317.2021.00065

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Finding energy levels of a quantum system is an important problem that can be solved using quantum computers. We showed that transition energies of a quantum system are related with the evolution of mean value of a physical quantity. In particular case when the operator of a physical quantity anticommutes with the Hamiltonian of a system we found that studies of the evolution of the mean value of this physical quantity give a possibility to determine the energy levels of the system and their degeneracy. On the basis of the results the energy levels of spin systems were detected on IBMs quantum computers ibmq manila and ibmq melbourne. It is worth noting that spin systems are one of the most suitable systems for modeling them on a quantum computer. We considered a spin in magnetic field, spin chain with Ising interaction, spin square-lattice with isotropic and spatially anisotropic Ising interactions. The results obtained on the quantum devices correspond to the analytical one. It is important to note that finding the maximal or the minimal energy of the Ising model on square lattice with spatially anisotropic interaction is not a trivial problem. Therefore, the proposed algorithm for obtaining the energy levels opens a possibility to achieve the quantum supremacy in solving this problem on many qubit quantum computers.
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.
The one-dimensional Ising model with its connections to several physical concepts plays a vital role in comprehension of several principles, phenomena and numerical methods. The Hamiltonian of a coupled one-dimensional dissipative spin system in the presence of magnetic field can be obtained from the Ising model. We simulate the above Hamiltonian by designing a quantum circuit with precise gate measurement and execute with the IBMQ experience platform through different [Formula: see text] states with controlled energy separation where we can check quantum synchronization in a dissipative lattice system. Our result shows the relation between various entangled states, the relation between the different energy separation ([Formula: see text]) with the spin–spin coupling ([Formula: see text]) in the lattice, along with fidelity calculations for several iterations of the model used. We also estimate the ground and first excited energy states of Ising-Hamiltonian using VQE algorithm and investigate the lowest energy values varying the number of layers of ansatz.
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 propose a detailed analysis of datasets generated from simulations of two-dimensional quantum spin systems using the quantum Ising model at absolute zero temperature. Our focus is on examining how fundamental physical properties, energy, magnetization, and entanglement entropy, evolve under varying external transverse magnetic fields and system sizes. From the Quantum Toolbox in Python (QuTiP), we simulate systems with 4, 8, and 16 spins arranged in square lattices, generating extensive datasets with 5000 samples per magnetic field value. The Hamiltonian operator incorporates quantum mechanical effects such as superposition and tunneling, challenging classical interpretations of spin states. We compute extended Pauli operators and construct the Hamiltonian to include spin-spin interactions and transverse field terms. Our analysis reveals that as the system size increases, fluctuations in energy and entanglement entropy become more evident, indicating lifted sensitivity to external perturbations and suggesting the onset of quantum phase transitions. Spin-spin correlation functions demonstrate that interactions are predominantly local, but larger systems exhibit more complex and fluctuating correlations. These findings provide valuable insights into the behavior of quantum spin systems and lay the groundwork for future machine learning applications aimed at predicting physical quantities and identifying phase transitions from a quantum perspective.
We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.