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Iranian Journal of Physics Research

Calculation of the ground and exited states of the Ising model using Quantum Approximate Optimization Algorithm(QAOA)

Davoodi Yeganeh, Hossein, Mahdian, Mahmoud, Mohammadi Aghdaragh, Reza

Published May 1, 2022• 0 citations
Publisher

Abstract

The one-dimensional quantum Ising model is used to describe many physical systems, therefore the calculation of ground and excited state energies of Ising model that has many applications in statistical mechanics, has been paid attention. We obtain the ground and excited states of this model by using the Quantum Approximate Optimization Algorithm(QAOA), which is a kind of hybrid quantum-classical algorithms. For this purpose, first, we find the ground state with quantum approximate optimization algorithm, then we generalize this algorithm by adding an overlap term to obtain excited states. We calculated ground state energy and excited states energies for different coupling coefficients which is compatible with exact calculation.

DOI
10.47176/ijpr.22.1.81301

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The one-dimensional quantum Ising model is used to describe many physical systems, therefore the calculation of ground and excited state energies of Ising model that has many applications in statistical mechanics, has been paid attention. We obtain the ground and excited states of this model by using the Quantum Approximate Optimization Algorithm(QAOA), which is a kind of hybrid quantum-classical algorithms. For this purpose, first, we find the ground state with quantum approximate optimization algorithm, then we generalize this algorithm by adding an overlap term to obtain excited states. We calculated ground state energy and excited states energies for different coupling coefficients which is compatible with exact calculation.
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Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ([Formula: see text]) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
Abstract The performance of the quantum approximate optimization algorithm is evaluated by using three different measures: the probability of finding the ground state, the energy expectation value, and a ratio closely related to the approximation ratio. The set of problem instances studied consists of weighted MaxCut problems and 2-satisfiability problems. The Ising model representations of the latter possess unique ground states and highly degenerate first excited states. The quantum approximate optimization algorithm is executed on quantum computer simulators and on the IBM Q Experience. Additionally, data obtained from the D-Wave 2000Q quantum annealer are used for comparison, and it is found that the D-Wave machine outperforms the quantum approximate optimization algorithm executed on a simulator. The overall performance of the quantum approximate optimization algorithm is found to strongly depend on the problem instance.
The quantum approximate optimization algorithm (QAOA) has proved to be an effective classical-quantum algorithm serving multiple purposes, from solving combinatorial optimization problems to finding the ground state of many-body quantum systems. Since the QAOA is an Ansatz-dependent algorithm, there is always a need to design Ans\"atze for better optimization. To this end, we propose a digitized version of the QAOA enhanced via the use of shortcuts to adiabaticity. Specifically, we use a counterdiabatic (CD) driving term to design a better Ansatz, along with the Hamiltonian and mixing terms, enhancing the global performance. We apply our digitized-CD QAOA to Ising models, classical optimization problems, and the PPP-spin model, demonstrating that it outperforms the standard QAOA in all cases we study.