The quantum approximate optimization algorithm (QAOA) is an efficient method for solving combinatorial optimization problems in quantum computing. These problems involve finding the best solution from a finite set of possibilities. At its core, the QAOA uses an Ansatz circuit composed of alternating unitary operators, the mixing and problem Hamiltonians, that are controlled by a set of parameters. Its goal is to find the optimal parameters so that the final quantum state of the circuit encodes the problem’s solution. While this parameter optimization is often handled by classical optimizers, including constrained optimization by linear approximations (COBYLA) and Nelder–Mead, these methods frequently present local extrema. Therefore, we developed a layerwise grid search (LGS) method as an alternative. Since a full grid search is too time-consuming, the LGS method significantly reduces the search time while still finding a good solution. To demonstrate its effectiveness, we present experimental results for the max-cut problem, comparing the performance of our LGS method against conventional classical optimizers.
The quantum approximate optimization algorithm (QAOA) is an efficient method for solving combinatorial optimization problems in quantum computing. These problems involve finding the best solution from a finite set of possibilities. At its core, the QAOA uses an Ansatz circuit composed of alternating unitary operators, the mixing and problem Hamiltonians, that are controlled by a set of parameters. Its goal is to find the optimal parameters so that the final quantum state of the circuit encodes the problem’s solution. While this parameter optimization is often handled by classical optimizers, including constrained optimization by linear approximations (COBYLA) and Nelder–Mead, these methods frequently present local extrema. Therefore, we developed a layerwise grid search (LGS) method as an alternative. Since a full grid search is too time-consuming, the LGS method significantly reduces the search time while still finding a good solution. To demonstrate its effectiveness, we present experimental results for the max-cut problem, comparing the performance of our LGS method against conventional classical optimizers.
Feature selection remains a challenge in high-dimensional data analysis. Identifying the optimal feature subset is a complex combinatorial problem often classified as nondeterministic polynomial-time-hard. Traditional classical heuristics frequently struggle to navigate the exponentially growing search space. In this paper, we propose a feature selection method utilizing the quantum approximate optimization algorithm (QAOA) integrated with a layerwise grid search (LGS) strategy. By formulating feature selection as a max-cut problem, we leverage QAOA to optimize the selection process. Unlike standard global optimization, the LGS approach sequentially optimizes parameters for each circuit layer. This systematic discretization of the parameter space ensures stable convergence and enhances overall performance. Experimental evaluations on the iris flower dataset demonstrate the effectiveness of the proposed method in addressing feature selection challenges.
Over the past few years, many researchers around the world have been keen to know the potential and efficiency of quantum computers. The researchers have focused on specific issues that classical computers cannot solve or issues that quantum computers can handle in a better way. Among these various attractive research topics in quantum computers, this paper introduces the Quantum Approximate Optimization Algorithm (QAOA) which guarantees relatively considerable performances in many combinatorial optimization problems. For the comprehensive understanding of QAOA, this paper also describes the approximate optimization, the Quantum Alternating Operator Ansatz, and applications. Besides the theories of QAOA and Quantum Alternating Operator Ansatz, this paper explains the applications of QAOA to major combinatorial optimization problems such as maximum cut (MaxCut) problem and the max-independent set (MIS) problem.
In the Noisy Intermediate-Scale Quantum (NISQ) era, the Quantum Approximate Optimization Algorithm (QAOA) has attracted significant attention as an algorithm for solving combinatorial optimization problems that are classically intractable. Various modifications and training strategies have been proposed to enhance the performance of QAOA. However, as the circuit depth increases, existing methods encounter greater training difficulty and reach a barren plateau in performance improvements. In this paper, we propose the Smart Insert Layerwise QAOA (SIL QAOA) to overcome these challenges. In SIL QAOA, at each stage, a new layer is inserted into all possible positions; each circuit is independently optimized, and the parameter set from the circuit yielding the highest expectation value is adopted as the initial parameter for the next stage. Applying the proposed method to the MaxCut problem, we achieved up to approximately 11% higher approximation ratios compared to conventional Layerwise learning QAOA, while successfully mitigating early performance saturation observed at shallow depths. These results demonstrate the superior performance improvement capability of SIL QAOA. Furthermore, we also discuss its potential applicability to more complex optimization problems and deeper variational quantum circuits in future work.
The quantum approximate optimization algorithm (QAOA) has numerous promising applications in solving the combinatorial optimization problems on near-term Noisy Intermediate Scalable Quantum (NISQ) devices. QAOA has a quantum-classical hybrid structure. Its quantum part consists of a parameterized alternating operator ansatz, and its classical part comprises an optimization algorithm, which optimizes the parameters to maximize the expectation value of the problem Hamiltonian. This expectation value depends highly on the parameters, this implies that a set of good parameters leads to an accurate solution. However, at large circuit depth of QAOA, it is difficult to achieve global optimization due to the multiple occurrences of local minima or maxima. In this paper, we propose a parameters fixing strategy which gives high approximation ratio on average, even at large circuit depths, by initializing QAOA with the optimal parameters obtained from the previous depths. We test our strategy on the Max-cut problem of certain classes of graphs such as the 3-regular graphs and the Erdös-Rényi graphs.
Farhi et al. recently proposed a class of quantum algorithms, the quantum approximate optimization algorithm (QAOA), for approximately solving combinatorial optimization problems (E. Farhi et al., arXiv:1411.4028; arXiv:1412.6062; arXiv:1602.07674). A level-p QAOA circuit consists of p steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As p increases, parameter optimization becomes inefficient due to the curse of dimensionality. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here we analytically and numerically study parameter setting for the QAOA applied to MaxCut. For the level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MaxCut, the ``ring of disagrees,'' or the one-dimensional antiferromagnetic ring, we provide an analysis for an arbitrarily high level. Using a fermionic representation, the evolution of the system under the QAOA translates into quantum control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of the QAOA for any p. It also greatly simplifies the numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional submanifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.
The Quantum Approximate Optimization Algorithm (QAOA) is one of the promising near-term algorithms designed to find approximate solutions for combinatorial optimization problems. The algorithm prepares a parametrized state that is aimed to maximize the expectation value of the objective function of the problem. The circuit for QAOA consists of p layers, and it depends on 2p parameters, the determination of which can be carried out using a variational quantum eigensolver (VQE) subroutine. This is a variational approach for finding optimal parameters, where the quantum state is prepared with a quantum computer, and a classical computer is used for calculating the corresponding objective function value and performing an outer loop optimization routine in order to find optimal parameters. Earlier work on the algorithm has shown that QAOA is closely related to the Quantum Adiabatic Algorithm, which results in noticeable patterns in the optimal parameters. To this purpose, two methods, INTERP and FOURIER, were proposed to exploit this structure by Zhou et al. (2018). In this thesis, I extend upon these results by examining the effectiveness of one of their methods, the INTERP method, on the Max-Cut problem from graph theory. The performance of QAOA using this method was studied on several classes of graphs and benchmarked against the currently best-known classical algorithm Goemans-Williamson, which achieves an approximation ratio of ρ ≈ 0.878 on arbitrary graphs. The classes of graphs examined include cyclic, weighted and unweighted 3-regular, and Erdős–Rényi graphs with edge probabilities of 0.5 and 0.75 and graph sizes ranging from 4 to 16. Moreover, it was investigated if the INTERP method offers an advantage compared to random initialization of parameters and whether or not the method is polynomial in p. It was found that different graph classes lead to similar but different parameter patterns. Furthermore, the performance of QAOA using the INTERP method was dependent on the graph class considered. For the small graphs investigated it was found that QAOA outperforms Goemans-Williamson on ER-0.50, ER-0.75 and weighted 3-regular graphs from p=7 and beyond with respect to how well the expectation of the objective function approximates the optimal value, while for unweighted 3-regular graphs p=6 suffices. However, the calculation of the parameters is costly and requires a number of function evaluations that is on the order of the number of possible bipartitions of small graph with sizes n ≤ 16. Moreover, the numerical results suggested that indeed the method is polynomial in p advocating that the algorithm might be a strong alternative to Goemans-Williamson for large graph sizes, once hardware allows it.