Abstract Quantum approximate optimization algorithm (QAOA) aims to solve discrete optimization problems by sampling bitstrings using a parameterized quantum circuit. The circuit parameters (angles) are optimized in the way that minimizes the cost Hamiltonian expectation value. Recently, general statistical properties of QAOA output probability distributions have begun to be studied. In contrast to the conventional approach, we analyse QAOA circuits with random angles. We provide analytical equations for probabilities and the numerical evidence that for unweighted Max-Cut problems on connected graphs such sampling always gives higher entropy of energy distribution than uniform random sampling of bitstrings. We also analyse the probability to obtain the global optima, which appears to be higher on average than for random sampling.
Abstract Quantum approximate optimization algorithm (QAOA) aims to solve discrete optimization problems by sampling bitstrings using a parameterized quantum circuit. The circuit parameters (angles) are optimized in the way that minimizes the cost Hamiltonian expectation value. Recently, general statistical properties of QAOA output probability distributions have begun to be studied. In contrast to the conventional approach, we analyse QAOA circuits with random angles. We provide analytical equations for probabilities and the numerical evidence that for unweighted Max-Cut problems on connected graphs such sampling always gives higher entropy of energy distribution than uniform random sampling of bitstrings. We also analyse the probability to obtain the global optima, which appears to be higher on average than for random sampling.
The quantum approximate optimization algorithm (QAOA) requires that circuit parameters are determined that allow one to sample from high-quality solutions to combinatorial optimization problems. Such parameters can be obtained using either costly outer-loop optimization procedures and repeated calls to a quantum computer or, alternatively, via analytical means. In this work, we consider a context in which one knows a probability density function describing how the objective function of a combinatorial optimization problem is distributed. We show that, if one knows this distribution, then the expected value of strings, sampled by measuring a Grover-driven, QAOA-prepared state, can be calculated independently of the size of the problem in question. By optimizing this quantity, optimal circuit parameters for average-case problems can be obtained on a classical computer. Such calculations can help deliver insights into the performance of and predictability of angles in QAOA in the limit of large problem sizes, in particular, for the number partitioning problem.
Abstract QAOA (Quantum Approximate Optimization Algorithms) is one of the most promising algorithms of Noisy Intermediate Scale Quantum (NISQ) era. The standard approach to QAOA involves the use of a hybrid quantum-classical optimization, although this approach was not considered as the main one in the original paper on QAOA. Recently, a new approach has emerged based on the hypothesis that optimal circuit parameters (angles) are close for a wide class of problems. However, the search for fixed angles itself remains a challenge with different approaches. We propose one specific method based on the use of a fixed training set and the special metric associated with increasing the probability of a correct answer. We carry out the analysis of the proposed method performance on the unweighted Max-Cut problems and random weighted QUBO (Quadratic Unconstrained Binary Optimization) problems of the special type.
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.
This paper presents an experimental study of the Quantum Approximate Optimization Algorithm (QAOA) applied to the Max-Cut problem on small graph instances. Using PennyLane and a quantum simulator, a modular QAOA framework is implemented and evaluated on a fixed set of graphs, including ring, complete, star, and random graphs. For each graph, QAOA circuits with one and two layers are optimized using gradient-based methods, and the optimized states are sampled to obtain candidate cut solutions. The analysis includes visualizations of graph structures, cut partitions, bitstring distributions, and quantum circuit diagrams. This study serves as a clear and reproducible demonstration of QAOA behavior on combinatorial optimization problems.
Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of L quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima. Published by the American Physical Society 2024
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.