Combinatorial optimization is among the main applications envisioned for near-term and fault-tolerant quantum computers. In this work, we consider a well-studied quantum algorithm for combinatorial optimization: the Quantum Approximate Optimization Algorithm (QAOA) applied to the MaxCut problem on 3-regular graphs. We explore the idea of improving the solutions returned by the simplest version of the algorithm (depth-1 QAOA) using a form of postselection that can be efficiently simulated by state preparation. We derive theoretical upper and lower bounds showing that a constant (though small) increase of the fraction of satisfied edges is indeed achievable. Numerical experiments on large problem instances (beyond classical simulatability) complement and support our bounds. We also consider a distinct technique: local updates, which can be applied not only to QAOA but any optimization algorithm. In the case of QAOA, the resulting improvement can be sharply quantified theoretically for large problem instances and in absence of postselection. Combining postselection and local updates, the theory is no longer tractable but numerical evidence suggests that improvements from both methods can be combined.
Combinatorial optimization is among the main applications envisioned for near-term and fault-tolerant quantum computers. In this work, we consider a well-studied quantum algorithm for combinatorial optimization: the Quantum Approximate Optimization Algorithm (QAOA) applied to the MaxCut problem on 3-regular graphs. We explore the idea of improving the solutions returned by the simplest version of the algorithm (depth-1 QAOA) using a form of postselection that can be efficiently simulated by state preparation. We derive theoretical upper and lower bounds showing that a constant (though small) increase of the fraction of satisfied edges is indeed achievable. Numerical experiments on large problem instances (beyond classical simulatability) complement and support our bounds. We also consider a distinct technique: local updates, which can be applied not only to QAOA but any optimization algorithm. In the case of QAOA, the resulting improvement can be sharply quantified theoretically for large problem instances and in absence of postselection. Combining postselection and local updates, the theory is no longer tractable but numerical evidence suggests that improvements from both methods can be combined.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
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.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.
The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by the limited quantum resources nowadays, QAOAs are infeasible to manipulate large-scale problems. To address this issue, here we revisit the MaxCut problem via the divide-and-conquer heuristic: seek the solutions of subgraphs in parallel and then merge these solutions to obtain the global solution. Because of the Z2 symmetry in MaxCut, we prove that the merging process can be further cast into a new MaxCut problem and thus be addressed by QAOAs or other MaxCut solvers. In view of this, we propose QAOA-in-QAOA (QAOA2) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the performance of QAOA2 is lower bounded with respect to the divide-and-conquer process. Experiment results illustrate that, under different graph settings, QAOA2 attains a competitive or even better performance over classical algorithms when the node count is around 2000. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale combinatorial optimization problems.
The solution of large-scale combinatorial optimization problems, which are exorbitantly complex and resource-consuming, is one of the most significant challenges that computing systems face. Traditional algorithms often provide suboptimal results when confronted with bigger issues. This research aims to bring attention to these problems by introducing the Extending Quantum Approximate Optimization Algorithm (EQAOA) framework, which improves scalability, parameter optimization, and error resilience. Fundamental to the EQAOA methodology is repeatedly optimizing solution states via adaptive-depth, parameter-tuned quantum circuits. EQAOA also employs the method of converting combinatorial problems into cost Hamiltonians. Similar problems are shown in the Max-Cut and Graph Coloring programs. The experiments were conducted on several benchmark circumstances using real NISQ hardware and quantum simulators. The enhanced EQAOA outperforms baseline versions in terms of convergence time and solution quality. The efficient integration of dynamic circuit depth with hybrid classical-quantum parameter approaches applied to increasingly complicated problem scenarios and yielded good results. According to the study's findings, applying the proposed QAOA extension to practical combinatorial contexts yields a quantum advantage. The future of this field is bright, with researchers aiming to improve its interaction with traditional optimizers and extend its usage to other practical optimization areas.
Abstract Maximum cut (MaxCut) on graphs is a classic NP-hard problem. In quantum computing, Farhi, Gutmann, and Goldstone proposed the Quantum Approximate Optimization Algorithm (QAOA) for solving the MaxCut problem. Its guarantee on cut fraction (the fraction of edges in the output cut over all edges) was mainly studied for high-girth graphs, i.e., graphs with only long cycles. On the other hand, low-girth graphs are ubiquitous in theoretical computer science, including expander graphs being outstanding examples with wide applications in theory and beyond. In this paper, we apply QAOA to MaxCut on a set of expander graphs proposed by Mohanty and O'Donnell known as additive product graphs. Additionally, we apply multi-angle QAOA (ma-QAOA) to better utilize the graph structure of additive product graphs in ansatz design. In theory, we derive an iterative formula to calculate the expected cut fraction of such graphs. This formula also extends to the quantum MaxCut problem. On the other hand, we conduct numerical experiments to compare between best-known classical local algorithms and QAOA with constant depth. Our results demonstrate that QAOA outperforms the best-known classical algorithms by 0.3% to 5.2% on several additive product graphs, while ma-QAOA further enhances this advantage by an additional 0.6% to 2.5%. In particular, we observe cases that ma-QAOA exhibits superiority over best-known classical algorithms but QAOA does not. Furthermore, we extend our experiments to planar graphs such as tiling grid graphs, where QAOA also demonstrates an advantage.