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arXiv:2205.11762 · Physical Review Applied

QAOA-in-QAOA: Solving Large-Scale MaxCut Problems on Small Quantum Machines

Z.-F. Zhou, Yuxuan Du, Xinmei Tian, Dacheng Tao

Published Feb 9, 2023• 77 citations
arXivHTMLPDF

Abstract

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 ${\mathbb{Z}}_{2}$ 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 (${\mathrm{QAOA}}^{2}$) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the performance of ${\mathrm{QAOA}}^{2}$ is lower bounded with respect to the divide-and-conquer process. Experiment results illustrate that, under different graph settings, ${\mathrm{QAOA}}^{2}$ 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.

DOI
10.1103/physrevapplied.19.024027

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Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising solution for combinatorial optimization problems using a hybrid quantum-classical framework. Among combinatorial optimization problems, the Maximum Cut (Max-Cut) problem is particularly important due to its broad applicability in various domains. While QAOA-based Max-Cut solvers have been developed, they primarily favor solution accuracy over execution efficiency, which significantly limits their practicality for large-scale problems. To address the limitation, we propose ParaQAOA, a parallel divide-and-conquer QAOA framework that leverages parallel computing hardware to efficiently solve large Max-Cut problems. ParaQAOA significantly reduces runtime by partitioning large problems into subproblems and solving them in parallel while preserving solution quality. This design not only scales to graphs with tens of thousands of vertices but also provides tunable control over accuracy-efficiency trade-offs, making ParaQAOA adaptable to diverse performance requirements. Experimental results demonstrate that ParaQAOA achieves up to 1,600x speedup over state-of-the-art methods on Max-Cut problems with 400 vertices while maintaining solution accuracy within 2% of the best-known solutions. Furthermore, ParaQAOA solves a 16,000-vertex instance in 19 minutes, compared to over 13.6 days required by the best-known approach. These findings establish ParaQAOA as a practical and scalable framework for large-scale Max-Cut problems under stringent time constraints.
Quantum approximate optimization algorithm (QAOA) is a promising hybrid quantum-classical algorithm for solving combinatorial optimization problems. However, it cannot overcome qubit limitation for large-scale problems. Furthermore, the simulation time of QAOA scales poorly with the problem size. We propose a divide-and-conquer QAOA (DC-QAOA) to address the above challenges for graph maximum cut (MaxCut) problem. The algorithm works by recursively partitioning a larger graph into smaller ones whose MaxCut solutions are obtained with small-size noisy intermediate-scale quantum computers. The overall solution is retrieved from the subsolutions by applying the combination policy of measurement distribution reconstruction (MDR). The solution quality depends on the graph partitioning algorithm and MDR policy. Multiple partitioning and reconstruction methods are proposed and compared. Results are evaluated by metrics, such as quantum program runtime, measurement expectation value (EV), and approximation ratio (AR). The results show that DC-QAOA achieves 97.14% AR (20.32% higher than classical counterpart), and 94.79% EV (15.80% higher than quantum annealing). DC-QAOA solves large-scale graph instances with a polynomial rate or returns unsuccessful partition if graph connectivity requirement is not fulfilled otherwise.
The Quantum approximate optimization algorithm (QAOA) is a leading hybrid classical-quantum algorithm for solving complex combinatorial optimization problems. QAOA-in-QAOA (QAOA2) uses a divide-and-conquer heuristic to solve large-scale Maximum Cut (MaxCut) problems, where many sub-graph problems can be solved in parallel. In this work, an implementation of the QAOA2method for the scalable solution of the MaxCut problem is presented, based on the Classiq platform. The framework is executed on an HPE-Cray EX supercomputer by means of the Message Passing Interface (MPI) and the SLURM workload manager. The limits of the Goemans-Williamson (GW) algorithm as a purely classical alternative to QAOA are investigated to understand if QAOA2could benefit from solving certain sub-graphs classically. Results from large-scale simulations of up to 33 qubits are presented, showing the advantage of QAOA in certain cases and the efficiency of the implementation, as well as the adequacy of the workflow in the preparation of real quantum devices. For the considered graphs, the best choice for the sub-graphs does not significantly improve results and is still outperformed by GW.
Quantum computers are devices which allow the solution of problems unsolvable to their classical counterparts. As an error-corrected quantum computer is still a decade away the quantum computing community has dedicated much attention to developing algorithms for currently available Noisy Intermediate-Scale Quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are exclusively tackled with the Quantum Approximate Optimization Algorithm (QAOA). This algorithm predominantly computes graph partitions with a maximal separation of edges (MaxCut), but can also be modified to calculate other properties of graphs. Here, I present a novel quantum optimization algorithm which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem. Such an improved performance allowed me to partition a graphs 32 nodes on publicly available 5 qubit gate-based solid state quantum devices without any preprocessing such as division of the graph into smaller subgraphs. This results represent a 40% increase in graph size as compared to state-of-art experiments on solid state devices such as Google Sycamore. The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal quantum simulators. Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with quantum simulators, again without any pre-division into smaller subproblems and a lower solution bound of 67.9% is achieved.
Quantum computers are devices, which allow more efficient solutions of problems as compared to their classical counterparts.As the timeline to developing a quantum-error corrected computer is unclear, the quantum computing community has dedicated much attention to developing algorithms for currently available noisy intermediate-scale quantum computers (NISQ).Thus far, within NISQ, optimization problems are one of the most commonly studied and are quite often tackled with the quantum approximate optimization algorithm (QAOA).This algorithm is best known for computing graph partitions with a maximal separation of edges (MaxCut), but can easily calculate other problems related to graphs.Here, I present a novel quantum optimization algorithm, which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem.Such an improved performance allowed me to partition graphs with 32 nodes on publicly available 5 qubit gate-based quantum computers without any preprocessing such as division of the graph into smaller subgraphs.These results represent a 40% increase in graph size as compared to state-of-art experiments on gate-based quantum computers such as Google Sycamore.The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal simulators of quantum computers.Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with simulators of quantum computers, again without any predivision into smaller subproblems and a lower solution bound of 67.9% is achieved.The study presented here paves way to using powerful genetic optimizer in synergy with quantum computers.
The transition from theoretical quantum computing to practical real-world applications heavily relies onhybrid quantum-classical architectures designed for NoisyIntermediate-Scale Quantum (NISQ) devices. Among these approaches, the Quantum Approximate Optimization Algorithm(QAOA) is a prominent strategy for addressing complex combinatorial optimization challenges, such as the NP-hard MaxCut problem. Instead of performing the entire computationon a quantum processor, QAOA utilizes a quantum circuit toevaluate a parameterized cost function, whereas a classical optimizer refines the parameters iteratively. This study presentsa rigorous empirical evaluation comparing the performanceof QAOA against established classical solvers on small-scaleMax-Cut instances. All experiments were implemented andexecuted using Python (Qiskit Aer) on a state-vector simulator.Using Erdős–Rényi random graphs and regular graphs with4 to 20 nodes, we benchmark QAOA at depths p = 2 andp = 3 against simulated annealing, the Goemans–WilliamsonSDP-based algorithm, and a linear-time greedy heuristic. Ourexperimental results demonstrate that low-depth QAOA implementations exhibit severe performance degradation, prohibitive computational overhead (O(2^n) scaling), and low convergencerates compared to mature classical algorithms. At 20 nodes,QAOA (p = 2) achieves only a 0.467 approximation ratio versus0.856 for Goemans–Williamson, with a 210× runtime slowdownover greedy heuristics