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arXiv:2509.17296

A Depth-Independent Linear Chain Ansatz for Large-Scale Quantum Approximate Optimization

Zixu Wang, Mandell, Jack, Yangyang Xu, Jian Shi

Published Sep 22, 2025• 0 citations
arXivHTMLPDF

Abstract

Combinatorial optimization lies at the heart of numerous real-world applications. For a broad category of optimization problems, quantum computing is expected to exhibit quantum speed-up over classic computing. Among various quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), as one of variational quantum algorithms, shows promise on demonstrating quantum advantage on noisy intermediate-scale quantum (NISQ) hardware. However, with increasing problem size, the circuit depth demanded by original QAOA scales rapidly and quickly surpasses the threshold at which meaningful results can be obtained. To address this challenge, in this work, we propose a variant of QAOA (termed linear chain QAOA) and demonstrate its advantages over original QAOA on paradigmatic MaxCut problems. In original QAOA, each graph edge is encoded with one entangling gate. In our ansatz, we locate a linear chain from the original MaxCut graph and place entangling gates sequentially along this chain. This linear-chain ansatz is featured by shallow quantum circuits and with the low execution time that scales independently of the problem size. Leveraging this ansatz, we demonstrate an approximation ratio of 0.78 (without post-processing) on non-hardware-native random regular MaxCut instances with 100 vertices in a digital quantum processor using 100 qubits. Our findings offer new insights into the design of hardware-efficient ansatz and point toward a promising route for tackling large-scale combinatorial optimization problems on NISQ devices.

DOI
10.48550/arxiv.2509.17296

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Combinatorial optimization lies at the heart of numerous real-world applications. For a broad category of optimization problems, quantum computing is expected to exhibit quantum speed-up over classic computing. Among various quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), as one of variational quantum algorithms, shows promise on demonstrating quantum advantage on noisy intermediate-scale quantum (NISQ) hardware. However, with increasing problem size, the circuit depth demanded by original QAOA scales rapidly and quickly surpasses the threshold at which meaningful results can be obtained. To address this challenge, in this work, we propose a variant of QAOA (termed linear chain QAOA) and demonstrate its advantages over original QAOA on paradigmatic MaxCut problems. In original QAOA, each graph edge is encoded with one entangling gate. In our ansatz, we locate a linear chain from the original MaxCut graph and place entangling gates sequentially along this chain. This linear-chain ansatz is featured by shallow quantum circuits and with the low execution time that scales independently of the problem size. Leveraging this ansatz, we demonstrate an approximation ratio of 0.78 (without post-processing) on non-hardware-native random regular MaxCut instances with 100 vertices in a digital quantum processor using 100 qubits. Our findings offer new insights into the design of hardware-efficient ansatz and point toward a promising route for tackling large-scale combinatorial optimization problems on NISQ devices.
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), which is a variational quantum algorithm, aims to give sub-optimal solutions of combinatorial optimization problems. It is widely believed that QAOA has the potential to demonstrate application-level quantum advantages in the noisy intermediate-scale quantum(NISQ) processors with shallow circuit depth. Since the core of QAOA is the computation of expectation values of the problem Hamiltonian, an important practical question is whether we can find an efficient classical algorithm to solve quantum mean value in the case of general shallow quantum circuits. Here, we present a novel graph decomposition based classical algorithm that scales linearly with the number of qubits for the shallow QAOA circuits in most optimization problems except for complete graph case. Numerical tests in Max-cut, graph coloring and Sherrington-Kirkpatrick model problems, compared to the state-of-the-art method, shows orders of magnitude performance improvement. Our results are not only important for the exploration of quantum advantages with QAOA, but also useful for the benchmarking of NISQ processors.
In recent years, significant progress has been made in building quantum computers by several companies. Despite the progress, these noisy intermediate-scale quantum (NISQ) computers still suffer from several noises and errors such as measurement errors, multi-qubit gate errors, and worse, short decoherence times. Though quantum computer vendors are releasing better quantum computers in terms of Quantum Volume, the quantum device still remains far from quantum supremacy in practical problems. The Quantum Approximate Optimization Algorithm (QAOA) was suggested to potentially demonstrate a computational advantage in combinatorial optimization problems on NISQ computers. In this paper, we optimize the QAOA circuits and to scale the problem size on IBM quantum processors. In addition, we study the effect of the length of the QAOA ansatz on IBM quantum processors and discuss optimal implementation methods for scalable QAOA. We test our implementations on the MaxCut problems.
The quantum approximate optimization algorithm (QAOA) promises to solve classically intractable computational problems in the area of combinatorial optimization. A growing amount of evidence suggests that the originally proposed form of the QAOA ansatz is not optimal, however. To address this problem, we propose an alternative ansatz, which we call QAOA+, that augments the traditional p=1 QAOA ansatz with an additional multiparameter problem-independent layer. The QAOA+ ansatz allows obtaining higher approximation ratios than p=1 QAOA while keeping the circuit depth below that of p = 2 QAOA, as benchmarked on the MaxCut problem for random regular graphs. We additionally show that the proposed QAOA+ ansatz, while using a larger number of trainable classical parameters than with the standard QAOA, in most cases outperforms alternative multiangle QAOA ansätze for fixed number of independent parameters.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.
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.