Scalable Clifford-Based Classical Initialization for the Quantum Approximate Optimization Algorithm

TL;DR

This work tackles the initialization bottleneck in QAOA by introducing SPIQ, a scalable framework that combines a relaxed ma-QAOA ansatz with Clifford-state search to produce high-quality, classically computable starting points. SPIQ uses two seed strategies (Fixed-Interval and K-GAPS) to seed multi-start optimization, enabling robust exploration and improved convergence on QUBO, PUBO, and PCBO problems, including Max-Cut, Knapsack, and medical high-order interaction tasks. Across diverse benchmarks and problem scales, SPIQ achieves near-ground-state initializations (up to 99.9% accuracy in some cases) and dramatically reduces the initial search space (up to 10^4× on average, with certain instances exceeding 10^4×), translating into substantial reductions in quantum resource costs. The framework generalizes beyond graph-structured problems, demonstrates scalability to tens–hundreds of qubits, and offers practical impact for near- and intermediate-term quantum devices by enabling faster, more reliable QAOA optimization and guiding future work on multi-start strategies and fault-tolerant deployments.

Abstract

Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA), offer a promising route to tackling combinatorial optimization problems on near and intermediate-term quantum devices. However, their performance critically depends on the choice of initial parameters, and the limited expressiveness of the QAOA ansatz makes identifying effective initializations both difficult and unscalable. To address this, we propose a framework, Scalable Parameter Initialization for QAOA (SPIQ), that employs a relaxed QAOA ansatz to enable classical search over a set of Clifford-preparable quantum states that yield high-quality solutions. These states serve as superior QAOA initializations, driving rapid convergence while significantly reducing the quantum circuit evaluations needed to reach high-quality solutions and consequently lowering quantum-device cost. We present a scalable, application-agnostic initialization framework that achieves an absolute accuracy improvement of up to 80% over state-of-the-art initialization and reduces initial-state diversity by up to 10,000x across QUBO, PUBO, and PCBO problems spanning tens to hundreds of qubits. We further benchmark its performance on a wide range of problem formulations and instances derived from real-world datasets, demonstrating consistent and scalable improvements. Furthermore, we introduce two complementary strategies for selecting high-quality Clifford points identified by our search procedure and using them to seed multi-start optimization, thereby enhancing exploration and improving solution quality.

Figures (14)

• Figure 1: (Left): High-level illustration of how initialization frameworks identify high-quality starting points across variational algorithms. The state-of-the-art Clifford approach (CAFQA) performs well for VQE but fails on standard QAOA due to its extremely limited optimization landscape. In contrast, SPIQ with QAOA yields robust and effective initialization. (Right): (a) Only a few, low-quality Clifford states are found with CAFQA, while (b) our framework, SPIQ, finds many high-quality Clifford points.
• Figure 2: (Left): Heatmap of unique Clifford states found by CAFQA, with both states yielding expectation values $\approx 0$, indicating that only a few states are explored and resulting in low-quality solutions. (Right): Comparison of best energies, showing that a random guess (best of 50) can outperform CAFQA initialization for QAOA.
• Figure 3: Overview of the QAOA + SPIQ workflow
• Figure 4: Illustration comparing two strategies for selecting points from high-quality SPIQ states: (a) naive selection may get stuck in local minima or barren plateaus, (b) choosing diverse points near steep curves helps avoid non-convex regions.
• Figure 5: Illustration of graphs used for Max-Cut with 6 nodes as an example: (a) Each node is connected to every other node. (b) Each node is connected to exactly three other nodes. (c) Central node and its neighbors.