Regularized Warm-Started Quantum Approximate Optimization and Conditions for Surpassing Classical Solvers on the Max-Cut Problem

TL;DR

Regularized Warm-Started QAOA (RWS-QAOA) is introduced, which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling, and proposes a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm.

Abstract

Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on $3$-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on $96$-node instances. Second, tensor-network simulations on graphs with up to $N{=}10{,}000$ nodes show that depth-$6$ RWS-QAOA, achieving an average cut fraction of $0.9167$, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below $0.2$ seconds on $3{,}000$-node graphs with fewer than $1.3$ million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.

Figures (13)

• Figure 1: Overview of the proposed RWS-QAOA algorithm. (A) Algorithmic pipeline of Goemans--Williamson (GW), Halperin--Livnat--Zwick (HLZ), Burer--Monteiro (BM) and RWS-QAOA. All four methods solve a continuous relaxation and then round to a bitstring. RWS-QAOA inserts a constant-depth QAOA circuit with a fixed, instance-independent parameter schedule between the relaxation and measurement steps. (B) RWS-QAOA circuit. A regularized warm-start optimization sets per-qubit $R_y(\theta)$ rotations, placing each qubit in a biased superposition. The state then undergoes $p$ alternating layers of cost and mixer evolution.
• Figure 2: Regularized Warm-started QAOA outperforms algorithms with best provable guarantees on hardware. Experimental results of Max-Cut on five $N=96$$3$-regular graphs executed on the Quantinuum trapped-ion device Helios. RWS‑QAOA, with and without a single‑step local search (LS), outperforms Goemans--Williamson (GW), Halperin--Livnat--Zwick (HLZ), and standard QAOA (each with and without LS) in both approximation ratio and success probability. HLZ natively includes a local search step. Shaded areas denote standard errors.
• Figure 3: RWS-QAOA outperforms classical heuristics on large graphs and conditions for runtime crossover. (A) Average cut fraction of fixed-depth RWS-QAOA ($p=6$) versus classical heuristics on 3-regular graphs, where local-search post-processing and, where applicable, iterative refinement are disabled for all solvers (see text for justification). (B) Fixed-depth RWS-QAOA consistently performs as well as or better than the restricted Burer--Monteiro algorithm on ten $N=2{,}000$ random graphs with different degrees. RWS-QAOA is simulated using our best known parameters (see \ref{['tab:fixed_parameters']}). (C) Estimated runtime for unrestricted Burer--Monteiro (MQLib and parallelized MQLib+) to match the solution quality of RWS-QAOA with $p=6$. For $N \leq 10{,}000$, the target cut fraction is obtained from tensor-network simulation; for $N > 10{,}000$ (faint and dashed lines), the target is held fixed at the $N{=}10{,}000$ value ($\approx 0.9167$). The RWS-QAOA runtime considers the overhead of fault-tolerant execution on a superconducting quantum computer. Shaded areas and error bars denote standard error over graph instances.
• Figure 4: QAOA parameter setting in RWS-QAOA. (A) The protocol for obtaining fixed parameters. We solve the regularized initial states for multiple graphs. Given a target $p$, we extract the subgraphs from the original graphs and sample a subset to conduct the parameter optimization. (B) Visualization of the fixed parameters and instance-wise optimized parameters at $p=1$. The per-instance parameters are numerically optimized for $N=1{,}000$ instances, while the fixed parameters are obtained through the proposed protocol. (C) Performance of RWS-QAOA for five $N=500$ instances under different parameter schedules up to $p=6$. The fixed parameter schedule and the per-instance optimized parameters perform very similarly, and both outperform the tree parameters. Error bars represent standard errors.
• Figure 5: Regularization strength in RWS-QAOA. (A) An illustrative example showing the impact of $\lambda$. With a proper choice of $\lambda$, the evolved quantum state concentrates towards the high-quality state quickly. (B) Cut fraction of five $N=2{,}000$$3$-regular graphs under different $\lambda$ for depths up to $p=4$; $p=0$ represents the initial state only. Here, the RWS-QAOA parameters are fixed to be the tree parameters wurtz2021fixed. Error bars represent standard errors. Empirically, $\lambda = 0.6$ performs well.