Theoretical Approximation Ratios for Warm-Started QAOA on 3-Regular Max-Cut Instances at Depth $p=1$

Abstract

We generalize Farhi et al.'s 0.6924-approximation result technique of the Max-Cut Quantum Approximate Optimization Algorithm (QAOA) on 3-regular graphs to obtain provable lower bounds on the approximation ratio for warm-started QAOA. Given an initialization angle $\theta$, we consider warm-starts where the initial state is a product state where each qubit position is angle $\theta$ away from either the north or south pole of the Bloch sphere; of the two possible qubit positions the position of each qubit is decided by some classically obtained cut encoded as a bitstring $b$. We illustrate through plots how the properties of $b$ and the initialization angle $\theta$ influence the bound on the approximation ratios of warm-started QAOA. We consider various classical algorithms (and the cuts they produce which we use to generate the warm-start). Our results strongly suggest that there does not exist any choice of initialization angle that yields a (worst-case) approximation ratio that simultaneously beats standard QAOA and the classical algorithm used to create the warm-start. Additionally, we show that at $\theta =60^{\circ }$, warm-started QAOA is able to (effectively) recover the cut used to generate the warm-start, thus suggesting that in practice, this value could be a promising starting angle to explore alternate solutions in a heuristic fashion.

arXiv commentThe first version of this arXiv submission, titled "Guarantees on Warm-Started QAOA: Single-Round Approximation Ratios for 3-Regular MAXCUT and Higher-Round Scaling Limits", has since been split into two parts: a longer part and a shorter part. The current version of this arXiv submission is the longer part. The shorter part can be found here: arxiv:2410.00027