Phantom Edges in the Problem Hamiltonian: A Method for Increasing Performance and Graph Visibility for QAOA

TL;DR

Phantom-QAOA introduces a modified graph with additional weighted edges to enhance the ability of QAOA to explore more of the graph's structure at low depths, improving approximation ratios for combinatorial problems.

Contribution

The paper proposes Phantom-QAOA, a novel ansatz adding a single parameter to expand QAOA's graph visibility without increasing circuit depth.

Findings

Improved approximation ratios for cycle graphs.

Significant enhancement in Max-Cut performance on random regular graphs.

Analytical proof of improvement at p=1.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm that can be used to approximately solve combinatorial optimization problems. However, a major limitation of QAOA is that it is a "local" algorithm for finite circuit depths, meaning it can only optimize over local properties of the graph. In this paper, we present Phantom-QAOA, a new QAOA ansatz that introduces only one additional parameter to the standard ansatz -- regardless of system size -- allowing QAOA to "see" more of the graph at a given depth $p$. We achieve this by modifying the target graph to include additional $\alpha$-weighted edges, with $\alpha$ serving as a tunable parameter. This modified graph is then used to construct the phase operator and allows QAOA to explore a wider range of the graph's features. We derive a general formula for our new ansatz at $p=1$ and analytically show an improvement in the approximation ratio for cycle graphs. We also provide numerical experiments that demonstrate significant improvements in the approximation ratio for the Max-Cut problem over the standard QAOA ansatz for $p=1$ and $p=2$ on random regular graphs up to 16 nodes.