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

TL;DR

Phantom-QAOA addresses the locality limitation of QAOA at finite depth by introducing a single tunable weight $\alpha$ on phantom edges in the phase operator, effectively expanding the graph seen by the algorithm. The authors derive a general $p=1$ Max-Cut expectation and prove analytical improvements on even-length cycles, supported by numerical gains on random regular graphs up to 16 nodes for $p=1$ and $p=2$. They propose two edge-placement strategies for the augmented graph, with the triangle-based approach yielding more robust gains while controlling circuit depth. The work demonstrates meaningful improvements in approximation ratios and outlines future directions for richer edge-weight schemes, alternative augmentations of $G'$, and hardware-aware optimization trade-offs.

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 $α$-weighted edges, with $α$ 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.

Figures (6)

• Figure 1: Visualization of graph $G'$, where the black lines are the edges from the original graph $G$ and additional $\alpha$-weighted edges are denoted by red lines. For the edge $(xy)$, $d = 2$ and $d' = 1$ denote two existing and one phantom neighbor of $x$, respectively. Similarly, for vertex $y$, $e = 1$ and $e' = 2$. The triangles formed around this edge include: one with vertex $u$ ($f = 1$), one with vertex $w$ ($f' = 1$), and another with vertex $v$ ($f" = 1$).
• Figure 2: Examples of 8-vertex cycle graphs $G$ shown in black and additional $\alpha$-weighted edges shown in red. The left image shows a $G'$ that is regular and triangle-free, while the right image shows a $G'$ that is also regular but contains triangles.
• Figure 3: The average approximation ratios on 16-node graphs with degrees 4, 8, and 12 across various QAOA ansatzes: the standard ansatz at p=1 and p=2 (denoted 'p=1' and 'p=2') and the Phantom-QAOA ansatz (denoted 'p=1 Phantom' and 'p=2 Phantom'). Here we used the Triangle method for Phantom-QAOA.
• Figure 4: Average improvement in approximation ratio vs degree for nodes 8-16: (a) p=1 Full method, (b) p=1 Triangle method, (c) p=2 Full method, (d) p=2 Triangle method
• Figure 5: Average optimal $\alpha$ for (a) Full method at $p=1$, (b) Triangle method at $p=1$, (c) Full method at $p=2$, and (d) Triangle method at $p=2$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof
• Corollary 1: Triangle-Free Case