Quantum approximate optimization algorithm with random and subgraph phase operators

TL;DR

An expected value formula is derived for QAOA with custom phase operators at p = 1 and it is shown numerically that some of these custom phase operators can achieve higher approximation ratio than the original algorithm implementation.

Abstract

The quantum approximate optimization algorithm (QAOA) is a promising quantum algorithm that can be used to approximately solve combinatorial optimization problems. The usual QAOA ansatz consists of an alternating application of the cost and mixer Hamiltonians. In this work, we study how using Hamiltonians other than the usual cost Hamiltonian, dubbed custom phase operators, can affect the performance of QAOA. We derive an expected value formula for QAOA with custom phase operators at $p = 1$ and show numerically that some of these custom phase operators can achieve higher approximation ratio than the original algorithm implementation. Out of all the graphs tested at $p=1$, 0.036\% of the random custom phase operators, 75.9\% of the subgraph custom phase operators, 95.1\% of the triangle-removed custom phase operators, and 93.9\% of the maximal degree edge-removed custom phase operators have a higher approximation ratio than the original QAOA implementation. Furthermore, we numerically simulate these phase operators for $p=2$ and $p=3$ levels of QAOA and find that there exist a large number of subgraph, triangle-removed, and maximal degree edge-removed custom phase operators that have a higher approximation ratio than QAOA at the same depth. These findings open up the question of whether better phase operators can be designed to further improve the performance of QAOA.

Figures (4)

• Figure 1: An eight-vertex graph where a subgraph phase operator achieves an approximation ratio of 1. The subset of edges included in the subgraph phase operator is marked by red color. The diamond and round vertices mark the two subsets that achieve the maximum cut. Note that the red edges form a perfect matching.
• Figure 2: The star graph, which consists of vertices and the solid edges. The dashed red edges are the terms included in the phase operator. This choice of phase operator yields an approximation ratio of 0.5.
• Figure 3: Comparison of the AR for each phase operator and $p$ tested. For each $p$, the Random phase operator performs the worst, while one of the TR-phase operators performs the best, with regular QAOA in the middle.
• Figure 4: The percentage of 8-vertex graphs that had at least one instance of the phase operator perform better than regular QAOA in terms of approximation ratio.