# Max Independent Set and Quantum Alternating Operator Ansatz

## TL;DR

This paper investigates the application of the quantum alternating operator ansatz to the maximum independent set problem, analyzing its performance and behavior through simulations on various graphs.

## Contribution

It provides the first detailed simulation analysis of the quantum alternating operator ansatz for MIS, highlighting the impact of circuit depth, initial states, and graph asymmetry.

## Key findings

- Feasible states distribution is asymmetric for MIS, unlike Max-Cut.
- Algorithm favors larger independent sets in asymmetric graphs.
- Approximation ratios depend on initial state choices.

## Abstract

The maximum independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, $K_{2,3}$, and $K_{3,3}$ graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.