Expert Commentary: Solving the Hamiltonian Cycle Problem using QAOA

In this article, the author discusses their approach to solving the Hamiltonian Cycle problem using the Quantum Approximate Optimization Algorithm (QAOA). They first define the QUBO form of the problem based on the work by Lucas and then transform it into a quantum circuit. This transformation involves encoding the problem of n vertices to an encoding of (n-1)^2 qubits, which allows for the representation of the Hamiltonian Cycle as a quantum problem.

To justify their approach, the author calculates the spectrum of the cost Hamiltonian for both the triangle case and the square case. This analysis provides insights into the effectiveness of their definition and allows for a deeper understanding of the problem at hand. Furthermore, they develop a Python program for automatically generating the cost Hamiltonian for finding the Hamiltonian Cycle in an arbitrary graph. This program enables them to test the correctness of their Hamiltonian formulation.

Due to the limitations imposed by the embedding of (n-1)^2 qubits, the author is only able to test the correctness of their approach on small and simple graphs in this project. Nonetheless, they proceed to implement the QAOA algorithm using Qiskit and run simulations for both the triangle case and the square case. They test their implementation both with and without noise to assess its performance under realistic conditions.

One intriguing result from their simulations is that in the square case, QAOA obtains better results on a noisy simulator compared to a noiseless one. This observation highlights the potential role of quantum noise in annealing algorithms like QAOA, suggesting that it may not always be detrimental but could actually aid in improving algorithm performance. This phenomenon warrants further investigation to understand its underlying mechanisms and implications.

The author also explores the use of two different mixers, namely the Rx mixer and the Ry circuit, in their simulations. They find that the Rx mixer outperforms the Ry mixer in this particular problem, indicating its suitability for manipulating the quantum state during the optimization process.

Overall, this article provides valuable insights into using QAOA to solve the Hamiltonian Cycle problem. The author’s approach, analysis, and experimental results contribute to advancing our understanding of quantum optimization algorithms and their potential applications. Future work could involve scaling up the simulations to larger graph sizes and further exploring the role of noise in QAOA optimization.
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