QAOA on Hamiltonian Cycle problem

TL;DR

This work implements a QAOA-based approach to the NP-complete Hamiltonian Cycle problem by translating graphs into a QUBO form (via Lucas) and encoding the solution in an $(n-1)^2$ qubit embedding. It validates the cost Hamiltonian spectrum for small graphs (triangle and square) and executes QAOA with different mixers in Qiskit, revealing that $R_X$ mixers perform better and that quantum noise can sometimes enhance optimization. A key observation is the presence of a large spectral gap between solution and non-solution states, which may influence QAOA performance. The authors provide reusable tooling to automatically generate the cost Hamiltonian for arbitrary graphs and demonstrate both noiseless and noisy simulations, highlighting practical limits and potential noise-assisted optimization avenues.

Abstract

I use QAOA to solve the Hamiltonian Circle problem. First, inspired by Lucas, I define the QUBO form of Hamiltonian Cycle and transform it to a quantum circuit by embedding the problem of $n$ vertices to an encoding of $(n-1)^2$ qubits. Then, I calcluate the spectrum of the cost hamiltonian for both triangle case and square case and justify my definition. I also write a python program to generate the cost hamiltonian automatically for finding the hamiltonian cycle in an arbitrary graph. I test the correctess of the hamailtonian by analyze their energy spectrums. Since the $(n-1)^2$ embedding limit my simulation of graph size to be less than $5$, I decide to test the correctness, only for small and simple graph in this project. I implement the QAOA algorithm using qiskit and run the simulation for the triangle case and the square case, which are easy to test the correctness, both with and without noise. A very interesting result I got is that for the square case, the QAOA get much better result on a noisy simulator than a noiseless simulator. The explanation for this phenomena require further investigation, perhaps quantum noise can actually be helpful, rather than harmful in the annealing algorithms. I also use two different kinds of mixer, $R_x$ mixer and $R_y$ circuit to run the simulation. It turns out that $R_x$ mixer performs much better than $R_y$ mixer in this problem.

Figures (18)

• Figure 1: The sketch of a triangle which we want to find a hamiltonian circle.
• Figure 2: The sketch of the cost function for the triangle. The lowest energy is $-4$, with eigenstates $\ket{0110}$ and $\ket{1001}$, which are the exact solution for hamiltonian cycle for a triangle. The spectrum justify the correctness of our definition.
• Figure 3: The sketch of a rectangle from where we want to find a hamiltonian circle.
• Figure 4: The output has proved the correctness of our hamiltonian, because the eigen states with the lowest energies are $\ket{001010100}$ and $\ket{100010001}$, which are the exact solution for the hamiltonian path of a square.
• Figure 5: Plot the energy spectrum of the cost hamiltonian. The minimum energy is -20, and the eigen states are $\ket{001010100}$ and $\ket{100010001}$. The energy spectrum justify the design of our hamiltonian for hamiltonian cycle because there is a huge energy gap between the solution we want and the other solutions.