QAOA on Hamiltonian Cycle problem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911744057671680 |
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| author | Ye, Zhuoyang |
| author_facet | Ye, Zhuoyang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00017 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | QAOA on Hamiltonian Cycle problem Ye, Zhuoyang Emerging Technologies 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. |
| title | QAOA on Hamiltonian Cycle problem |
| topic | Emerging Technologies |
| url | https://arxiv.org/abs/2401.00017 |