Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm
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arXiv
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| Format: | Preprint |
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2025
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| author | Boulebnane, Sami Khan, Abid Liu, Minzhao Larson, Jeffrey Herman, Dylan Shaydulin, Ruslan Pistoia, Marco |
| author_facet | Boulebnane, Sami Khan, Abid Liu, Minzhao Larson, Jeffrey Herman, Dylan Shaydulin, Ruslan Pistoia, Marco |
| contents | The Quantum Approximate Optimization Algorithm (QAOA) achieves monotonically improving performance with circuit depth $p$, yet the study of the high-depth regime has been obstructed by the exponential in $p$ cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-$p$ QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to $p$ bosonic modes. We simulate the spin-boson system using matrix product states and provide numerical evidence that QAOA obtains a $(1-ε)$ approximation to the optimal energy of the SK model with circuit depth $O(n/ε^{1.13})$ in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves $\varepsilon\lesssim 2.2\%$ at $p=160$ in the infinite-size limit, extending far beyond $p\leq 20$ accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm Boulebnane, Sami Khan, Abid Liu, Minzhao Larson, Jeffrey Herman, Dylan Shaydulin, Ruslan Pistoia, Marco Quantum Physics The Quantum Approximate Optimization Algorithm (QAOA) achieves monotonically improving performance with circuit depth $p$, yet the study of the high-depth regime has been obstructed by the exponential in $p$ cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-$p$ QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to $p$ bosonic modes. We simulate the spin-boson system using matrix product states and provide numerical evidence that QAOA obtains a $(1-ε)$ approximation to the optimal energy of the SK model with circuit depth $O(n/ε^{1.13})$ in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves $\varepsilon\lesssim 2.2\%$ at $p=160$ in the infinite-size limit, extending far beyond $p\leq 20$ accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation. |
| title | Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2505.07929 |