Symmetries and Dimension Reduction in Quantum Approximate Optimization Algorithm
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917571257696256 |
|---|---|
| author | Tsvelikhovskiy, Boris Safro, Ilya Alexeev, Yuri |
| author_facet | Tsvelikhovskiy, Boris Safro, Ilya Alexeev, Yuri |
| contents | In this paper, the Quantum Approximate Optimization Algorithm (QAOA) is analyzed by leveraging symmetries inherent in problem Hamiltonians. We focus on the generalized formulation of optimization problems defined on the sets of $n$-element $d$-ary strings. Our main contribution encompasses dimension reductions for the originally proposed QAOA. These reductions retain the same problem Hamiltonian as the original QAOA but differ in terms of their mixer Hamiltonian, and initial state. The vast QAOA space has a daunting dimension of exponential scaling in $n$, where certain reduced QAOA spaces exhibit dimensions governed by polynomial functions. This phenomenon is illustrated in this paper, by providing partitions corresponding to polynomial dimensions of the corresponding subspaces. As a result, each reduced QAOA partition encapsulates unique classical solutions absent in others, allowing us to establish a lower bound on the number of solutions to the initial optimization problem. Our novel approach opens promising practical advantages in accelerating the algorithm. Restricting the algorithm to Hilbert spaces of smaller dimension may lead to significant acceleration of both quantum and classical simulation of circuits and serve as a tool to cope with barren plateaus problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13787 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symmetries and Dimension Reduction in Quantum Approximate Optimization Algorithm Tsvelikhovskiy, Boris Safro, Ilya Alexeev, Yuri Quantum Physics In this paper, the Quantum Approximate Optimization Algorithm (QAOA) is analyzed by leveraging symmetries inherent in problem Hamiltonians. We focus on the generalized formulation of optimization problems defined on the sets of $n$-element $d$-ary strings. Our main contribution encompasses dimension reductions for the originally proposed QAOA. These reductions retain the same problem Hamiltonian as the original QAOA but differ in terms of their mixer Hamiltonian, and initial state. The vast QAOA space has a daunting dimension of exponential scaling in $n$, where certain reduced QAOA spaces exhibit dimensions governed by polynomial functions. This phenomenon is illustrated in this paper, by providing partitions corresponding to polynomial dimensions of the corresponding subspaces. As a result, each reduced QAOA partition encapsulates unique classical solutions absent in others, allowing us to establish a lower bound on the number of solutions to the initial optimization problem. Our novel approach opens promising practical advantages in accelerating the algorithm. Restricting the algorithm to Hilbert spaces of smaller dimension may lead to significant acceleration of both quantum and classical simulation of circuits and serve as a tool to cope with barren plateaus problem. |
| title | Symmetries and Dimension Reduction in Quantum Approximate Optimization Algorithm |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2309.13787 |