Quantum Computational Complexity and Symmetry
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912219994783744 |
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| author | Rethinasamy, Soorya LaBorde, Margarite L. Wilde, Mark M. |
| author_facet | Rethinasamy, Soorya LaBorde, Margarite L. Wilde, Mark M. |
| contents | Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for BQP, QMA, QSZK, QIP(2), QIP_EB(2), and QIP, thus spanning the prominent classes of the quantum interactive proof hierarchy and forging a non-trivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and QAM, while leaving it as an intriguing open question to determine whether these problems are complete for these classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10081 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum Computational Complexity and Symmetry Rethinasamy, Soorya LaBorde, Margarite L. Wilde, Mark M. Quantum Physics Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for BQP, QMA, QSZK, QIP(2), QIP_EB(2), and QIP, thus spanning the prominent classes of the quantum interactive proof hierarchy and forging a non-trivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and QAM, while leaving it as an intriguing open question to determine whether these problems are complete for these classes. |
| title | Quantum Computational Complexity and Symmetry |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2309.10081 |