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arXiv · 2309.10081

Quantum Computational Complexity and Symmetry

Soorya Rethinasamy·Margarite L. LaBorde·Mark M. Wilde
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Abstract

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.

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Soorya Rethinasamy, Margarite L. LaBorde, Mark M. Wilde. 2023-09-18. Quantum Computational Complexity and Symmetry. https://doi.org/10.1139/cjp-2023-0260

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