Testing symmetry on quantum computers

Margarite L. LaBorde1, Soorya Rethinasamy2,1, and Mark M. Wilde3,1

1Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14850, USA
3School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA

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Abstract

Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.

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[6] Denis Lacroix, Edgar Andres Ruiz Guzman, and Pooja Siwach, "Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers", European Physical Journal A 59 1, 3 (2023).

[7] Chung-Yun Hsieh, Matteo Lostaglio, and Antonio Acín, "Quantum channel marginal problem", Physical Review Research 4 1, 013249 (2022).

[8] Jonathan Z. Lu, Rodrigo A. Bravo, Kaiying Hou, Gebremedhin A. Dagnew, Susanne F. Yelin, and Khadijeh Najafi, "Learning quantum symmetries with interactive quantum-classical variational algorithms", arXiv:2206.11970, (2022).

[9] Zachary P. Bradshaw, Margarite L. LaBorde, and Mark M. Wilde, "Cycle index polynomials and generalized quantum separability tests", Proceedings of the Royal Society of London Series A 479 2274, 20220733 (2023).

[10] Quynh T. Nguyen, "The mixed Schur transform: efficient quantum circuit and applications", arXiv:2310.01613, (2023).

[11] Margarite L. LaBorde, "A Menagerie of Symmetry Testing Quantum Algorithms", arXiv:2305.14560, (2023).

[12] Shrigyan Brahmachari, Roberto Rubboli, and Marco Tomamichel, "A fixed-point algorithm for matrix projections with applications in quantum information", arXiv:2312.14615, (2023).

[13] Zachary P. Bradshaw and Christophe Vignat, "Dubious Identities: A Visit to the Borwein Zoo", arXiv:2307.05565, (2023).

[14] Zachary P. Bradshaw and Margarite L. LaBorde, "Quantum entanglement & purity testing: A graph zeta function perspective", Physics Letters A 481, 128993 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-18 02:06:32) and SAO/NASA ADS (last updated successfully 2024-06-18 02:06:33). The list may be incomplete as not all publishers provide suitable and complete citation data.