Computational complexity of unitary and state design propertie
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| Název: | Computational complexity of unitary and state design propertie |
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| Autoři: | Nakata, Yoshifumi, Kliesch, Martin, Darmawan, Andrew |
| Zdroj: | PRX quantum. 6(3) |
| Informace o vydavateli: | 2025. |
| Rok vydání: | 2025 |
| Témata: | Quantum algorithms computation |
| Popis: | We investigate unitary and state 𝑡-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) 𝑡-designs. We present a quantum algorithm for computing frame potentials and establish the following: (1) exact computation can be achieved by a single query to a #𝖯 oracle and is #𝖯-hard; (2) for state vectors, deciding whether the frame potential is larger than or smaller than certain values is 𝖡𝖰𝖯-complete, provided that the promise gap between the two values is inverse polynomial in the number of qubits; and (3) for both state vectors and unitaries, this promise problem is 𝖯𝖯-complete if the promise gap is exponentially small. Second, we address the promise problem of deciding whether or not a given set is a good approximation to a design. Given a certain promise gap that could be constant, we show that this problem is 𝖯𝖯-hard, highlighting the inherent computational difficulty of determining properties of unitary and state designs. We further identify implications of our results, including variational methods for constructing designs, diagnosing quantum chaos, and exploring emergent designs in Hamiltonian systems. |
| Druh dokumentu: | Article |
| Jazyk: | English |
| ISSN: | 2691-3399 |
| DOI: | 10.15480/882.15885 |
| Rights: | CC BY |
| Přístupové číslo: | edsair.dris...01170..3df2d18bce49f636df758d9748c4a3dd |
| Databáze: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | We investigate unitary and state 𝑡-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) 𝑡-designs. We present a quantum algorithm for computing frame potentials and establish the following: (1) exact computation can be achieved by a single query to a #𝖯 oracle and is #𝖯-hard; (2) for state vectors, deciding whether the frame potential is larger than or smaller than certain values is 𝖡𝖰𝖯-complete, provided that the promise gap between the two values is inverse polynomial in the number of qubits; and (3) for both state vectors and unitaries, this promise problem is 𝖯𝖯-complete if the promise gap is exponentially small. Second, we address the promise problem of deciding whether or not a given set is a good approximation to a design. Given a certain promise gap that could be constant, we show that this problem is 𝖯𝖯-hard, highlighting the inherent computational difficulty of determining properties of unitary and state designs. We further identify implications of our results, including variational methods for constructing designs, diagnosing quantum chaos, and exploring emergent designs in Hamiltonian systems. |
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| ISSN: | 26913399 |
| DOI: | 10.15480/882.15885 |