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Computational complexity of unitary and state design propertie

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Názov: Computational complexity of unitary and state design propertie
Autori: Nakata, Yoshifumi, Kliesch, Martin, Darmawan, Andrew
Zdroj: PRX quantum. 6(3)
Informácie o vydavateľovi: 2025.
Rok vydania: 2025
Predmety: 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 𝖡⁢𝖰⁢𝖯-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 𝖯⁢𝖯-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 𝖯⁢𝖯-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
Prístupové číslo: edsair.dris...01170..3df2d18bce49f636df758d9748c4a3dd
Databáza: OpenAIRE
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  Data: Computational complexity of unitary and state design propertie
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  Data: <i>PRX quantum</i>. 6(3)
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  Data: 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 𝖡⁢𝖰⁢𝖯-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 𝖯⁢𝖯-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 𝖯⁢𝖯-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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