Computational Complexity of Unitary and State Design Properties

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 compu...

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Veröffentlicht in:PRX quantum Jg. 6; H. 3; S. 030345
Hauptverfasser: Nakata, Yoshifumi, Takeuchi, Yuki, Kliesch, Martin, Darmawan, Andrew
Format: Journal Article
Sprache:Englisch
Veröffentlicht: American Physical Society 09.09.2025
ISSN:2691-3399, 2691-3399
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Abstract 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.
AbstractList 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.
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.
ArticleNumber 030345
Author Darmawan, Andrew
Kliesch, Martin
Takeuchi, Yuki
Nakata, Yoshifumi
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  surname: Takeuchi
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  surname: Kliesch
  fullname: Kliesch, Martin
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  givenname: Andrew
  surname: Darmawan
  fullname: Darmawan, Andrew
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Snippet We investigate unitary and state t -designs from a computational complexity perspective. First, we address the problems of computing frame potentials that...
We investigate unitary and state t-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that...
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Title Computational Complexity of Unitary and State Design Properties
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