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Computational complexity of unitary and state design propertie
Citation Link: https://doi.org/10.15480/882.15885
Publikationstyp
Journal Article
Date Issued
2025-09-09
Sprache
English
Author(s)
TORE-DOI
Journal
Volume
6
Issue
3
Article Number
030345
Citation
PRX quantum 6 (3): 030345 (2025)
Publisher DOI
Publisher
American Physical Society
Peer Reviewed
true
We investigate unitary and state ๐ก-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) ๐ก-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 #๐ฏ oracle and is #๐ฏ-hard; (2) for state vectors, deciding whether the frame potential is larger than or smaller than certain values is ๐กโข๐ฐโข๐ฏ-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 ๐ฏโข๐ฏ-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 ๐ฏโข๐ฏ-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.
Subjects
Quantum algorithms computation
DDC Class
004: Computer Sciences
530: Physics
Publication version
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