Local equivalence, surface-code states, and matroids

verfasst von: Pradeep Sarvepalli, Robert Raussendorf
Abstract: Recently, Ji disproved the local-unitary-local Clifford (LU-LC) conjecture and showed that the local unitary (LU) and local Clifford (LC) equivalence classes of the stabilizer states are not always the same. Despite the fact that this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is not known. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states that violate the LU-LC conjecture. In particular, we show that for all numbers of qubits n≥28, there exist distance-two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that, for all odd n≥195, there exist stabilizer states with distance greater than two that are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the Calderbank-Shor-Steane (CSS) surface-code states in terms of a class of minor closed binary matroids. In addition to making a connection to an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.
Externe Organisation(en): University of British Columbia
Typ: Artikel
Journal: Physical Review A - Atomic, Molecular, and Optical Physics
Band: 82
ISSN: 1050-2947
Publikationsdatum: 05.08.2010
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Atom- und Molekularphysik sowie Optik
Elektronische Version(en): https://doi.org/10.48550/arXiv.0911.1571 (Zugang: Offen)
https://doi.org/10.1103/PhysRevA.82.022304 (Zugang: Geschlossen)

BibTeX

@article{b7546896deff431385d9cbf411d9acc6,
title = "Local equivalence, surface-code states, and matroids",
abstract = "Recently, Ji disproved the local-unitary-local Clifford (LU-LC) conjecture and showed that the local unitary (LU) and local Clifford (LC) equivalence classes of the stabilizer states are not always the same. Despite the fact that this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is not known. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states that violate the LU-LC conjecture. In particular, we show that for all numbers of qubits n≥28, there exist distance-two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that, for all odd n≥195, there exist stabilizer states with distance greater than two that are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the Calderbank-Shor-Steane (CSS) surface-code states in terms of a class of minor closed binary matroids. In addition to making a connection to an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.",
author = "Pradeep Sarvepalli and Robert Raussendorf",
year = "2010",
month = aug,
day = "5",
doi = "10.48550/arXiv.0911.1571",
language = "English",
volume = "82",
journal = "Physical Review A - Atomic, Molecular, and Optical Physics",
issn = "1050-2947",
publisher = "American Physical Society",
number = "2",
}