Details
| Original language | English |
|---|---|
| Journal | Quantum |
| Volume | 7 |
| Publication status | Published - 13 Apr 2023 |
| Externally published | Yes |
Abstract
A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function-an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces-possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n ≥ 2. We establish that the obstructions to the existence of such Wigner functions are cohomological.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Atomic and Molecular Physics, and Optics
- Physics and Astronomy(all)
- Physics and Astronomy (miscellaneous)
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In: Quantum, Vol. 7, 13.04.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The role of cohomology in quantum computation with magic states
AU - Raussendorf, Robert
AU - Okay, Cihan
AU - Zurel, Michael
AU - Feldmann, Polina
N1 - Funding Information: Acknowledgments. RR and PF are funded from the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. MZ is funded by the National Science and Engineering Research Council of Canada. CO is supported by the US Air Force Office of Scientific Research under award number FA9550-21-1-0002.
PY - 2023/4/13
Y1 - 2023/4/13
N2 - A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function-an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces-possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n ≥ 2. We establish that the obstructions to the existence of such Wigner functions are cohomological.
AB - A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function-an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces-possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n ≥ 2. We establish that the obstructions to the existence of such Wigner functions are cohomological.
UR - http://www.scopus.com/inward/record.url?scp=85163782989&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2110.11631
DO - 10.48550/arXiv.2110.11631
M3 - Article
AN - SCOPUS:85163782989
VL - 7
JO - Quantum
JF - Quantum
SN - 2521-327X
ER -