The role of cohomology in quantum computation with magic states
- URL: http://arxiv.org/abs/2110.11631v3
- Date: Sat, 8 Apr 2023 21:16:12 GMT
- Title: The role of cohomology in quantum computation with magic states
- Authors: Robert Raussendorf, Cihan Okay, Michael Zurel, Polina Feldmann
- Abstract summary: A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality.
We extend this web to quantum computation with magic states.
In this scheme, the negativity of certain quasiprobability functions is an indicator for quantumness.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- 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.
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