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In several papers, including Wikipedia's quantum contextuality, contextuality is identified with quantum speed-up. However, due to Gottesmann-Knill theorem, for quantum speed-up we also need magic states (or some non-Clifford gate). A highly cited paper equates magic states with contextuality: Contextuality supplies the magic for quantum computation and Contextuality of Physical Theories as an Axiom. The former claims:

Consequently, for qudits, any contextuality is necessarily state-dependent and our results show that this contextuality has an operational meaning as necessary and possibly sufficient for the “magic” that makes quantum computers tick. In the qubit case, it is a pressing open question whether a suitable operationally motivated refinement or quantification of contextuality can align more precisely with the potential to provide a quantum speed-up.

What is the relation here between contexuality, magic and speed up? Clearly contextuality or magic alone does not offer speed up. But does magic necessarily entails contextuality? A GHZ state is contextual and fully Clifford, so it is not a one way street.

glS
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Mauricio
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1 Answers1

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Quantum contextuality is the impossibility of assigning outcome values independent of measurement context and is treated as a fundamental axiom of quantum theory (Cabello et al. 2010) arXiv:1010.2163. Yet circuits that stay inside the Clifford/stabilizer framework (e.g. those that create GHZ states) remain classically efficiently simulatable, so contextuality by itself yields no computational advantage. The missing ingredient is a magic state: any pure state outside the stabilizer set necessarily has Wigner-function negativity and therefore exhibits contextuality (Gross 2006). Injecting such a state (or an equivalent non-Clifford gate) lifts a Clifford circuit to full universality. I

Howard proved an operational equivalence between the onset of contextuality and the ability to distil magic states, establishing contextuality as necessary and (in this distillation model) sufficient for quantum speed-up (Howard et al. 2014)Contextuality supplies the ‘magic’ for quantum computation Thus, every magic state is contextual, whereas contextuality guarantees an advantage only when it is present in a consumable resource. This explains why the GHZ state is contextual but gives only advantage when magic is included.

GHZ non negative: In the phase-space formalisms used in the literature (Gross 2006 for odd-prime qudits; Kocia & Love 2017 for qubits) every pure stabiliser state is represented by a uniform, non-negative quasiprobability distribution that is supported on an affine, maximally isotropic subspace (“a line”) of the discrete phase space.For a single qubit the phase space has points; a stabiliser state (say) is non-zero on the two points of one vertical line, with value on each.

In general, an qubit stabiliser state lives on points one line per qubit tensor-factor, each carrying weight. Because every Clifford operator acts as a permutation of phase-space points. it can only reshuffle the values of ; it can never change the set of values . The GHZ-preparation circuit is a sequence of Cliffords that maps the line supporting onto another line, so inherits exactly the same non-negative weights (1/8 on eight points, 0 elsewhere).

This illustrates two distinct facts: All stabiliser states (including GHZ) are non-negative in these Wigner representations (Kocia & Love 2017)https://arxiv.org/abs/1705.08869

Covariance of the Wigner function under Clifford gates ensures that Clifford circuits cannot create negativity (Gross 2006, §III.B)(Physical Review). https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.021003?

Bram
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