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### Rui Soares Barbosa

International Iberian Nanotechnology Laboratory, Portugal, rui.soaresbarbosa@inl.int

### Causal contextuality and adaptive MBQC

(joint work with Cihan Okay)

#### contextuality vs causality

Contextuality has been linked to quantum advantage in various setups.
In particular, Raussendorf [1] considered a specific model of measurement-based quantum computation
where a (possibly adaptive) classical control is restricted to performing $\mathbb{Z}_2$-linear operations,
being supplemented by access to a resource in the form of an empirical model (i.e.~correlation table) in an $(n,2,2)$ Bell-type scenario ($n$ sites, $2$ measurement settings, $2$ outcomes).
It is shown that if the MBQC program \textit{deterministically} implements a non-linear Boolean function then the resource correlation must be strongly contextual.
However, in the presence of adativity -- and if one is interested in a specific computation with a given dependency structure between sites rather than in the whole class of computations achievable with a single resource -- there is arguably no reason to assume no-signalling to sites in the past, or to expect a classical counterpart to yield measurement outcomes independently of such prior measurements.
For a fixed causal/adaptivity structure, we give sufficient conditions on the computed function that imply the resource displays strong causal non-classicality in the sense of Gogioso and Pinzani [2]. This result generalises Raussendorf's in the case of a flat causal order.

[1] R. Raussendorf, Contextuality in measurement-based quantum computation, Physical Review A 88: 022322 (2013).

[2] S. Gogioso and N. Pinzani, The sheaf-theoretic structure of definite causality, 18th International Conference on Quantum Physics and Logic (QPL 2021), M. Backens and C. Heunen (eds),
Electronic Proceedings in Theoretical Computer Science 343: 301–324 (2021).