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It is in general believed that contextuality is one of the quantum resource that provides the quantum advantage. A context is usually defined in terms of a set of commuting observables. The quantum algorithms are usually describe employing the circuit model. I am curious how can one define contextuality in the within the circuit model?

To be concrete, we may consider Peres-Mermin Square and define a circuit representing each observable at each spot. I think the above definition of contextuality require some upgrading while combining all the gates to implement quantum contextuality for Peres-Mermin square. Please share any suggestion or any reference that can be help as a starting point.

Spekken (2005) generalizes the above definition of contextuality. Could we apply Spekken definition for a circuit model too?

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The commuting model only differs from the spatial (tensor-product) model when one considers infinite-dimensional Hilbert spaces, so one would have to start with considering circuits with an infinite number of degrees of freedom.

That being said, such a model has been considered in this paper and is called the $C^*$-circuit model.

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