Comparing two cohomological obstructions for contextuality, and a generalised construction of quantum advantage with shallow circuits

We present two results on the subject of quantum contextuality and cohomology, and non-locality and quantum advantage with shallow circuits. Abramsky et al. showed that a range of examples of quantum contextuality is detected by a cohomological invariant based on \v{C}ech cohomology. However, the approach does not give a complete cohomological characterisation of contextuality. A different cohomological approach to contextuality was introduced by Okay et al. Their approach exploits the algebraic structure of the Pauli operators and their qudit generalisations known as Weyl operators. We give an abstract account of this structure, then generalise their approach to any example of contextuality with this structure. We prove at this general level that the approach does not give a more complete characterisation of contextuality than the \v{C}ech cohomology approach. Bravyi, Gosset, and K\"{o}nig (BGK) gave the first unconditional proof that a restricted class of quantum circuits is more powerful than its classical analogue. The result, for the class of circuits of bounded depth and fan-in (shallow circuits), exploits a particular family of examples of contextuality. BGK's quantum circuit and computational problem are derived from a family of non-local games related to the well-known GHZ non-local game. We present a generalised version of their construction. A systematic way of taking examples of contextuality and producing unconditional quantum advantage results with shallow circuits.