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From Contextuality to Phase Cohomology: A Computable Bridge Between Bohrification and Geometric Quantization

Zhou-Li Chen

Published May 1, 2026• 0 citations
Publisher
DOI
10.5281/zenodo.20072818

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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 Čech 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 Čech cohomology approach. Bravyi, Gosset, and König (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.

•Jan 1, 2021•0 citations
Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computation. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to the Cech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof that Cech cohomology is at least as powerful.
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 Čech 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 Čech cohomology approach. Bravyi, Gosset, and König (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.