A Quantum Algorithm For Computing Contextuality Bounds

Quantum contextuality is a limitation on deterministic hidden variable models, testable in measurement scenarios where outcomes differ under quantum or classical descriptions due to a common set of constraints. When considering measurements of $N$-qubit spin operators, constraints arise from commutation relations and classical bounds are determined by the $\textit{degree}$ of contextuality, an NP-hard quantity to compute in general, related to the larger class of optimisation problems known as $\texttt{MaxLin2}$. In this work we give a quantum algorithm based on Grover's search algorithm, computing the degree of contextuality in $O(\sqrt{n} \log\log{n})$ in $n$ states, a speedup over classical brute force method. We also study variations of Grover which encode the relevant information in the phases of the basis states, reducing circuit width and depth requirements with indicative complexity of $O(n^{\frac{1}{3}}(\log{n})^{2}\log\log{n})$. Testing on contemporary quantum backends with the IBM quantum experience gives inconclusive outputs due to noise-induced errors, an issue hopefully fixed by future hardware.