The power of qutrits for non-adaptive measurement-based quantum computing

Abstract: Non-locality is not only one of the most prominent quantum features but can also serve as a resource for various information-theoretical tasks. Analysing it from an information-theoretical perspective has linked it to applications such as non-adaptive measurement-based quantum computing (NMQC). In this type of quantum computing the goal is to output a multivariate function. The success of such a computation can be related to the violation of a generalised Bell inequality. So far, the investigation of binary NMQC with qubits has shown that quantum correlations can compute all Boolean functions using at most $2^n-1$ qubits, whereas local hidden variables (LHVs) are restricted to linear functions. Here, we extend these results to NMQC with qutrits and prove that quantum correlations enable the computation of all ternary functions using the generalised qutrit Greenberger-Horne-Zeilinger (GHZ) state as a resource and at most $3^n-1$ qutrits. This yields a corresponding generalised GHZ type paradox for any ternary function that LHVs cannot compute. We give an example for an $n$-variate function that can be computed with only $n+1$ qutrits, which leads to convenient generalised qutrit Bell inequalities whose quantum bound is maximal. Finally, we prove that not all functions can be computed efficiently with qutrit NMQC by presenting a counterexample.