Generalized Mermin Inequalities for Benchmarking Large-Scale GHZ States

As quantum processors scale to dozens of qubits, benchmarking their performance becomes increasingly difficult. Traditional multipartite Bell tests, such as the standard Mermin inequality, often struggle with noise and the exponential growth of required measurement terms. This paper investigates whether modifying the Bell functional—specifically by increasing the number of measurement settings—can provide a more robust and scalable benchmark for noisy Greenberger-Horne-Zeilinger (GHZ) states.

The authors propose a family of generalized Mermin inequalities where the number of measurement settings $m$ acts as a certification parameter complementary to the system size $n$. By choosing $m$ equatorial measurement directions, they derive analytical classical bounds that decrease as $m$ increases, while the ideal quantum value for the GHZ state remains constant. This effectively widens the gap between quantum and classical predictions. They test this on the Zuchongzhi 3.1 superconducting processor, preparing GHZ states of up to 80 qubits and using randomized sampling to estimate the Bell operator without relying on readout correction, tomography, or model-based mitigation.

The experiments demonstrate that increasing the number of measurement settings $m$ significantly strengthens the Bell-violation ratio for the same noisy GHZ state. For an 80-qubit state, the authors observe that the Bell ratio increases monotonically as $m$ is raised from 2 to 32. Furthermore, this generalized approach allows for the certification of higher nonlocality depth—ruling out classical grouping models that the standard two-setting Mermin test cannot exclude. The results show that the exponential scaling of the Bell ratio is improved, providing a sharper benchmark for large-scale quantum processors.