Fermionic advantage in shallow circuits

Fermionic systems can be mapped to qubit or bosonic models, but these mappings do not in general preserve locality, circuit depth, or the operational structure of the original dynamics. This raises the question of whether fermionic statistics can lead to computational behaviours that cannot be reproduced by bosonic circuits under the same structural constraints. In this work, we investigate this question in the setting of shallow quantum circuits, using a quantum network game whose correlations are sensitive to fermionic exchange statistics. We construct constant-depth fermionic circuits that realise the target distributions and study whether they can be reproduced by bosonic circuits within the same local interaction architecture. By combining locality constraints with self-testing properties of the underlying network correlations, we prove that there exist fermionic shallow circuits whose output distributions cannot be reproduced by bosonic circuits of the same depth and architecture. Our results show that fermionic exchange statistics can constitute an operational resource for quantum information processing, rather than merely a feature to be removed through a fermion-to-qubit encoding. They also motivate the study of how this separation scales with circuit depth and graph size, and whether it can ultimately be related to a computational advantage.