On the locality of qubit encodings of local fermionic modes

Abstract: Known mappings that encode fermionic modes into a bosonic qubit system are non-local transformations. In this paper we establish that this must necessarily be the case, if the locality graph is complex enough (for example for regular 2$d$ lattices). In particular we show that, in case of exact encodings, a fully local mapping is possible if and only if the locality graph is a tree. If instead we allow ourselves to also consider operators that only act fermionically on a subspace of the qubit Hilbert space, then we show that this subspace must be composed of long range entangled states, if the locality graph contains at least two overlapping cycles. This implies, for instance, that on 2$d$ lattices there exist states that are of low depth from the fermionic point of view, while in any encoding require a circuit of depth at least proportional to the system size to be prepared.