Toward a Unified Picture of Fermion-to-Qubit Transforms

Several mappings of fermions to qubits analogous to Jordan-Wigner have been proposed, including the Bravyi-Kitaev transformation, as well as the Ternary Tree and Parity mappings. Using graph- and group-theoretic methods, we present progress toward a unified framework for all such mappings. In particular, we describe the set of such mappings whose qubit representations of creation and annihilation operators are single Pauli strings. We endow this set with a group structure, divide it into equivalence classes based on operator locality properties, and show that these classes can be labeled by certain orbits of the symplectic group over $mathbb{F}_2$. For encodings with non-entangled basis states, we find a labeling in terms of nonsingular directed graphs. We present considerations for optimization over this space of equivalence classes, with applications to near-term quantum simulation.