Clifford Transformations for Fermionic Quantum Systems: From Paulis to Majoranas to Fermions

Abstract: Clifford gates and transformations, which map products of elementary Pauli or Majorana operators to other such products, are foundational in quantum computing, underpinning the stabilizer formalism, error-correcting codes, magic state distillation, quantum communication and cryptography, and qubit tapering. Moreover, circuits composed entirely of Clifford gates are classically simulatable, highlighting their computational significance. In this work, we extend the concept of Clifford transformations to fermionic systems. We demonstrate that fermionic Clifford transformations are generated by half-body and pair operators, providing a systematic framework for their characterization. Additionally, we establish connections with fermionic mean-field theories and applications in qubit tapering, offering insights into their broader implications in quantum computing.