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

TL;DR

This work extends Clifford transformations from Pauli and Majorana operator algebras to general fermionic systems by showing fermionic Clifford unitaries are generated by anti-Hermitian/Hermitian linear combinations of half-body and pair operators with angles restricted to discrete values; these transformations preserve many-body rank and fermionic parity in broad cases while enabling particle-hole conjugation and index-swapping actions. It connects these fermionic Clifford unitaries to fermionic mean-field theories, revealing a bridge between Clifford dynamics and Hartree–Fock, Bogoliubov, and Fukutome-type mean-field frameworks, and highlights their role in qubit tapering when mapping to second-quantized Hamiltonians. The paper also demonstrates practical consequences for tapering H$_2$/MBS, showing Pauli Clifford gates translate into fermionic Clifford constructs under Jordan–Wigner, though leading to increased term counts and symmetry-breaking in the fermionic space. Finally, it provides a Lie-algebraic classification of the operator sets generating these transformations, clarifying the underlying group-theoretical structure of fermionic Clifford operations and their connection to standard many-body formalisms.

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.

Figures (8)

• Figure 1: Schematic illustration of the normalizer $\mathcal{N}_\mathcal{G}(\mathcal{S})$ of a subset $\mathcal{S}$ of a group $\mathcal{G}$. Elements of $\mathcal{N}_\mathcal{G}(\mathcal{S})$ map under conjugation an element of $\mathcal{S}$ to another element of $\mathcal{S}$. Note that if $\mathcal{S}$ is a subgroup of $\mathcal{G}$, then it is a subgroup of $\mathcal{N}_\mathcal{G}(\mathcal{S})$ as well.
• Figure 2: Illustration of the continuous rotation of the Pauli string $O$ into $\mathrm{i} OP$ under the unitary transformation $\exp(-\mathrm{i} \theta P) O \exp(\mathrm{i} \theta P)$, with $\{O,P\}=0$.
• Figure 3: Illustration of the continuous rotation of the Majorana string $O$ into $\mathrm{i} O\Gamma$ under the unitary transformation $\exp(-\mathrm{i} \theta \Gamma) O \exp(\mathrm{i} \theta \Gamma)$, with $\Gamma^\dagger = \Gamma$ and $\{O,\Gamma\}$. For an anti-Hermitian $\Gamma$, the imaginary unit is dropped.
• Figure 4: Molecular orbital diagram of the H2 molecule in a minimum basis.
• Figure 5: Schematic representation of the ground electronic state of the H2 molecule as described by a minimum basis set. Before tapering, the state is a linear combination of two Slater determinants, $\ket{\Psi} = c_0 \ket{1100} + c_1 \ket{0011}$, while after the tapering of the last three spinorbitals it becomes $\ket{\Psi} = c_0 \ket{1} + c_1 \ket{0}$.