Tetron code[1–4]

Cousins

• Hamiltonian-based code— Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [2] and other qubit Hamiltonians on certain graphs [5,6].
• Qubit stabilizer code— Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[2n,k,2d]]_{f}\) Majorana stabilizer code by concatenating with the tetron code [2][3; Lemma 1]. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [2] and other qubit Hamiltonians on certain graphs [5,6].
• Qubit CSS code— Any \([[n,k,d]]\) stabilizer code can be mapped into a \([[4n,2k,2d]]\) self-dual CSS code via an intermediate tetron Majorana stabilizer code [8][3; Corr. 1], which preserves geometric locality of a code up to a constant factor.
• Toric code— Coherent physical errors in the toric code are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed by embedding each physical qubit into two fermions via the tetron code [9] as well as deriving analytical bounds [10].
• Kitaev honeycomb code— Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [2] and other qubit Hamiltonians on certain graphs [5,6]. It also allows the logical subspace of the Kitaev honeycomb model to be formulated as a joint eigenspace of certain Majorana operators [11; Sec. 4.1], which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. When done in reverse, this embedding can be thought of as a 2D bosonization fermion-into-qubit encoding by converting to a relabeled square lattice and performing single-qubit rotations [12][13; Sec. IV.B].