Kitaev chain code[1]

Description

An \([[n,1]]_{f}\) Majorana stabilizer code obtained from the ground-state subspace of the Kitaev Majorana chain in its fermionic topological phase [1]. Its codespace is stabilized by nearest-neighbor Majorana bilinears, while two unpaired edge Majoranas furnish one logical fermionic mode. Under parity-preserving noise, it behaves as the Majorana analogue of the repetition code [2].

At the fixed-point limit, the code Hamiltonian is proportional to \(-\sum_{j=1}^{n-1} i \gamma_{2j}\gamma_{2j+1}\), so the codespace is the common \(+1\) eigenspace of the stabilizers \(S_j=i\gamma_{2j}\gamma_{2j+1}\). The two unpaired edge operators \(\gamma_{1}\) and \(\gamma_{2n}\) are the Majorana zero modes (MZMs) or Majorana edge modes (MEMs); they commute with all stabilizers and define a logical fermionic mode \(f_{\mathrm{L}}=(\gamma_{1}+i\gamma_{2n})/2\). Via the Jordan-Wigner transformation, the model maps to the 1D quantum Ising chain in its symmetry-breaking phase. The code can be thought of as the Majorana stabilizer analogue of the quantum repetition code: parity-preserving dephasing operators \(Z_j=i\gamma_{2j-1}\gamma_{2j}\) play the role of repetition-code bit flips, while the logical Majorana operators have odd weight and therefore encode a logical fermion [2].

The two basis states of a single chain have opposite fermionic parity. Therefore, a single chain does not by itself furnish a fixed-parity qubit encoding; coherent superpositions between the two basis states are not directly accessible in an isolated fermionic system with parity superselection. One can combine two such code blocks to form a Majorana box qubit, which is the fixed-parity subspace of the combined codespace. Odd numbers of code blocks also contain fixed-parity logical subspaces in their codespace.