Majorana stabilizer code[1]

Cousins

• Dual linear code— Classical self-orthogonal codes can be used to construct Majorana stabilizer codes [2,10,11]. The direct relationship between the two codes follows from expressing the Majorana strings as binary vectors – akin to the symplectic representation – and observing that the binary stabilizer matrix \(S\) for such a Majorana stabilizer code satisfies \(S\cdot S^T=0\) because it has commuting stabilizers, which is precisely the condition \(G\cdot G^T=0\) on the generator matrix \(G\) of a self-orthogonal classical code. A self-orthogonal classical code \(C\) with parameters \([2N,k,d]\) yields a Majorana stabilizer code with parameters \([[N,N-k,d^\perp]]_f\), where \(d^\perp\) is the code distance of the dual code \(C^\perp\).
• Qubit CSS code— Every \([[n,k,d]]_f\) Majorana stabilizer code is associated with a \([[2n,2k,d]]\) qubit CSS code whose \(X\)- and \(Z\)-check supports coincide [1; Lemma 2]. This map assigns one qubit to each Majorana mode and is the Majorana form of symplectic doubling. An odd-length self-dual CSS code can be converted into a complex-fermion code by replacing qubit \(Z\)-type and \(X\)-type operators with \(\gamma\)-type and \(\tilde{\gamma}\)-type Majorana operators, respectively [3].
• \([[7,1,3]]\) Steane code— Applying the CSS-to-Majorana map of [1; Lemma 2] to the \([[7,1,3]]\) Steane code yields a seven-Majorana code encoding half a qubit; pairing two such odd-length copies gives a physical Majorana stabilizer code with odd logical operators [1; Sec. 8].
• Linear binary code— When constructing a Majorana stabilizer code from a self-orthogonal classical code with an odd number of bits and generator matrix \(G\), a more complex procedure must be applied to ensure that the fermion code has an even number of Majorana zero modes, and thus a physical Hilbert space [1,2]. Rather than taking \(G\) to be the stabilizer matrix as in the even case, we take \(G\oplus G\). This is a concatenation of classical codes as in the CSS construction and it yields a mapping \([2n-1,k,d]\rightarrow [[2n-1,2n-1-k,d^\perp]]_f\). This procedure may be further generalized by concatenating two different self-orthogonal classical codes with an odd number of bits, as is often done in the CSS construction.
• Cyclic linear binary code— Cyclic binary linear codes can be used to construct translation-invariant Majorana stabilizer codes, provided that they are also self-orthogonal [2].
• Stabilizer code— Majorana stabilizer codes are useful for Majorana-based architectures, where the degrees of freedom are electrons, and the notion of locality is different than all other code kingdoms.
• Modular-qudit stabilizer code— Majorana stabilizer codes can be extended to modular qudits, yielding parafermion stabilizer codes [12].
• Jordan-Wigner transformation code— A Majorana stabilizer code is a stabilizer code whose stabilizers are composed of Majorana fermion operators, which are in turn realizable using Pauli strings via the Jordan-Wigner mapping.
• Honeycomb Floquet code— The Honeycomb code admits a convenient representation in terms of Majorana fermions. This leads to a possible physical realization of the code in terms of tetrons [13], where each physical qubit is composed of four Majorana modes.
• Dynamical code— Dynamical codes are viable candidates for storage in Majorana-qubit devices [14].
• \([[5,1,3]]\) Five-qubit perfect code— The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [15].
• Transverse-field Ising model (TFIM) code— The TFIM code stabilizers can be expressed in terms of Majorana operators.
• Quantum parity code (QPC)— QPCs for \(m_1=m_2\) can be conveniently expressed in terms of mutually commuting Majorana operators [16].
• Self-dual CSS code— An odd-length self-dual CSS code can be converted into a complex-fermion code by replacing qubit \(Z\)-type and \(X\)-type operators with \(\gamma\)-type and \(\tilde{\gamma}\)-type Majorana operators, respectively [3].
• Pastawski-Yoshida-Harlow-Preskill (HaPPY) code— The pentagon HaPPY code Hamiltonian can be expressed in terms of mutually commuting weight-two (two-body) Majorana operators [17].
• Majorana subsystem stabilizer code— A Majorana subsystem stabilizer code with no gauge qubits is a Majorana stabilizer code.