Constant-excitation (CE) code[1][2][3]

Code whose codewords lie in an excited-state eigenspace of a Hamiltonian governing the total energy or total number of excitations of the underlying quantum system. For qubit codes, such a Hamiltonian is often the total spin Hamiltonian, \(H=\sum_i Z_i\). For spin-\(S\) codes, this generalizes to \(H=\sum_i J_z^{(i)}\), where \(J_z\) is the spin-\(S\) \(Z\)-operator. For bosonic codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\).

One of the first such codes [1] is a \(((8,1,3))\) qubit code, with codewords \begin{align} \begin{split} |\overline{0}\rangle&= |00001111\rangle + |11101000\rangle − |10010110\rangle − |01110001\rangle\\ & +|11010100\rangle + |00110011\rangle + |01001101\rangle + |10101010\rangle\\ |\overline{1}\rangle&= X^{\otimes 8} |\overline{0}\rangle~. \end{split} \tag*{(1)}\end{align} Each logical state is a superposition of computational basis states with four excitations.