Constant-excitation (CE) code[13] 


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.


Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the pure-loss Gaussian channel [4].


  • Hamiltonian-based code — Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.




M. B. Plenio, V. Vedral, and P. L. Knight, “Quantum error correction in the presence of spontaneous emission”, Physical Review A 55, 67 (1997) arXiv:quant-ph/9603022 DOI
P. Zanardi and M. Rasetti, “Noiseless Quantum Codes”, Physical Review Letters 79, 3306 (1997) arXiv:quant-ph/9705044 DOI
D. A. Lidar, D. Bacon, and K. B. Whaley, “Concatenating Decoherence-Free Subspaces with Quantum Error Correcting Codes”, Physical Review Letters 82, 4556 (1999) arXiv:quant-ph/9809081 DOI
M. S. Winnel et al., “Achieving the ultimate end-to-end rates of lossy quantum communication networks”, npj Quantum Information 8, (2022) arXiv:2203.13924 DOI
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Zoo Code ID: constant_excitation

Cite as:
“Constant-excitation (CE) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_constant_excitation, title={Constant-excitation (CE) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Constant-excitation (CE) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.