Description
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.
Rate
Parent
- Hamiltonian-based code — Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.
Children
- Chuang-Leung-Yamamoto (CLY) code — Chuang-Leung-Yamamoto codewords are constructed out of Fock states with the same total excitation number.
- Ouyang-Chao constant-excitation permutation-invariant code
- One-hot quantum code
- Very small logical qubit (VSLQ) code
Cousins
- Constant-weight code — Constant-weight codes are classical analogues of qubit constant-excitation codes.
- Constant-energy code
References
- [1]
- 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
- [2]
- P. Zanardi and M. Rasetti, “Noiseless Quantum Codes”, Physical Review Letters 79, 3306 (1997) arXiv:quant-ph/9705044 DOI
- [3]
- 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
- [4]
- 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
Page edit log
- Victor V. Albert (2022-03-01) — most recent
Cite as:
“Constant-excitation (CE) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/constant_excitation