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 (and, similarly, for fermion) codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\).Protection
CE codewords have to lie in the same excitation subspace in order to protect against changes in the total excitation number. Fock-state CE codes are protected from identical AD acting on all modes because the damping acts on all codewords identically [4,5]. The all-zero AD Kraus operator acts identically on every state and so can be exactly correctable in the case of Fock-state CE codes. For example, this operator's acting on a Fock state \(|\boldsymbol{m}\rangle\) depends only on the total occupation number \(|\boldsymbol{m}|=\sum_j m_j\) and not on the individual occupation numbers \(m_j\), \begin{align} E_{0}^{\otimes n}|\boldsymbol{m}\rangle=\left(1-\gamma\right)^{|\boldsymbol{m}|/2}|\boldsymbol{m}\rangle~. \tag*{(1)}\end{align} This effect extends to the damping portion, \(\left(1-\gamma\right)^{\hat{n}/2}\), of any \(\ell\neq 0\) AD Kraus operators.
In similar fashion, qubit CE codes are protected from coherent noise in the form of transversal \(Z\)-rotations because such rotations act identically on all codewords [6,7]. In the case of CSS codes, all codes oblivious to such rotations are CE codes [6,7]. Stabilizer codes can be extended to codes that are protected against such coherent noise via an enlargement procedure [7].
Rate
Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the AD Gaussian channel [8].Cousins
- Qubit CSS code— Qubit CE codes are protected from coherent noise in the form of transversal \(Z\)-rotations because such rotations act identically on all codewords [6,7]. In the case of qubit CSS codes, all codes oblivious to such rotations are CE codes [6,7]. Any \([[n,k,d]]\) CSS code can be made into an \([[mn,k,>d]]\) CE code [6].
- Five-qubit perfect code— The five-qubit code can be concatenated with a particular decoherence-free subspace (DFS) [9–12] to yield a 20-qubit CE code [3,13].
- Constant-weight code— Constant-weight codes are classical analogues of qubit constant-excitation codes.
- Fermion code— Fermion codewords lying in a fixed fermion-number subspace have to lie in the same subspace in order to protect against changes in fermion number [14].
- Quantum parity code (QPC)— QPCs for even \(m_1\) can be made into CE codes by a Pauli transformation (e.g., \(XIXI\cdots XI\)) applied to each block of \(m_1\) qubits.
- Qubit stabilizer code— Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [15] that protects against \(d-1\) AD errors [16].
Primary Hierarchy
References
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- J. Hu, Q. Liang, N. Rengaswamy, and R. Calderbank, “CSS Codes that are Oblivious to Coherent Noise”, 2021 IEEE International Symposium on Information Theory (ISIT) 1481 (2021) DOI
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- A. Schuckert, E. Crane, A. V. Gorshkov, M. Hafezi, and M. J. Gullans, “Fermion-qubit fault-tolerant quantum computing”, (2024) arXiv:2411.08955
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- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [16]
- R. Duan, M. Grassl, Z. Ji, and B. Zeng, “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
Page edit log
- Yinchen Liu (2024-03-15) — most recent
- Victor V. Albert (2022-03-01)
Cite as:
“Constant-excitation (CE) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/constant_excitation