## 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)/\sqrt{8}\\ |\overline{1}\rangle&=X^{\otimes8}|\overline{0}\rangle~. \end{split} \tag*{(1)}\end{align} Each logical state is a superposition of computational basis states with four excitations.

## Protection

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*{(2)}\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

## Parents

- Hamiltonian-based code — Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.
- Amplitude-damping (AD) code — Fock-state (and qubit) CE codes exactly protect against the AD Kraus operator \(E_{0}^{\otimes n}\) because it acts identically on all Fock (and qubit) states with the same excitation number [4,5].

## 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 PI code
- Very small logical qubit (VSLQ) code
- Jump code

## 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].
- Constant-weight code — Constant-weight codes are classical analogues of qubit constant-excitation codes.
- 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 [9] that protects against \(d-1\) AD errors [10].

## 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]
- D. W. Leung et al., “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
- [5]
- I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997) DOI
- [6]
- J. Hu et al., “CSS Codes that are Oblivious to Coherent Noise”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) DOI
- [7]
- J. Hu et al., “Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers”, IEEE Transactions on Information Theory 68, 1795 (2022) arXiv:2011.00197 DOI
- [8]
- 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
- [9]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [10]
- R. Duan et al., “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