Constant-excitation (CE) code[13] 

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 amplitude damping acting on all modes because the damping acts on all codewords identically [4]. Similarly, qubit CE codes are protected from coherent noise in the form of transversal \(Z\)-rotations because such rotations act identically on all codewords [5,6]. In the case of CSS codes, all codes oblivious to such rotations are CE codes [5,6]. Stabilizer codes can be extended to codes that are protected against such coherent noise via an enlargement procedure [6].

Rate

Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the pure-loss Gaussian channel [7].

Parent

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

Children

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 [5,6]. In the case of qubit CSS codes, all codes oblivious to such rotations are CE codes [5,6]. Any \([[n,k,d]]\) CSS code can be made into an \([[mn,k,>d]]\) CE code [5].
  • Quantum spherical code (QSC) — QSC (CE) codewords are superpositions of coherent states (Fock states) with the same energy.
  • Qubit stabilizer code — Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [8].
  • 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.

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]
I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997) DOI
[5]
J. Hu et al., “CSS Codes that are Oblivious to Coherent Noise”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) DOI
[6]
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
[7]
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
[8]
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 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.), 2024. https://errorcorrectionzoo.org/c/constant_excitation
BibTeX:
@incollection{eczoo_constant_excitation, title={Constant-excitation (CE) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/constant_excitation} }
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“Constant-excitation (CE) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/constant_excitation

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/hamiltonian/constant_excitation.yml.