Chebyshev code[1]
Description
Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. Codewords for the \(s\)th-order Chebyshev code are \begin{align} \begin{split} \ket{\overline 0} &=\sum_{k \text{~even}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2\left( k\pi/{2s}\right) \right\rfloor},\\ \ket{\overline 1} &= \sum_{k \text{~odd}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2 \left(k\pi/{2s}\right) \right\rfloor}, \end{split} \tag*{(1)}\end{align} where \(\tilde{c}_k>0\) can be obtained by solving a system of order \(O(s^2)\) linear equations, and where \(\lfloor x \rfloor\) is the floor function. The code approaches optimality for sensing the signal Hamiltonian as \(M\) increases.
Protection
The \(s\)th-order code corrects errors from the set \(\{I,a,a^{\dagger},{\hat n},{\hat n}^2,\cdots,{\hat n}^{s-1}\}\).
Parents
Cousin
- Binomial code — Chebyshev codes resemble binomial codes, and a class of binomial codes have similar error-correcting properties [1].
References
- [1]
- D. Layden et al., “Ancilla-Free Quantum Error Correction Codes for Quantum Metrology”, Physical Review Letters 122, (2019) arXiv:1811.01450 DOI
Page edit log
- Victor V. Albert (2022-07-27) — most recent
Cite as:
“Chebyshev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/chebyshev