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} \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). DOI; 1811.01450

Zoo code information

Internal code ID: chebyshev

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Zoo Code ID: chebyshev

Cite as:
“Chebyshev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/chebyshev
BibTeX:
@incollection{eczoo_chebyshev, title={Chebyshev code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/chebyshev} }
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Permanent link:
https://errorcorrectionzoo.org/c/chebyshev

Cite as:

“Chebyshev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/chebyshev

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/fock_state/chebyshev.yml.