## Description

Approximate code whose codewords lie in the low-energy subspace of a conformal field theory, e.g., the quantum Ising model at its critical point [1,2]. Its encoding is argued to perform source coding (i.e., compression) as well as channel coding (i.e., error correction) [1].

## Protection

Code performance is quantified by a lower bound on the entanglement fidelity in terms of the conditional mutual information [1; Eq. (9)]; see also [3; Appx. A]. The coherent information of the combined noise and recover channel can also be perturbatively expanded [2].

Certain CFT codes have indefinite codespace complexity, and their protection depends on the minimum scaling dimension of the underlying CFT [4].

## Code Capacity Threshold

Threshold under dephasing depends on the structure of the conformal field theory, with the 1D critical Ising model admitting a finite threshold against certain dephasing noise [2].

## Parents

- Qubit code
- Hamiltonian-based code — CFT codewords lie in the low-energy subspace of a conformal field theory (CFT), e.g., the quantum Ising model at its critical point.
- Approximate quantum error-correcting code (AQECC)
- Holographic code — CFT codewords lie in the low-energy subspace of a conformal field theory (CFT), e.g., the quantum Ising model at its critical point.

## References

- [1]
- F. Pastawski, J. Eisert, and H. Wilming, “Towards Holography via Quantum Source-Channel Codes”, Physical Review Letters 119, (2017) arXiv:1611.07528 DOI
- [2]
- S. Sang, T. H. Hsieh, and Y. Zou, “Approximate quantum error correcting codes from conformal field theory”, (2024) arXiv:2406.09555
- [3]
- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
- [4]
- J. Yi et al., “Complexity and order in approximate quantum error-correcting codes”, (2024) arXiv:2310.04710

## Page edit log

- Victor V. Albert (2024-07-02) — most recent

## Cite as:

“Conformal-field theory (CFT) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/cft

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/holographic/cft.yml.