Low-depth random Clifford-circuit qubit code[1] 

Description

An \([[n,k]]\) qubit stabilizer code whose encoder is a random Clifford circuit of depth \(O(\log^3 n)\).

An \(n\)-qubit quantum encoding circuit with \(O(n^2 \log n)\)) random two-qubit Clifford gates yields a code with distance \(d\) with high probability, granted that \begin{equation} \frac{k}{n} < 1 - \frac{d}{n}\log_2 3 - h\left(\frac{d}{n}\right)~, \tag*{(1)}\end{equation} where \(h\) is the entropy function. Noting that two gates acting on disjoint qubits could be executed simultaneously, the depth of a circuit with such a size is typically of order \(O(\log^3 n)\).'

Rate

The achievable distance of these codes is asymptotically the same as a code whose encoder is a random (not necessarily log-depth) general Clifford unitary [1].

Encoding

Random \(\log^3\)-depth Clifford circuit.

Decoding

Minimum-weight decoding via using tropical tensor networks [2].

Fault Tolerance

Fault-tolerant state preparation [2].

Parents

References

[1]
W. Brown and O. Fawzi, “Short random circuits define good quantum error correcting codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1312.7646 DOI
[2]
J. Nelson et al., “Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes”, (2023) arXiv:2311.17985
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Zoo Code ID: nonlocal_lowdepth

Cite as:
“Low-depth random Clifford-circuit qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/nonlocal_lowdepth
BibTeX:
@incollection{eczoo_nonlocal_lowdepth, title={Low-depth random Clifford-circuit qubit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/nonlocal_lowdepth} }
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Cite as:

“Low-depth random Clifford-circuit qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/nonlocal_lowdepth

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/dynamic/random/nonlocal_lowdepth.yml.