Low-depth random Clifford-circuit code[1]

Description

An encoder for an \([[n,k]]\) quantum error correcting code, is an \(n\)-qubit unitary transformation that takes a \(k\)-qubit state as input (with \(k\leq n\), and the remaining \(n-k\) qubits initialized to \(|0\rangle^{\otimes n-k}\) ) to give a corresponding state in the codespace as the output. An n-qubit quantum circuit with random 2-qubit Clifford gates can act as an encoder into a code with distance \(d\) with high probability, with a size (i.e. number of gates in the circuit) at most \(O(n^2 log n)\)). Noting that two gates acting on disjoint qubits could in fact be executed simultaneously, this is equivalent to the depth (number of time steps in the circuit) being at most \(O(log^3 n)\).

Protection

Creates a random \([[n,k,d]]\) stabilizer code that detects errors on \(d-1\) qubits, and corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.

Parent

References

[1]
W. Brown and O. Fawzi, “Short random circuits define good quantum error correcting codes”, 2013 IEEE International Symposium on Information Theory (2013). DOI; 1312.7646
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Zoo code information

Internal code ID: nonlocal_lowdepth

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Zoo Code ID: nonlocal_lowdepth

Cite as:
“Low-depth random Clifford-circuit 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 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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Permanent link:
https://errorcorrectionzoo.org/c/nonlocal_lowdepth

Cite as:

“Low-depth random Clifford-circuit 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/tree/main/codes/quantum/qubits/nonlocal_lowdepth.yml.