Approximate secret-sharing code[1]

Description

A family of \( [[n,k,d]]_{GF(q)} \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as an \(t\)-error tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction.

Protection

Corrects up to \(\lfloor (n-1)/2 \rfloor\) errors with fidelity exponentially lose to 1.

Encoding

Uses a quantum authentication scheme, which is a keyed system in which a valid state has high fidelity, and a classical secret-sharing scheme.

Decoding

Decoding is analagous to reconstruction in a secret sharing scheme and is done in polynomial time. The only required operations are verification of quantum authentication, which is a pair of polynomial-time quantum algorithms that check if the fidelity of the received state is close to 1, and erasure correction for a stabilizer code, which involves solving a system of linear equations.

Parent

  • Galois-qudit CSS code — The code required to construct this code must be a non-degenerate Galois-qubit CSS code.

Cousins

References

[1]
Claude Crepeau, Daniel Gottesman, and Adam Smith, “Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes”. quant-ph/0503139
[2]
R. Cleve, D. Gottesman, and H.-K. Lo, “How to Share a Quantum Secret”, Physical Review Letters 83, 648 (1999). DOI; quant-ph/9901025
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Zoo code information

Internal code ID: quantum_secret_sharing

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Zoo Code ID: quantum_secret_sharing

Cite as:
“Approximate secret-sharing code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_secret_sharing
BibTeX:
@incollection{eczoo_quantum_secret_sharing, title={Approximate secret-sharing code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_secret_sharing} }
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https://errorcorrectionzoo.org/c/quantum_secret_sharing

Cite as:

“Approximate secret-sharing code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_secret_sharing

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits_galois/quantum_secret_sharing.yml.