# Approximate secret-sharing code[1]

## Description

A family of \( [[n,k,d]]_q \) 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

- Approximate quantum error-correcting code (AQECC) — Secret-sharing codes approximately correct errors on up to \(\lfloor (n-1)/2 \rfloor\) errors.
- Galois-qudit RS code — Polynomial codes can be used for a specific construction of this code.
- Reed-Solomon (RS) code — The classical information in this code is encoded using an RS code.
- Purity-testing stabilizer code — The purity-testing protocol of Ref. [2] can be improved using approximate codes similar the approximate secret-sharing codes [3].
- Three-qutrit code — The three-qutrit code defines a minimal secret-sharing scheme [4] that is substantially generalized by approximate secret-sharing codes.
- Singleton-bound approaching AQECC — Quantum secret-sharing codes have asymptotically decaying rate and require qudit dimension to increase exponentially with \(n\), while Singleton-bound approaching AQECCs have constant rate and qudit dimension.

## References

- [1]
- C. Crepeau, D. Gottesman, and A. Smith, “Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes”, (2005) arXiv:quant-ph/0503139
- [2]
- H. Barnum et al., “Authentication of quantum messages”, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. arXiv:quant-ph/0205128 DOI
- [3]
- M. Ben-Or et al., “Secure Multiparty Quantum Computation with (Only) a Strict Honest Majority”, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06) (2006) arXiv:0801.1544 DOI
- [4]
- R. Cleve, D. Gottesman, and H.-K. Lo, “How to Share a Quantum Secret”, Physical Review Letters 83, 648 (1999) arXiv:quant-ph/9901025 DOI

## Page edit log

- Victor V. Albert (2021-12-15) — most recent
- Manasi Shingane (2021-12-14)

## Cite as:

“Approximate secret-sharing code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_secret_sharing