Approximate secret-sharing code
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.
Corrects up to \(\lfloor (n-1)/2 \rfloor\) errors with fidelity exponentially lose to 1.
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 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.
- Galois-qudit CSS code — The code required to construct this code must be a non-degenerate Galois-qubit CSS code.
- 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 a Reed-Solomon code.
- Three-qutrit code — Three-qutrit code defines a minimal secret-sharing scheme  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.
“Approximate secret-sharing code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_secret_sharing