Verifiable secret sharing and multiparty protocols with honest majority
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Codes for interactive authentication
CRYPTO '93 Proceedings of the 13th annual international cryptology conference on Advances in cryptology
Fault-tolerant quantum computation with constant error
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On sharing secrets and Reed-Solomon codes
Communications of the ACM
Communications of the ACM
Secure multi-party quantum computation
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Quantum computation and quantum information
Quantum computation and quantum information
Secret Sharing Schemes with Detection of Cheaters for a General Access Structure
Designs, Codes and Cryptography
Authentication of Quantum Messages
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
On the Cost of Reconstructing a Secret, or VSS with Optimal Reconstruction Phase
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Efficient multiparty computations secure against an adaptive adversary
EUROCRYPT'99 Proceedings of the 17th international conference on Theory and application of cryptographic techniques
On quantum fidelities and channel capacities
IEEE Transactions on Information Theory
Classical cryptographic protocols in a quantum world
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
On non-binary quantum BCH codes
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
Conditions for the approximate correction of algebras
TQC'09 Proceedings of the 4th international conference on Theory of Quantum Computation, Communication, and Cryptography
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It is a standard result in the theory of quantum error- correcting codes that no code of length n can fix more than n/4 arbitrary errors, regardless of the dimension of the coding and encoded Hilbert spaces. However, this bound only applies to codes which recover the message exactly. Naively, one might expect that correcting errors to very high fidelity would only allow small violations of this bound. This intuition is incorrect: in this paper we describe quantum error-correcting codes capable of correcting up to $\lfloor(n - 1)/2\rfloor$ arbitrary errors with fidelity exponentially close to 1, at the price of increasing the size of the registers (i.e., the coding alphabet). This demonstrates a sharp distinction between exact and approximate quantum error correction. The codes have the property that any t components reveal no information about the message, and so they can also be viewed as error-tolerant secret sharing schemes. The construction has several interesting implications for cryptography and quantum information theory. First, it suggests that secret sharing is a better classical analogue to quantum error correction than is classical error correction. Second, it highlights an error in a purported proof that verifiable quantum secret sharing (VQSS) is impossible when the number of cheaters t is n/4. In particular, the construction directly yields an honest-dealer VQSS scheme for $t= \lfloor(n - 1)/2\rfloor$. We believe the codes could also potentially lead to improved protocols for dishonest-dealer VQSS and secure multi-party quantum computation. More generally, the construction illustrates a difference between exact and approximate requirements in quantum cryptography and (yet again) the delicacy of security proofs and impossibility results in the quantum model.