dorsal/arxiv
View SchemaApproximate Quantum Error-Correcting Codes and Secret Sharing Schemes
| Authors | Claude Crepeau, Daniel Gottesman, Adam Smith |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0503139 |
| URL | https://arxiv.org/abs/quant-ph/0503139 |
| Journal | Preliminary version in proceedings of "Advances in Cryptology -- EUROCRYPT 2005" |
Abstract
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 (n-1)/2 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. 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.
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"abstract": "It is a standard result in the theory of quantum error-correcting codes that\nno code of length n can fix more than n/4 arbitrary errors, regardless of the\ndimension of the coding and encoded Hilbert spaces. However, this bound only\napplies to codes which recover the message exactly. Naively, one might expect\nthat correcting errors to very high fidelity would only allow small violations\nof this bound. This intuition is incorrect: in this paper we describe quantum\nerror-correcting codes capable of correcting up to (n-1)/2 arbitrary errors\nwith fidelity exponentially close to 1, at the price of increasing the size of\nthe registers (i.e., the coding alphabet). This demonstrates a sharp\ndistinction between exact and approximate quantum error correction. The codes\nhave the property that any $t$ components reveal no information about the\nmessage, and so they can also be viewed as error-tolerant secret sharing\nschemes.\n The construction has several interesting implications for cryptography and\nquantum information theory. First, it suggests that secret sharing is a better\nclassical analogue to quantum error correction than is classical error\ncorrection. Second, it highlights an error in a purported proof that verifiable\nquantum secret sharing (VQSS) is impossible when the number of cheaters t is\nn/4. More generally, the construction illustrates a difference between exact\nand approximate requirements in quantum cryptography and (yet again) the\ndelicacy of security proofs and impossibility results in the quantum model.",
"arxiv_id": "quant-ph/0503139",
"authors": [
"Claude Crepeau",
"Daniel Gottesman",
"Adam Smith"
],
"categories": [
"quant-ph",
"cs.CR"
],
"journal_ref": "Preliminary version in proceedings of \"Advances in Cryptology --\n EUROCRYPT 2005\"",
"title": "Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes",
"url": "https://arxiv.org/abs/quant-ph/0503139"
},
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