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Good Quantum Error-Correcting Codes Exist

A. R. Calderbank·Peter W. Shor
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Abstract

A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.

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A. R. Calderbank, Peter W. Shor. 1996-04-16. Good Quantum Error-Correcting Codes Exist. https://doi.org/10.1103/physreva.54.1098

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