Calderbank, AR; Shor, PW, *Good quantum error-correcting codes exist*,
Physical Review A, vol. 54 no. 2
(August, 1996),
pp. 1098-1105 [doi] .
**Abstract:**

*A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-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 = 1-2H2(2t/n) where H2(P) is the binary entropy function -plog2p-(1 -p)log2(l - p). Upper bounds on this asymptotic rate are given.*