Good quantum error-correcting codes exist

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=1-2${\mathit{H}}_{2}$(2t/n) where ${\mathit{H}}_{2}$(p) is the binary entropy function -p${\mathrm{log}}_{2}$p-(1-p)${\mathrm{log}}_{2}$(1-p). Upper bounds on this asymptotic rate are given. \textcopyright{} 1996 The American Physical Society.