The paper by Calderbank and Shor introduces the concept of quantum error-correcting codes, which are unitary mappings of $k$ qubits into a subspace of the quantum state space of $n$ qubits. These codes enable the recovery of the original quantum state even if any $t$ of the $n$ qubits undergo arbitrary decoherence. The authors show that such codes exist with an asymptotic rate of $1 - 2H_2(2t/n)$, where $H_2(p)$ is the binary entropy function. They provide upper bounds on this rate and demonstrate that these codes can be constructed using classical error-correcting codes. The paper also discusses the decoding process for these quantum codes and analyzes their performance in the context of noisy quantum channels, showing that they can achieve high fidelity transmission under certain conditions. The results highlight the potential for quantum error correction to improve the stability of quantum information and the feasibility of quantum computing.The paper by Calderbank and Shor introduces the concept of quantum error-correcting codes, which are unitary mappings of $k$ qubits into a subspace of the quantum state space of $n$ qubits. These codes enable the recovery of the original quantum state even if any $t$ of the $n$ qubits undergo arbitrary decoherence. The authors show that such codes exist with an asymptotic rate of $1 - 2H_2(2t/n)$, where $H_2(p)$ is the binary entropy function. They provide upper bounds on this rate and demonstrate that these codes can be constructed using classical error-correcting codes. The paper also discusses the decoding process for these quantum codes and analyzes their performance in the context of noisy quantum channels, showing that they can achieve high fidelity transmission under certain conditions. The results highlight the potential for quantum error correction to improve the stability of quantum information and the feasibility of quantum computing.