How Long Can a Quantum Memory Withstand Depolarizing Noise?

We investigate the possibilities and limitations of passive Hamiltonian protection of a quantum memory against depolarizing noise. Without protection, the lifetime of a single qubit is independent of $N$, the number of qubits composing the memory. In the presence of a protecting Hamiltonian, the lifetime increases at most logarithmically with $N$. We construct an explicit time-independent Hamiltonian which saturates this bound, exploiting the noise itself to achieve the protection.