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.

Figure 1

Decoding a nested QECC. The “discarded” qubits carry most of the entropy and are not used further.

Figure 2

A steplike trajectory in light gray illustrates the two ways of leaving region (3) of good trajectories (dashed lines): either a spin flip can take the polarization out of the thickly marked regions, or polarization may leave region (3) as time passes without a spin flip (hollow dots).