Limits to Error Correction in Quantum Chaos

We study the correction of errors that have accumulated in an entangled state of spins as a result of unknown local variations in the Zeeman energy ( $B$) and spin-spin interaction energy ( $J$). A nondegenerate code with error rate $\kappa$ can recover the original state with high fidelity within a time $t_{\mathrm{R}}\simeq ħ{\kappa }^{1/2}/\max (B,J)$—independent of the number of encoded qubits. Whether the Hamiltonian is chaotic or not does not affect this time scale, but it does affect the complexity of the error-correcting code.