  Surface codes$\unicode{x2014}$leading candidates for quantum error correction
(QEC)$\unicode{x2014}$and entanglement phases$\unicode{x2014}$a key notion for
many-body quantum dynamics$\unicode{x2014}$have heretofore been unrelated.
Here, we establish a link between the two. We map two-dimensional (2D) surface
codes under a class of incoherent or coherent errors (bit flips or uniaxial
rotations) to $(1+1)$D free-fermion quantum circuits via Ising models. We show
that the error-correcting phase implies a topologically nontrivial area law for
the circuit's 1D long-time state $|\Psi_\infty\rangle$. Above the error
threshold, we find a topologically trivial area law for incoherent errors and
logarithmic entanglement in the coherent case. In establishing our results, we
formulate 1D parent Hamiltonians for $|\Psi_\infty\rangle$ via linking Ising
models and 2D scattering networks, the latter displaying respective insulating
and metallic phases and setting the 1D fermion gap and topology via their
localization length and topological invariant. We expect our results to
generalize to a duality between the error-correcting phase of ($d+1$)D
topological codes and $d$-dimensional area laws; this can facilitate assessing
code performance under various errors. The approach of combining Ising models,
scattering networks, and parent Hamiltonians can be generalized to other
fermionic circuits and may be of independent interest.
