  Quantum error correction was invented to allow for fault-tolerant quantum
computation. Systems with topological order turned out to give a natural
physical realization of quantum error correcting codes (QECC) in their
groundspaces. More recently, in the context of the AdS/CFT correspondence, it
has been argued that eigenstates of CFTs with a holographic dual should also
form QECCs. These two examples raise the question of how generally eigenstates
of many-body models form quantum codes. In this work we establish new
connections between quantum chaos and translation-invariance in many-body spin
systems, on one hand, and approximate quantum error correcting codes (AQECC),
on the other hand. We first observe that quantum chaotic systems exhibiting the
Eigenstate Thermalization Hypothesis (ETH) have eigenstates forming approximate
quantum error-correcting codes. Then we show that AQECC can be obtained
probabilistically from translation-invariant energy eigenstates of every
translation-invariant spin chain, including integrable models. Applying this
result to 1D classical systems, we describe a method for using local symmetries
to construct parent Hamiltonians that embed these codes into the low-energy
subspace of gapless 1D quantum spin chains. As explicit examples we obtain
local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and
the Motzkin spin chain model with periodic boundary conditions, thereby
yielding non-stabilizer codes in the ground space and low energy subspace of
physically plausible 1D gapless models.
