  We study the fundamental limits on the reliable storage of quantum
information in lattices of qubits by deriving tradeoff bounds for approximate
quantum error correcting codes. We introduce a notion of local approximate
correctability and code distance, and give a number of equivalent formulations
thereof, generalizing various exact error-correction criteria. Our tradeoff
bounds relate the number of physical qubits $n$, the number of encoded qubits
$k$, the code distance $d$, the accuracy parameter $\delta$ that quantifies how
well the erasure channel can be reversed, and the locality parameter $\ell$
that specifies the length scale at which the recovery operation can be done. In
a regime where the recovery is successful to accuracy $\epsilon$ that is
exponentially small in $\ell$, which is the case for perturbations of local
commuting projector codes, our bound reads $kd^{\frac{2}{D-1}} \le O\bigl(n
(\log n)^{\frac{2D}{D-1}} \bigr)$ for codes on $D$-dimensional lattices of
Euclidean metric. We also find that the code distance of any local approximate
code cannot exceed $O\bigl(\ell n^{(D-1)/D}\bigr)$ if $\delta \le O(\ell
n^{-1/D})$. As a corollary of our formulation of correctability in terms of
logical operator avoidance, we show that the code distance $d$ and the size
$\tilde d$ of a minimal region that can support all approximate logical
operators satisfies $\tilde d d^{\frac{1}{D-1}}\le O\bigl( n
\ell^{\frac{D}{D-1}} \bigr)$, where the logical operators are accurate up to
$O\bigl( ( n \delta / d )^{1/2}\bigr)$ in operator norm. Finally, we prove that
for two-dimensional systems if logical operators can be approximated by
operators supported on constant-width flexible strings, then the dimension of
the code space must be bounded. This supports one of the assumptions of
algebraic anyon theories, that there exist only finitely many anyon types.
