  We ask whether there are fundamental limits on storing quantum information
reliably in a bounded volume of space. To investigate this question, we study
quantum error correcting codes specified by geometrically local commuting
constraints on a 2D lattice of finite-dimensional quantum particles. For these
2D systems, we derive a tradeoff between the number of encoded qubits k, the
distance of the code d, and the number of particles n. It is shown that
kd^2=O(n) where the coefficient in O(n) depends only on the locality of the
constraints and dimension of the Hilbert spaces describing individual
particles. We show that the analogous tradeoff for the classical information
storage is k\sqrt{d} =O(n).
