Logical-operator tradeoff for local quantum codes

We study the structure of logical operators in local D-dimensional quantum codes, considering both subsystem codes with geometrically local gauge generators and codes defined by geometrically local commuting projectors. We show that if the code distance is d, then any logical operator can be supported on a set of specified geometry containing (d) over tilde qubits, where (d) over tilded(1/(D-1)) = O(n) and n is the code length. Our results place limitations on partially self-correcting quantum memories, in which at least some logical operators are protected by energy barriers that grow with system size. We also show that for any two-dimensional local commuting projector code there is a nontrivial logical string operator supported on a narrow strip, where the operator is only slightly entangling across any cut through the strip.