Logical-operator tradeoff for local quantum codes
Jeongwan Haah, John P. Preskill · Physical Review A · 2012
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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$ qubits, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}{d}^{1/(D\ensuremath{-}1)}=O(n)$ 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.