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 $\tilde{d}$ qubits, where $\tilde{d}{d}^{1/(D-1)}=O\left(n\right)$ 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.

Figure 1

Lattice covering used in the proof of Theorem 1, shown in two dimensions. Each gray square is $l\times l$ and the white gap between squares has width $w-1$. The solid blue curve represents the support of a nontrivial logical operator; because the square $M_i$ is correctable, this square can be “cleaned”; we can find an equivalent logical operator supported on $M_i^c$, the complement of $M_i$. When all squares are cleaned, the logical operator is supported on the narrow strips between the squares.