Samuel Dai, Ray Li, Eugene Tang (Mar 31 2025).

Abstract: We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any $D$-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters $[[n,k,d]]$ into $\mathbb{R}^D$ must have at least $M^*$ interactions of length at least $\ell^*$, where [M^* = \Omega(\max(k,d)), \quad\textand\quad \ell^* = \Omega\bigg(\max\bigg(\fracdn^\fracD-1D, \bigg(\frackd^\frac1D-1n\bigg)^\fracD-1D\bigg)\bigg). ]We also give tradeoffs between the locality and parameters of commuting projector codes in $D$-dimensions, generalizing a result of Dai and Li. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length.

Arxiv: https://arxiv.org/abs/2503.22651