Locality vs Quantum Codes

This paper proves optimal tradeoffs between the locality and parameters of quantum error-correcting codes. Quantum codes give a promising avenue towards quantum fault tolerance, but the practical constraint of locality limits their quality. The seminal Bravyi-Poulin-Terhal (BPT) bound says that a $[[n,k,d]]$ quantum stabilizer code with 2D-locality must satisfy $kd^2\le O(n)$. We answer the natural question: for better code parameters, how much "non-locality" is needed? In particular, (i) how long must the long-range interactions be, and (ii) how many long-range interactions must there be? We give a complete answer to both questions for all $n,k,d$: above the BPT bound, any 2D-embedding must have at least $\Omega(\#^*)$ interactions of length $\Omega(\ell^*)$, where $\#^*= \max(k,d)$ and $\ell^*=\max\big(\frac{d}{\sqrt{n}}, \big( \frac{kd^2}{n} \big)^{1/4} \big)$. Conversely, we exhibit quantum codes that show, in strong ways, that our interaction length $\ell^*$ and interaction count $\#^*$ are asymptotically optimal for all $n,k,d$. Our results generalize or improve all prior works on this question, including the BPT bound and the results of Baspin and Krishna. One takeaway of our work is that, for any desired distance $d$ and dimension $k$, the number of long-range interactions is asymptotically minimized by a good qLDPC code of length $\Theta(\max(k,d))$. Following Baspin and Krishna, we also apply our results to the codes implemented in the stacked architecture and obtain better bounds. In particular, we rule out any implementation of hypergraph product codes in the stacked architecture.