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Quantum low-density parity-check (LDPC) codes are a promising avenue toreduce the cost of constructing scalable quantum circuits. However, it isunclear how to implement these codes in practice. Seminal results of Bravyi &Terhal, and Bravyi, Poulin & Terhal have shown that quantum LDPC codesimplemented through local interactions obey restrictions on their dimension $k$and distance $d$. Here we address the complementary question of how manylong-range interactions are required to implement a quantum LDPC code withparameters $k$ and $d$. In particular, in 2D we show that a quantum LDPC withdistance $n^{1/2 + \epsilon}$ code requires $\Omega(n^{1/2 + \epsilon})$interactions of length $\widetilde{\Omega}(n^{\epsilon})$. Further a codesatisfying $k \propto n$ with distance $d \propto n^\alpha$ requires$\widetilde{\Omega}(n)$ interactions of length$\widetilde{\Omega}(n^{\alpha/2})$. Our results are derived using bounds onquantum codes from graph metrics. As an application of these results, weconsider a model called a stacked architecture, which has previously beenconsidered as a potential way to implement quantum LDPC codes. In this model,although most interactions are local, a few of them are allowed to be verylong. We prove that limited long-range connectivity implies quantitative boundson the distance and code dimension.

Quantifying nonlocality: how outperforming local quantum codes is expensive