Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive

Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi et al. [Phys. Rev. Lett. 104, 050503 (2010)] have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension k and distance d. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters k and d. In particular, in 2D we show that a quantum LDPC code with distance d∝n1/2+ϵ requires Ω(n1/2+ϵ) interactions of length Ω(nϵ). Further, a code satisfying k∝n with distance d∝n^α requires Ω(n) interactions of length Ω(n^α/2). As an application of these results, we consider a model called a stacked architecture, which has previously been considered 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 very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.