Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive
Nouédyn Baspin, Anirudh Krishna
Abstract
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 $d\ensuremath{\propto}{n}^{1/2+ϵ}$ requires $\mathrm{\ensuremath{\Omega}}({n}^{1/2+ϵ})$ interactions of length $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Omega}}}({n}^{ϵ})$. Further, a code satisfying $k\ensuremath{\propto}n$ with distance $d\ensuremath{\propto}{n}^{\ensuremath{\alpha}}$ requires $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Omega}}}(n)$ interactions of length $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Omega}}}({n}^{\ensuremath{\alpha}/2})$. 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.