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 and distance . Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters and . In particular, in 2D we show that a quantum LDPC code with distance requires interactions of length . Further, a code satisfying with distance requires interactions of length . 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.
- Received 1 December 2021
- Accepted 26 May 2022
DOI:https://doi.org/10.1103/PhysRevLett.129.050505
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