  Quantum low-density parity-check (qLDPC) codes can achieve high encoding
rates and good code distance scaling, providing a promising route to
low-overhead fault-tolerant quantum computing. However, the long-range
connectivity required to implement such codes makes their physical realization
challenging. Here, we propose a hardware-efficient scheme to perform
fault-tolerant quantum computation with high-rate qLDPC codes on reconfigurable
atom arrays, directly compatible with recently demonstrated experimental
capabilities. Our approach utilizes the product structure inherent in many
qLDPC codes to implement the non-local syndrome extraction circuit via atom
rearrangement, resulting in effectively constant overhead in practically
relevant regimes. We prove the fault tolerance of these protocols, perform
circuit-level simulations of memory and logical operations with these codes,
and find that our qLDPC-based architecture starts to outperform the surface
code with as few as several hundred physical qubits at a realistic physical
error rate of $10^{-3}$. We further find that less than 3000 physical qubits
are sufficient to obtain over an order of magnitude qubit savings compared to
the surface code, and quantum algorithms involving thousands of logical qubits
can be performed using less than $10^5$ physical qubits. Our work paves the way
for explorations of low-overhead quantum computing with qLDPC codes at a
practical scale, based on current experimental technologies.
