  Quantum error correction becomes a practical possibility only if the physical
error rate is below a threshold value that depends on a particular quantum
code, syndrome measurement circuit, and decoding algorithm. Here we present an
end-to-end quantum error correction protocol that implements fault-tolerant
memory based on a family of LDPC codes with a high encoding rate that achieves
an error threshold of $0.8\%$ for the standard circuit-based noise model. This
is on par with the surface code which has remained an uncontested leader in
terms of its high error threshold for nearly 20 years. The full syndrome
measurement cycle for a length-$n$ code in our family requires $n$ ancillary
qubits and a depth-7 circuit composed of nearest-neighbor CNOT gates. The
required qubit connectivity is a degree-6 graph that consists of two
edge-disjoint planar subgraphs. As a concrete example, we show that 12 logical
qubits can be preserved for nearly one million syndrome cycles using 288
physical qubits in total, assuming the physical error rate of $0.1\%$. We argue
that achieving the same level of error suppression on 12 logical qubits with
the surface code would require nearly 3000 physical qubits. Our findings bring
demonstrations of a low-overhead fault-tolerant quantum memory within the reach
of near-term quantum processors.
