  Quantum error correction protects logical quantum information against
environmental decoherence by encoding logical qubits into entangled states of
physical qubits. One of the most important near-term challenges in building a
scalable quantum computer is to reach the break-even point, where logical
quantum circuits on error-corrected qubits achieve higher fidelity than
equivalent circuits on uncorrected physical qubits. Using Quantinuum's H2
trapped-ion quantum processor, we encode the GHZ state in four logical qubits
with fidelity $ 99.5 \pm 0.15 \% \le F \le 99.7 \pm 0.1\% $ (after
postselecting on over 98% of outcomes). Using the same quantum processor, we
can prepare an uncorrected GHZ state on four physical qubits with fidelity
$97.8 \pm 0.2 \% \le F\le 98.7\pm 0.2\%$. The logical qubits are encoded in a
$[\![ 25,4,3 ]\!]$ Tanner-transformed long-range-enhanced surface code. Logical
entangling gates are implemented using simple swap operations. Our results are
a first step towards realizing fault-tolerant quantum computation with logical
qubits encoded in geometrically nonlocal quantum low-density parity check
codes.
