Entangling four logical qubits beyond break-even in a nonlocal code

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 ± 0.15 % ≤ F ≤ 99.7 ± 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 ± 0.2 % ≤ F≤ 98.7± 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.