Protecting Expressive Circuits with a Quantum Error Detection Code

Quantum error correction opens the way for quantum computers to speed up relevant tasks like simulating quantum systems. However, fully fault-tolerant quantum error correction imposes heavy demands on quantum hardware, making it practically unreachable for existing quantum computers. In this context we develop the $[[k+2,k,2]]$ quantum error detection code towards implementation on existing trapped-ion computers. Encoding $k$ logical qubits into $k+2$ physical qubits, this code provides fault-tolerant state initialisation and syndrome measurement circuits that can detect any single-qubit error. The code has a universal set of local and global logical rotations that, notably, have physical support on only two qubits. A high-fidelity -- though non fault-tolerant -- compilation of this universal gate set is possible thanks to the two-qubit physical rotations present in trapped-ion computers with all-to-all connectivity. Given the particular structure of the logical operators, we nickname it the Iceberg code. On the 12-qubit Quantinuum H1-2 hardware we demonstrate the protection of circuits of 8 logical qubits with up to 256 layers, saturate the logical quantum volume of $2^8$, and show the positive effect of increasing the number of syndrome measurements. These results demonstrate the practical usefulness of the Iceberg code for early fault-tolerant computation.