Quantum LDPC codes can protect many logical qubits with far fewer physical qubits than the surface code. To run a computation on them, though, you need logical gates, logical measurements and a way to bring in non-Clifford resources such as T states.
In this work, continuing our collaboration with Guido Pupillo’s group at CESQ, we develop a complete toolbox for logical computation with La-cross codes: native Clifford gates, fast logical measurement gadgets, and T-state injection. Together they give a universal Clifford+T instruction set across multiple encoded qubits.
The primitives mirror those of the familiar surface-code architecture, while needing only low connectivity. This makes La-cross codes a practical route to logical computation on near-term neutral-atom hardware.
Technical abstract
Quantum LDPC codes promise to reduce the overhead of fault-tolerant quantum computation, but their practical use requires efficient logical gates, measurements, and access to non-Clifford resources. Here, we develop a complete set of computational primitives for La-cross hypergraph product codes. For a subclass of these codes, we show that seed automorphisms together with fold-transversal gates generate the full logical Clifford group up to Pauli corrections. We further prove that every rank one logical operator admits D pairwise disjoint physical representatives, enabling parallel gauging and a space-time tradeoff for logical measurements. Finally, we construct an addressable surface code interface for injecting externally prepared resource states into selected logical qubits. Together, these results establish La-cross codes as a simple, low-connectivity generalization of surface code computation, retaining analogous computational primitives while providing a complete Clifford+T instruction set across multiple encoded qubits.