Towards fault-tolerance implementation of the [[7,1,3]] code in 2D local quantum architectures

Recent experimental advancements have demonstrated the implementation of logical operations between encoded qubits of the [[7,1,3]] code using quantum architectures with all-to-all connectivity, such as neutral atom and ion trap platforms. In these systems, non-local interactions are achieved through the coherent transport of ions and atoms across two spatial dimensions. While the mobility of ions and atoms presents significant potential for enhanced connectivity within larger quantum registers, the impact of excess heating and atom loss during movement on the performance of state preparation and quantum error correction (QEC) for small-distance logical qubits has not been thoroughly investigated. Specifically, it remains unclear whether such dynamic connectivity is more beneficial or detrimental compared to architectures restricted to local interactions with nearest-neighbor qubits.

In this work, we evaluate the performance of various fault-tolerant encoding circuits and QEC protocols for the [[7,1,3]] code mapped onto a two-dimensional square lattice with limited local connectivity. We demonstrate how fault-tolerant designs for level-1 encoding (a 7-qubit logical state) and QEC can be achieved using only two-qubit gates between nearest-neighbor qubits, mid-circuit measurements, and post-selection. Furthermore, we illustrate how fault-tolerant transport can be realized in a static array through the use of SWAP operations and CNOT gate teleportation, enabling logical operations between encoded qubits and facilitating the preparation of larger logical states.