Directional Codes: a new family of quantum LDPC codes on hexagonal- and square-grid connectivity hardware

Utility-scale quantum computing requires quantum error correction (QEC) to protect quantum information against noise. Currently, superconducting hardware is a promising candidate for achieving fault tolerance due to its fast gate times and feasible scalability. However, it is often restricted to two-dimensional nearest-neighbour connectivity, which is thought to be incapable of accommodating high-rate quantum low-density parity-check (qLDPC) codes that promise to greatly reduce the number of physical qubits needed to encode logical qubits. In this paper we construct a new family of qLDPC codes, which we call ``Directional Codes'', that outperforms the rotated planar code (RPC) while naturally meeting the connectivity requirements of the widely adopted square-grid, and some even the sparser hexagonal-grid. The key idea is to utilise the iSWAP gate -- a natural native gate for superconducting qubits -- to construct circuits that measure the stabilisers of these qLDPC codes without the need for any long-range connections or an increased degree of connectivity. We numerically evaluate the performance of directional codes, encoding four, six and twelve logical qubits, using a common superconducting-inspired circuit-level Pauli noise model. We also compare them to the RPC and to the bivariate bicycle (BB) codes, currently the two most popular quantum LDPC code families. As a concrete example, directional codes outperform the RPC by achieving approximately the same logical error probability at physical error rate $p=10^{-3}$ using only $18.75-45\%$ of the physical qubits at distance up to $10$. Our discovery represents a breakthrough in QEC code design that suggests complex long-range, high-connectivity hardware may not be necessary for low-overhead fault-tolerant quantum computation.