LDPC codes use way fewer physical qubits than surface codes. But there is currently no known logical gate set for LDPC codes. It was just shown that you can correct the identity gate, making a quantum memory (https://arxiv.org/abs/2308.07915). This makes gives people hope that a logical gate set might be exist. The trouble is that it's not know how to construct one. This is an interesting open question.
LDPC codes use way fewer physical qubits than surface codes. But there is currently no known logical gate set for LDPC codes. It was just shown that you can correct the identity gate, making a quantum memory (https://arxiv.org/abs/2308.07915). This makes gives people hope that a logical gate set might be exist. The trouble is that it's not know how to construct one. This is an interesting open question.