No-go theorems for logical gates on product quantum codes

Understand

Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation.

• Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes.
• Based on extensions of the Bravyi--K\"{o}nig theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes.
• Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits.