Tailored rotated Toric codes for biased noise

Quantum error correction lies at the heart of fault-tolerant quantum computing. The performance of error-correcting codes can be greatly boosted when tailored for realistic types of noise, such as noise biased towards dephasing. We consider a family of generalized rotated toric codes (RTCs) that are defined on a square lattice whose plaquettes represent the XZZX type of check operators, with periodic boundary conditions specified by two (not necessarily orthogonal) periodicity vectors. We show that by adaptively choosing the code layout and tailoring the decoder, the RTCs exhibit high threshold and favorable sub-threshold resource scaling in the biased-noise regime. With perfect measurements, the code threshold saturates the hashing bound. Furthermore, the RTCs have a remarkable fault-tolerant threshold using practical decoders under a phenomenological noise model.  We show that the optimally-chosen RTCs, encoding a single logical qubit, can outperform the planar surface codes when using the same amount of qubit resource across different bias regimes, therefore enabling more efficient quantum error correction.