Tailoring surface codes for highly biased noise

The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high thresholds. We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases, and show how to exploit these features to achieve significant improvement in logical failure rate. First, we consider the infinite bias limit, meaning pure dephasing. We prove that the error threshold of the modified surface code for pure dephasing noise is $50\%$, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time decoding algorithm. We demonstrate that the sub-threshold behavior of the code depends critically on the precise shape and boundary conditions of the code. That is, for rectangular surface codes with standard rough/smooth open boundaries, it is controlled by the parameter $g=\gcd(j,k)$, where $j$ and $k$ are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for co-prime codes that have $g=1$, and closely-related rotated codes, which have a modified boundary. The effect is dramatic: the same logical failure rate achievable with a square surface code and $n$ physical qubits can be obtained with a co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards dephasing. In particular, comparing with a square surface code, we observe a significant improvement in logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.