Ultrahigh Error Threshold for Surface Codes with Biased Noise

We show that a simple modification of the surface code can exhibit an enormous gain in the error correction threshold for a noise model in which Pauli $Z$ errors occur more frequently than $X$ or $Y$ errors. Such biased noise, where dephasing dominates, is ubiquitous in many quantum architectures. In the limit of pure dephasing noise we find a threshold of 43.7(1)% using a tensor network decoder proposed by Bravyi, Suchara, and Vargo. The threshold remains surprisingly large in the regime of realistic noise bias ratios, for example 28.2(2)% at a bias of 10. The performance is, in fact, at or near the hashing bound for all values of the bias. The modified surface code still uses only weight-4 stabilizers on a square lattice, but merely requires measuring products of $Y$ instead of $Z$ around the faces, as this doubles the number of useful syndrome bits associated with the dominant $Z$ errors. Our results demonstrate that large efficiency gains can be found by appropriately tailoring codes and decoders to realistic noise models, even under the locality constraints of topological codes.

Figure 1

Threshold error rate $p_c$ as a function of bias $\eta$. The dark gray line is the zero-rate hashing bound for the associated Pauli error channel. Lighter gray lines show the hashing bound for rates $R=0.001$ and 0.01 for comparison; the surface code family has rate $1/n$ for $n$ qubits. Blue points show the estimates for the threshold using the fitting procedure described in the main text together with 1-standard-deviation error bars. The point at the largest bias value corresponds to infinite bias, i.e., only $Z$ errors.

Figure 2

The modified surface code, tailored for biased $Z$ noise, with logical operators given by a product of $Y$ along the top edge and a product of $X$ along the left edge. The stabilizers are shown at right.

Figure 3

Exponential decay of the logical failure rate $f$ with respect to code distance $d$ in the regime $p<p_c$ for $\eta =100$ and $\chi =48$. We observe scaling behavior of the form $f\sim \exp (-\alpha d)$ where $\alpha$ depends on the bias and is an increasing function of $(p_c-p)$. In this bias regime, the decoder performance is likely farthest from optimal, but the decay is still clearly exponential over this range. Other values of $\eta$ show the same general scaling behavior, though with different decay rates $\alpha$. The statistical error bars from 30 000 trials per point are smaller than the individual plot points in every case.