Random Clifford Deformations of the Surface Code

Clifford deformations of the surface code have been shown to have high error correction thresholds for noise biased towards dephasing. Moreover, careful tuning of the lattice geometry, i.e. shape and boundary conditions, can dramatically improve the subthreshold performance of the code with biased noise. Inspired by the improved code performance via the choice of Clifford deformation and lattice geometry, we consider random Clifford deformations of the surface code on the rotated layout. We demonstrate in the case of perfect syndrome measurements that certain random codes can outperform the best known translation-invariant deformations, i.e., the so-called ``XY'' code and ``XZZX'' codes at finite bias, in terms of subthreshold performance, while their code capacity thresholds are close to the previously highest values obtained from the XZZX code. We consider the statistical-mechanical mapping of random codes and conjecture a phase diagram of random codes with 50% thresholds at infinite bias through percolation theory arguments. We support this conjecture via tensor network decoder numerics on a large set of random code ensembles. It follows from results in bond-percolation, that for a linear system size L and at the critical point with 50% probability of Hadamard deformations, the infinite-bias distance scales like O(L^1.1)