Coherent-Error Threshold for Surface Codes from Majorana Delocalization

Statistical mechanics mappings provide key insights on quantum error correction. However, existing mappings assume incoherent noise, thus ignoring coherent errors due to, e.g., spurious gate rotations. We map the surface code with coherent errors, taken as $X$ or $Z$ rotations (replacing bit or phase flips), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network. Our mappings reveal both commonalities and qualitative differences in correcting coherent and incoherent errors. For both, the error-correcting phase maps, as we explicitly show by linking 2D networks to 1D fermions, to a ${\mathbb{Z}}_2$-nontrivial 2D insulator. However, beyond a rotation angle ${\varphi }_{\mathrm{th}}$, instead of a ${\mathbb{Z}}_2$-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This ${\varphi }_{\mathrm{th}}$ is the theoretically achievable storage threshold. We numerically find ${\varphi }_{\mathrm{th}}\approx 0.14\pi$. The corresponding bit-flip rate ${\sin }^2({\varphi }_{\mathrm{th}})\approx 0.18$ exceeds the known incoherent threshold $p_{\mathrm{th}}\approx 0.11$.

Figure 1

Left panel: bulk patch of a surface code. Black dots are qubits, and white and gray disks, respectively, show stabilizers $S_v^X$ and $S_w^Z$. The red line marks the logical $\overline{X}$’s path (for suitable boundary conditions). In the RBIM, the $S_v^X$ become spins interacting with their nearest neighbors through couplings set by the errors. Right panel: in the network model, the Ising bonds become junctions scattering incoming into outgoing Majorana modes. Solid and dashed lines show, respectively, the modes’ propagation direction for coherent and incoherent errors. In the transfer matrix, the junctions are quantum gates acting on pairs of Majorana operators.

Figure 2

Conductivity $g$ for the coherent-error network on a cylinder of length $L$ and circumference $M=5L$, averaged over 500 to ${10}^5$ syndrome realizations. Error bars ($2\times \text{standard}$ error) are imperceptible. The data following scaling curves $g[L/\ell (\varphi )]$ shows that $\varphi$ enters via a length scale $\ell (\varphi )$ [47]. For the insulator, $g[L/\ell (\varphi )]$ decays with $L$ and $g\propto e^{-2L/\ell (\varphi )}$ for $L\gg \ell (\varphi )$. For the metal, $g$ increases; the $g\propto \ln (L)/\pi$ class $D$ asymptote [37, 42] is not yet reached for the accessible range of $L$. The insulator-metal transition is at ${\varphi }_{\mathrm{th}}=(0.14\pm 0.005)\pi$. We observe $\mathcal{Q}=-1$ throughout the insulating phase.

Figure 3

Figure of merit $\mathrm{\Delta }$ for the $L\times L$ planar geometry of recent experiments [10, 11] (cf. inset for $L=3$, 5). We averaged over 250 to $2\times {10}^5$ syndrome realizations; error bars ($2\times \text{standard}$ error) are imperceptible. $\mathrm{\Delta }$ decays exponentially (dashed) with $L$ for $\varphi <{\varphi }_{\mathrm{th}}$. Above ${\varphi }_{\mathrm{th}}$, the data are consistent with $\mathrm{\Delta }$ decaying as a power law to ${\mathrm{\Delta }}_{\infty }(\varphi )<{\mathrm{\Delta }}_{\text{u}}=(\pi -2)/2\pi$.