Non-linear Sigma Model for the Surface Code with Coherent Errors

Abstract

The surface code is a promising platform for a quantum memory, but its threshold under coherent errors remains incompletely understood. We study maximum-likelihood decoding of the square-lattice surface code in the presence of single-qubit unitary rotations that create electric anyon excitations. We microscopically derive a non-linear sigma model with target space $\mathrm{SO}(2n)/\mathrm{U}(n)$ as the effective long-distance theory of this decoding problem, with distinct replica limits: $n\to1$ for optimal decoding, which assumes knowledge of the coherent rotation angle, and $n\to0$ for suboptimal decoding with imperfect angle information. This exposes a sharp distinction between the two decoders. The suboptimal decoder supports a ``thermal-metal'' phase, a non-decodable regime that is qualitatively distinct from the conventional non-decodable phase of the surface code under incoherent Pauli errors. By contrast, the metal phase cannot arise in optimal decoding, since the metallic fixed-point becomes unstable in the $n\to 1$ replica limit. We argue that optimal decoding may be possible up to the maximally-coherent rotation angle. Within the sigma model description, we show that the decoding fidelity is related to twist defects of the order-parameter field, yielding quantitative predictions for its system-size dependence near the metallic fixed point for both decoders. We examine our analytic predictions for the decoding fidelity as well as other physical observables with extensive numerical simulations. We discuss how the symmetries and the target space for the sigma model rely on the lattice of the surface code, and how a stable thermal metal phase can arise in optimal decoding when the syndromes reside on a non-bipartite lattice.

Figures (18)

• Figure 1: Proposed phase diagrams for optimal (a) and suboptimal (b) Pauli decoding in the square-lattice surface code subject to single-qubit unitary rotations with a uniform rotation angle $\theta$. For optimal decoding, the effective description near $\theta=\pi/4$ is close to an unstable metallic fixed point, resulting in a large crossover length-scale, after which we believe an asymptotic decodable phase is reached.
• Figure 2: The decoding fidelity is related to a twist of the order-parameter field $Q \rightarrow \Lambda Q\Lambda$. In one picture, this twist is inserted along the open direction of the cylinder (a), while in a dual description, the twist is inserted along the compact direction of the cylinder (b).
• Figure 3: Square-lattice surface code on the cylinder of circumference $L$ and length $T$ terminated with rough boundaries at the top and bottom. The code is defined with $X$-vertex stabilizers $A_v$ and $Z$-plaquette stabilizers $B_p$. Logical $X_L$ and $Z_L$ operators traverse the horizontal and vertical directions, respectively, encoding one qubit of quantum information.
• Figure 4: The partition sum $\mathcal{Z}_{\alpha,s}$ (\ref{['eqn:single_replica_complex_rbim']}) is defined for Ising spins on the dual square lattice of the surface code. A reference error string $\mathcal{C}_{z}^{\mathrm{ref}}$ for a syndrome intersects bonds of this lattice, and fixes a configuration of the Ising interactions $\eta_{\boldsymbol{r}\boldsymbol{r}'}$. On the right, this partition sum is represented as a product of transfer matrices, as in Eq. (\ref{['eqn:single_copy_rbim_prob_amplitude']}).
• Figure 5: Fidelity of the optimal decoder. (a) $\mathcal{F}_\mathrm{opt}$ as a function of $\theta$ at a fixed $\kappa = 4$ for various $L$. For small $\theta$, the fidelity is very close to $1$ while for larger $\theta$, it increases with $L$, consistent with a decodable phase. (b, c) Decoding infidelity $1 - \mathcal{F}_\mathrm{opt}$ as a function of $\kappa$. The plots demonstrate that increasing $\kappa$ for fixed $L$, or increasing $L$ for fixed $\kappa > 1$, increases $\mathcal{F}_\mathrm{opt}$. Furthermore, increasing $g_R^{-1}(L)$ through $\theta$ leads to increased fidelity for $\kappa < 1$ but decreased fidelity for $\kappa > 1$. This behavior agrees with our predictions. Plots generated with $900$ to $15000$ samples. The error bars are within the size of the markers.