The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology

TL;DR

This paper analyzes the surface code under the most general single-qubit X-errors combining coherent and incoherent components, linking the residual logical-noise coherence to information-theoretic diagnostics and decoding thresholds. By mapping decoding to a classical random-bond Ising model and developing a transfer-matrix/MPS-based syndrome-sampling method, it reveals that logical coherence γ_L decays exponentially with code distance for any nonzero incoherent noise, enabling S_rel and I_C to signal thresholds only at sufficiently large distances. A phenomenological errorfield double field theory explains the instability of the fully coherent above-threshold regime when incoherent perturbations are present. The work provides a practical framework for simulating non-Pauli errors, computing maximum-likelihood thresholds, and understanding the concept of information-theoretic efficiency in fault-tolerant quantum memories under realistic noise. Overall, it advances the theoretical and numerical toolkit for evaluating QEC performance beyond Pauli channels and clarifies the role of coherence in logical noise and information measures.

Abstract

We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit X-error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.

Figures (7)

• Figure 1: (a) Approximate QEC phase diagram where bottom and top boundaries correspond to the incoherent ($\gamma=0$) and coherent ($\gamma=1$) limits. The dotted line denotes the maximum-likelihood threshold separating the error-correcting phase (QEC✓) from a non-correcting phase (QEC✗), where black dots represent numerically calculated values including error bars. Teal crosses show the minimum weight perfect matching threshold. Green and blue arrows show the incoherent Dennis:2002ds and coherent Venn:2023fp maximum-likelihood thresholds, respectively. (b) Part of the phase diagram, indicated in (a) as a gray rectangle. Olive and black diamonds denote the parameters used in panels (c) and (d). (c) Coherent information $I_\text{C}$ (solid lines) and quantum relative entropy $S_\mathrm{rel}$ (dashed lines and crosses) as a function of code distance $d$ for $p=0.1$, $\gamma= 0.995$ (olive markers) and $p=0.105$, $\gamma = 0.99$ (black markers). (d) Logical noise coherence $\gamma_\text{L}$ versus $d$ for the same $p$ and $\gamma$ as in (c). (For coherent errors, $\gamma_\text{L}=1$.) Data are averaged over 1000 to 10000 syndromes and the error bars showing the standard error of the mean are imperceptible.
• Figure 2: (a) Surface code ($L=M=5$) with physical qubits on the vertices (black discs), and alternating $S_v^X$ and $S_p^Z$ stabilizers on the faces of the lattice. The logical $X_\text{L}$ and $Z_\text{L}$ are denoted by blue and red dashed lines, respectively. We map the surface code to a complex RBIM with two Ising spins $\sigma_v$ and $\bar{\sigma}_v$ on each $S_v^X$ site. The dashed black lines connect $S_v^X$ sites, constituting vertical slices of the RBIM. (b) Quantum circuit with many-body gates $\hat{V}_{l,m}^{(q\bar{q},s)}$ on physical qubits, where diamonds denote state projection or initialization to $2\ket{++}$ (hence adjacent diamonds are projective $x$-basis measurements with outcome $+1$). The $\hat{V}_{l,m}^{(q\bar{q},s)}$ alternate with layers $\hat{H}$ and $\hat{H}'$, denoted by dashed rectangles. (c) To sample an error string's weight on site $j$ via $\eta_j = \pm 1$ [shown here: $j=22$ with the gate $\hat{T}_{22}^{(\eta_{22})}$ in red; $\hat{T}^{(\eta_j)}_j$ is a shorthand notation for $\hat{V}_{l,m}^{(qq,s)}$, see the discussion of Eq. \ref{['eq:Tnotation']}], we evaluate its conditional probability $P_j \propto \braket{ \omega_{j+1} | \hat{T}_j^{(\eta_j)} | \phi_{j-1}^{\{\eta\}} }$ [Eq. \ref{['eq:conditional']}] using that $\ket{\omega_{j+1}}$ is a product state of $2\ket{++}$ states (denoted by diamonds) and $\ket{00} + \ket{11}$ Bell states (denoted by rectangles). The state $\ket{\omega_{j+1}}$ is the superposition of all $\{\eta\}$ configurations of the state $T_{j+1}^{(\eta_{j+1})} \dots T_N^{(\eta_N)} \ket{\phi_0}$, shown on the right, cf. Eq. \ref{['eq:omega_sampling']}.
• Figure 3: (a)--(e) Logical error rate, (f)--(j) MWPM error rate, (k)--(o) quantum relative entropy, and (p)--(t) coherent information as a function of $p$ for fixed $\gamma$, which increases from left to right. The colors denote different system sizes. Results are averaged over 1000 to 10000 syndromes, and the error bars show the standard error of the mean. The black dashed line shows the approximate maximum likelihood threshold, the red dotted line the MWPM threshold, and the gray dashed line in panels (p)--(t) the perfect-recoverability limit $I_\text{C} = \ln 2$ as a guide for the eyes.
• Figure 4: (a)--(c) Logical error rate and (d)--(f) coherent information as a function of $-\log(1-\gamma)$ for fixed $p$, which increases from left to right. Note that we have used an unusual scaling of the $x$-axis with $\gamma$ to highlight features visible for large $\gamma$ close to the coherent limit $\gamma = 1$. ($\gamma$ itself is shown on the second $x$ axis on top of the panel.) The colors denote different system sizes. Results are averaged over 1000 to 10000 syndromes, and the error bars show the standard error of the mean. The gray dashed line in panels (d)--(f) shows $\ln 2$ as a guide for the eyes.
• Figure 5: (a)--(e) Entanglement entropy and (f)--(j) sample standard deviation of half-system entanglement entropy of the quantum circuit. The colors denote different system sizes, and the black dashed line the maximum likelihood threshold. Results are averaged over 1000 to 10000 syndromes, and the error bars show the standard error of the mean.