Color code thresholds under circuit-level noise beyond the Pauli framework

TL;DR

This study addresses the problem of estimating quantum error-correction thresholds for the color code under realistic, non-Pauli circuit-level noise. It introduces a Tree Tensor Network (TTN) framework coupled with quantum trajectories to simulate open-system dynamics with coherent over-rotations and amplitude-damping noise injected throughout circuit execution. Threshold estimates reveal that coherent SRX errors yield higher logical failure rates than their Pauli-twirled equivalents, while amplitude-damping noise closely matches the Pauli-twirling approximation; results are obtained for code distances up to $d=7$ (73 qubits). The work demonstrates the viability and limits of tensor-network methods for realistic QECC evaluation and highlights the need to go beyond Pauli twirling for accurate threshold assessments in future quantum hardware and full-stack error-correction studies.

Abstract

A quantum error correction code is assessed over its ability to correct errors in noisy quantum circuits. This task requires extensive simulations of faulty quantum circuits, which are often made tractable by considering stochastic Pauli noise models, as they are compatible with efficient classical simulation techniques. However, such noise models do not fully capture the variety of physical error mechanisms encountered in realistic quantum platforms. In this work, we extend circuit-level noise modeling beyond the Pauli framework by estimating the threshold of the color code under more general noise models. Specifically, we consider two representative non-Pauli error channels: a systematic $X$-rotation model that introduces coherent over-rotations, and an amplitude damping channel that captures relaxation processes. These models are incorporated at the circuit level into color code circuits using a Tree Tensor Network ansatz. Our simulations demonstrate that tensor network simulations enable accurate threshold estimation under non-Pauli noise for color codes up to distance $d=7$ (73 qubits). Comparing our results with the Pauli twirling approximations of the noise models, we find that coherent over-rotations yield systematically higher error rates, deviating from the Pauli twirling approximation as the code distance increases.

Figures (8)

• Figure 1: The color code.(a) Honeycomb lattice with qubits placed on its vertices. A plaquette $p$ is highlighted. (b) The stabilizer operators of the color code acting on $p$. (c) Color code on a finite patch of the hexagon lattice. The cut shown in the figure is a $[[7,1,3]]$ code, the smallest in the color code family. Notice that the stabilizers on the boundary qubits act on 4 qubits rather than 6.
• Figure 2: Summary of the simulation setup and assumptions of the circuit-level noise model.(a) The simulated circuit consists of $C$ quantum error correction cycles of the color code. As a reference, we consider the $[[7,1,3]]$ color code, which requires $6$ ancilla qubits for syndrome measurement, leading to a circuit of $7+6$ qubits. The schematic illustrates an example of $s_p^X$ and $s_p^Z$ stabilizer measurements for the blue plaquette; the other plaquettes' syndromes are treated analogously. The scheduling of syndrome measurement operations across all plaquettes follows Ref. Lee2025colorcodedecoder. The striped regions denote idle qubits. (b) Circuit-level noise assumption: every gate is made noisy by executing the ideal gate $G$, followed by some error operation $E$. Moreover, we assume that the error acts on the idle qubits and on the qubit initialization/reset. (c) Gates $G$ acting on more than one qubit are made noisy by applying errors on each of the qubits $G$ acts on. (d) Noise on idle qubits is treated as for a multiple-qubit gate. (e) The error operations considered in this work are determined by the SRX (systematic rotations around $X$) and AD (amplitude damping) error models, defined respectively by Eqs. \ref{['eq:SRX']} and \ref{['eq:AD']}.
• Figure 3: Threshold estimation for one error correction cycle ($C=1$). The top panel corresponds to the SRX noise model, while the bottom panel shows results for the AD model. Each data point represents the logical failure rate $p_{\mathrm{fail}}$ estimated from $10^4$ independent decoding trials, where each trial involves a full circuit simulation with injected physical noise. Shaded regions around each curve delimit the 99% confidence intervals, computed assuming binomial statistics for the decoder outcomes (success or failure).
• Figure 4: Threshold estimation for repeated error correction cycles ($C=d$). The evaluation of the logical failure rate $p_{\mathrm{fail}}$ follows the same procedure as described in Figure \ref{['fig:T1-SRX+AD']}, based on $10^4$ decoding trials per data point with binomial confidence intervals.
• Figure 5: Comparison with Pauli Twirling Approximation of the noise models. We simulate a single round of error correction ($C=1$) at fixed noise strengths: the SRX model (left) with $\theta/\pi = 10^{-2}$ and the AD model (right) with $\gamma=4\cdot10^{-3}$. In both cases, we compare the logical failure rate obtained from TTN simulations with that predicted by the Pauli twirling approximation, evaluated using a Clifford simulator. The shaded area represents 99% confidence interval. The results show good agreement between the exact and twirled models for AD, while for SRX the discrepancy grows with code distance.