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arXiv:2511.05719 · Quantum Science and Technology

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

Francesco Pio Barone, Daniel Jaschke, Ilaria Siloi, Simone Montangero

quant-phSubmitted Nov 7, 2025• Announced Nov 11, 2025• 0 citations
v1 · Nov 7, 2025v2 · Sep 14, 2026
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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.

Journal reference
Quantum Sci. Technol. 11 035065, 2026
DOI
10.1088/2058-9565/ae8df5

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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 XXX-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=7d=7d=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.
Quantum error correcting (QEC) codes protect quantum information from decoherence as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, suboptimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered, we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close agreement with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models. Published by the American Physical Society 2024
Quantum noise is a central challenge in quantum computing across many applications. Extensive work has examined how qubits couple to their environment, leading to decoherence and irreversible relaxation. This work studies a continuous coherent noise model for quantum circuits and compares it with a discrete Pauli model. The focus is on small, coherent gate errors that accumulate across circuit depth. These errors are modeled as random rotations on the Bloch sphere using a von Mises-Fisher distribution. In the small-angle limit, the model reduces to an isotropic Gaussian distribution. We test the model on quantum error-correction circuits based on the [[5,1,3]] and [[7,1,3]] codes. A variant of Grover's search circuit with different qubit counts is also examined. To enable fair comparison, we introduce a model-independent matching scheme. Pauli and continuous noise channels are aligned using the binary entropy at readout. This isolates the effect of noise structure at fixed uncertainty. An approximate analytical method for the propagation of coherent errors is also developed. The method tracks error distributions both on Clifford and non-Clifford circuits without full Monte Carlo sampling. It reduces simulation cost while preserving accuracy for circuit-level error estimates. The approximation is validated against brute-force simulations, identifying its regime of validity with Clifford and non-Clifford circuits under error correction. Our results show that continuous coherent noise and Pauli noise lead to comparable logical-error trends in the stabilizer-code circuits studied here. This behavior is consistent with syndrome extraction projecting a general single-qubit error onto Pauli-error subspaces before recovery. They also show where simplified propagation models work well and where their accuracy is reduced.
Quantum error correction (QEC), the lynchpin of fault-tolerant quantum computing (FTQC), is designed and validated against well-behaved Pauli stochastic error models. But in real-world deployment, QEC protocols encounter a vast array of other errors -- coherent and non-Pauli errors -- whose impacts on quantum circuits are vastly different than those of stochastic Pauli errors. The impacts of these errors on QEC and FTQC protocols have been largely unpredictable to date due to exponential classical simulation cost. Here, we show how to accurately and efficiently model the effects of coherent and non-Pauli errors on FTQC, and we study the effects of such errors on syndrome extraction for surface and bivariate bicycle codes, and on magic state cultivation. Our analysis suggests that coherent error can shift fault-tolerance thresholds, increase the space-time cost of magic state cultivation, and can increase logical error rates by an order of magnitude compared to equivalent stochastic errors. These analyses are enabled by a new technique for mapping any Markovian circuit-level error model with sufficiently small error rates onto a detector error model (DEM) for an FTQC circuit. The resulting DEM enables Monte Carlo estimation of logical error rates and noise-adapted decoding, and its parameters can be analytically related to the underlying physical noise parameters to enable approximate strong simulation.
Quantum error correcting (QEC) codes protect quantum information from decoherence, as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, sub-optimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close accordance with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive practical tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.
Stabilizer-based simulation of quantum error-correcting codes typically relies on the Pauli-twirling approximation (PTA) to render non-Clifford noise classically tractable, but PTA can distort the behavior of physically relevant channels such as thermal relaxation. Physically accurate noise simulation is needed to train decoders and understand the noise suppression capabilities of quantum error correction codes. In this work, we develop an exact and stabilizer-compatible model of qubit thermal relaxation noise and show that the combined amplitude damping and dephasing channel admits a fully positive probability decomposition into Clifford operations and reset whenever T2⩽T1T_2 \leqslant T_1T2​⩽T1​. For T2>T1T_2 > T_1T2​>T1​, the resulting decomposition is negative, but allows a smaller sampling overhead versus independent channels. We further introduce an approximated error channel with reset that removes the negativity of the decomposition while achieving higher channel fidelity to the true thermal relaxation than PTA, and extend our construction to finite temperature relaxation. We apply the exact combined model to investigate large surface codes and bivariate bicycle codes on superconducting platforms with realistic thermal relaxation error. The differing logical performances across code states further indicate that noise-model-informed decoders will be essential for accurately capturing thermal-noise structure in future fault-tolerant architectures.
The rapidly growing landscape of quantum error-correction (QEC) protocols has produced a wealth of numerical data, but comparatively few heuristics for understanding and predicting their performance. Here, we develop a simple entropy-based proxy that predicts the thresholds of a variety of QEC protocols, ranging from the code-capacity setting of Clifford-deformed surface codes with biased Pauli or erasure noise to the circuit-level noise model of the surface or color codes with flag qubits. Our proxy estimates the threshold by locally comparing the noise entropy with the error information gained via stabilizer measurements (or spacetime detectors and flag outcomes in the circuit-level settings). Despite neglecting correlations between stabilizer outcomes and the contribution from code degeneracy, the proxy captures the main trends across diverse settings and yields threshold estimates in good agreement with numerical results. Our work develops much-needed phenomenology that enables simple back-of-the-envelope estimates of QEC performance, recovering and providing an explanation for the results of computationally intensive detailed simulations.