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arXiv:2603.14670

Computing logical error thresholds with the Pauli Frame Sparse Representation

Thomas Tuloup, Thomas Ayral

quant-phSubmitted Mar 15, 2026• Announced Mar 17, 2026• 0 citations• Revised Apr 9, 2026 (v2)
v1 · Mar 15, 2026v2 · Apr 9, 2026
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Abstract

We introduce a sparse classical representation, a truncation strategy and a shot-efficient sampling method to push the classical prediction of quantum error correction thresholds beyond Clifford operations and Pauli errors. As two illustrations of the potential of our method, we first show that coherent noise error thresholds, when computed at the circuit level (i.e taking into account full syndrome circuits) for distances up to d=9, are systematically overestimated (by a factor of about 4) by a Pauli-twirling approximation of the noise. We then apply our method to the recently introduced magic-state cultivation protocol. We show, through shot-efficient importance sampling, that, at distance d=5, the multiplicative factor between the T-gate and the S-gate injection error rate is not the one conjectured from low-d computations: it can be as large as 7.

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25 pages, 13 figures

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quant-pharXiv:2609.17706•Sep 17, 2026•0 citations
We introduce a sparse classical representation, a truncation strategy and a shot-efficient sampling method to push the classical prediction of quantum error correction thresholds beyond Clifford operations and Pauli errors. As two illustrations of the potential of our method, we first show that coherent noise error thresholds, when computed at the circuit level (i.e taking into account full syndrome circuits) for distances up to d=9, are systematically overestimated (by a factor of about 4) by a Pauli-twirling approximation of the noise. We then apply our method to the recently introduced magic-state cultivation protocol. We show, through shot-efficient importance sampling, that, at distance d=5, the multiplicative factor between the T-gate and the S-gate injection error rate is not the one conjectured from low-d computations: it can be as large as 7.
The preparation of high-fidelity logical magic states is a crucial subroutine for universal fault-tolerant quantum computation (FTQC). Predicting the performance of FTQC and developing improved protocols rely on numerical methods to classically simulate logical magic state preparation in the presence of noise. Clifford logic on Pauli-stabilizer codes with circuit-level Pauli errors can be efficiently simulated using Pauli-stabilizer formalism, but the non-Clifford operations required to prepare logical magic states render generic simulation inefficient. We introduce Clifford-stabilizer simulation, an exact and efficient algorithm based on updating a Clifford-stabilizer group to simulate noisy preparation protocols for a broad class of logical magic states used to implement non-Clifford gates in the third level of the Clifford hierarchy under circuit-level Pauli errors. Clifford-stabilizer simulation applies to a range of operations that commonly appear in preparation protocols for such logical magic states, including Pauli-stabilizer measurements, logical Clifford measurements, and transversal non-Clifford gates. Our algorithm for Clifford-stabilizer simulation maps a non-Clifford circuit with sampled circuit-level Pauli errors to a Clifford circuit that exactly reproduces its measurement outcome distribution, achieving time and space complexities polynomial in relevant protocol parameters. We perform exact simulation of magic state cultivation up to fault distance 7 by Clifford-stabilizer simulation. Our method provides a route to perform exact benchmarking of large-scale logical magic state preparation protocols required for useful FTQC.
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.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
We compute exactly the acceptance and logical error rates for the distance d=3d=3d=3 and d=5d=5d=5 magic state cultivation circuits from Clifft [arXiv:2604.27058] and SOFT [arXiv:2512.23037] using Pauli propagation and binary tensor contraction. Actual TTT-gates are studied, not the SSS-gate proxy used for sampling. The calculation includes every fault order at several circuit-level noise strengths (ppp). We provide a series expansion form to the logical error rates, through order (p/(1−p))10(p/(1-p))^{10}(p/(1−p))10. The analytical results recover the numerical values from Clifft and SOFT at both d=3d=3d=3 and d=5d=5d=5 to within their sampling uncertainty. Then, we show that the d=3d=3d=3 and d=5d=5d=5 circuits actually have fault distances of dfault=2d_{\text{fault}}=2dfault​=2 and dfault=3d_{\text{fault}}=3dfault​=3 respectively, explaining the similar distance degrading effects from the companion code of [Quantum 10, 2134 (2026)].
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 hexagonal 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.
T-state cultivation is a resource-efficient protocol producing logical T states but recent benchmarks show that its logical error rate is considerably higher than intended i.e. than S-state cultivation, which is the analogous protocol for producing logical S states. In this paper, we explain this T-S discrepancy by analytically showing, under circuit-level depolarising noise, that distance-3 (-5) T-state cultivation has fault distance 2 (3) due to Pauli hook errors that propagate to coherent Clifford errors after its final double-check circuit; such errors remain Pauli in S-state cultivation, which consequently retains fault distance 3 (5). As part of our analysis we derive a general formula, and an O(n3)\mathcal O(n^3)O(n3)-time algorithm for fixed logical-qubit count, for the acceptance probability of a logical mixed state afflicted with a Clifford error, where nnn is the physical qubit count. We then design flags that detect the malignant hook errors in cultivation, improving the pre-escape logical error rate from Θ(p3)Θ(p^3)Θ(p3) to Θ(p5)Θ(p^5)Θ(p5). At noise level p=10−3p =10^{-3}p=10−3, this is a 4.9×\times× improvement, costing a 1.36×\times× increase in attempts per accepted shot.