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arXiv:2609.18922

Exact logical error rates for magic state cultivation

Kwok Ho Wan, Ainhoa Zapirain

quant-phSubmitted Sep 16, 2026• Announced Sep 17, 2026• 0 citations• Revised Sep 29, 2026 (v3)
v1 · Sep 16, 2026v2 · Sep 24, 2026v3 · Sep 29, 2026
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Abstract

We compute exactly the acceptance and logical error rates for the distance $d=3$ and $d=5$ magic state cultivation circuits from Clifft [arXiv:2604.27058] and SOFT [arXiv:2512.23037] using Pauli propagation and binary tensor contraction. Actual $T$-gates are studied, not the $S$-gate proxy used for sampling. The calculation includes every fault order at several circuit-level noise strengths ($p$). We provide a series expansion form to the logical error rates, through order $(p/(1-p))^{10}$. The analytical results recover the numerical values from Clifft and SOFT at both $d=3$ and $d=5$ to within their sampling uncertainty. Then, we show that the $d=3$ and $d=5$ circuits actually have fault distances of $d_{\text{fault}}=2$ and $d_{\text{fault}}=3$ respectively, explaining the similar distance degrading effects from the companion code of [Quantum 10, 2134 (2026)].

Comments
v3 - expanded section on Pauli semiwebs and implementation available at: https://github.com/kh428/exact_ler_for_msc

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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)].
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.
Magic state cultivation prepares high-fidelity magic states at low expected space-time costs; however, the exact performance of some schemes is unsettled due to the difficulty in simulating non-Clifford circuits. Here, we use matrix-product states (MPS) based methods to compute the exact performance of two types of fold-transversal cultivation schemes: (i) the Sahay et al method based on the regular surface code S gate, and (ii) a method we propose based on a partially fault-tolerant fold-transversal S gate. We show that for the former protocol at d=5d=5d=5, the ∣T⟩|T\rangle∣T⟩ output reaches similar logical error rates to the ∣S⟩|S\rangle∣S⟩ output, traditionally used as a cheap full Clifford proxy. This contrasts with the ∼10×\sim10\times∼10× discrepancy reported for the d=5d=5d=5 colour-code scheme of Gidney et al. We also find that our new d=5d=5d=5 scheme has ∼1.3×\sim1.3\times∼1.3× lower expected space-time cost while still reaching 10−910^{-9}10−9 logical error rate. We show that MPS and Clifford-augmented MPS (CAMPS) perform on par with or even better than the recently introduced near-Clifford simulator Clifft on the hardest d=5d=5d=5 regular surface code scheme. Additionally, to speed up simulation, we propose a new pre-screening method based on simple Pauli propagation, lowering by up to three orders of magnitude the required number of exact simulations, and use several simulator-agnostic sampling methods such as subset sampling.
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.
Circuit simulation tools are critical for developing and assessing quantum-error-correcting and fault-tolerant strategies. In this work, we present SOFT, a high-performance SimulatOr for universal Fault-Tolerant quantum circuits. Integrating the generalized stabilizer formalism and highly optimized GPU parallelization, SOFT enables the simulation of noisy quantum circuits containing non-Clifford gates at a scale not accessible with existing tools. To provide a concrete demonstration, we simulate the state-of-the-art magic state cultivation (MSC) protocol at code distance d=5d=5d=5, involving 42 qubits, 72 TTT / T†T^\daggerT† gates, and mid-circuit measurements. Using only modest GPU resources, SOFT performs over 200 billion shots and achieves the first ground-truth simulation of the cultivation protocol at a non-trivial scale. This endeavor not only certifies the MSC's effectiveness for generating high-fidelity logical TTT-states, but also reveals a large discrepancy between the actual logical error rate and the previously reported values. Our work demonstrates the importance of reliable simulation tools for fault-tolerant architecture design, advancing the field from simulating quantum memory to simulating a universal quantum computer.
We refine ideas from Knill 1996, Jones 2016, Chamberland 2020, Gidney 2023+2024, Bombin 2024, and Hirano 2024 to efficiently prepare good ∣T⟩|T\rangle∣T⟩ states. We call our construction "magic state cultivation" because it gradually grows the size and reliability of one state. Cultivation fits inside a surface code patch and uses roughly the same number of physical gates as a lattice surgery CNOT gate of equivalent reliability. We estimate the infidelity of cultivation (from injection to idling at distance 15) using a mix of state vector simulation, stabilizer simulation, error enumeration, and Monte Carlo sampling. Compared to prior work, cultivation uses an order of magnitude fewer qubit-rounds to reach logical error rates as low as 2⋅10−92 \cdot 10^{-9}2⋅10−9 when subjected to 10−310^{-3}10−3 uniform depolarizing circuit noise. Halving the circuit noise to 5⋅10−45 \cdot 10^{-4}5⋅10−4 improves the achievable logical error rate to 4⋅10−114 \cdot 10^{-11}4⋅10−11. Cultivation's efficiency and strong response to improvements in physical noise suggest that further magic state distillation may never be needed in practice.
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.