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arXiv:2603.05429 · Physical Review Letters

Constant depth magic state cultivation with Clifford measurements by gauging

Bence Hetényi, Benjamin J. Brown, Dominic J. Williamson

quant-phSubmitted Mar 5, 2026• Announced Mar 6, 2026• 0 citations
v1 · Mar 5, 2026v2 · Sep 29, 2026
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Abstract

Magic states are a scarce resource for two-dimensional qubit stabilizer codes. Magic state cultivation was recently proposed to reduce the cost of magic state preparation by measuring the transversal Clifford operator of the color code. Cultivation achieves $\sim 10^{-9}$ logical error rates for the $d=5$ color code, with substantially lower space-time overhead than magic state distillation. However, due to the $\mathcal{O}(d)$ depth of the Clifford measurement circuit, magic state cultivation becomes impractical for $d>5$. Here, we perform logical $XS^\dagger$ measurements on the color code by gauging a transversal Clifford gate, resulting in a constant-depth logical measurement circuit. We employ repeated gauging measurements with post-selection rather than performing error correction on the Clifford stabilizer code that emerges during the gauging protocol, thus gaining simplicity at the cost of scalability. Our protocol requires a regular square grid connectivity and yields logical error rates comparable to magic state cultivation. The $d=7$ version of our protocol gives access to the $10^{-12}$ logical error rate regime at $0.05\%$ physical error rate while retaining more than $1\%$ of the shots after the equivalent of the cultivation stage.

Journal reference
Phys. Rev. Lett. 137, 140601 (2026)
DOI
10.1103/f7fz-6zsy

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Magic states are a scarce resource for two-dimensional qubit stabilizer codes. Magic state cultivation was recently proposed to reduce the cost of magic state preparation by measuring the transversal Clifford operator of the color code. Cultivation achieves ∼10−9\sim 10^{-9}∼10−9 logical error rates for the d=5d=5d=5 color code, with substantially lower space-time overhead than magic state distillation. However, due to the O(d)\mathcal{O}(d)O(d) depth of the Clifford measurement circuit, magic state cultivation becomes impractical for d>5d>5d>5. Here, we perform logical XS†XS^\daggerXS† measurements on the color code by gauging a transversal Clifford gate, resulting in a constant-depth logical measurement circuit. We employ repeated gauging measurements with post-selection rather than performing error correction on the Clifford stabilizer code that emerges during the gauging protocol, thus gaining simplicity at the cost of scalability. Our protocol requires a regular square grid connectivity and yields logical error rates comparable to magic state cultivation. The d=7d=7d=7 version of our protocol gives access to the 10−1210^{-12}10−12 logical error rate regime at 0.05%0.05\%0.05% physical error rate while retaining more than 1%1\%1% of the shots after the equivalent of the cultivation stage.
Magic state cultivation is a promising method for preparing high-fidelity non-Clifford resource states - a central component of fault-tolerant quantum computing - with low spacetime overhead. In benchmarking cultivation, existing resource estimates assume access to noiseless transversal measurement of the output magic state to allow for postselection during the final escape stage of the cultivation process. However, such measurements would destroy the magic state and are therefore unavailable during computation. We develop an open-boundary postselection method, Caliper, that uses only nondestructive mid-circuit syndrome information and substantially outperforms existing methods operating under these constraints. In some regimes, our postselection method uniquely recovers the resource efficiency of prior estimates. However, in other cases, we observe a tradeoff. At spacetime costs reported in previous works, the achievable logical error rates can be orders of magnitude higher, and recovering the reported logical error rate requires substantially larger codes and longer escape stages. Our results highlight how practical considerations involving mid-circuit postselection can impact the design and performance of magic state cultivation protocols.
Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.
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.
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.
Magic-state preparation is a central component of fault-tolerant quantum computing. Recent theoretical and experimental successes in code-switch-based magic-state preparation have underscored the promise of these methods for quantum error correction. Similarly, magic-state cultivation has likewise been demonstrated in both numerical and experimental settings. However, a thorough comparison between magic-state cultivation and code-switch-based magic-state factories is still missing. In this work, we carry out end-to-end simulations of magic-state preparation using code switching and compare its resource requirements and performance against magic-state cultivation. As part of this analysis, we develop a lattice-surgery protocol for transfer between the doubled color code and the rotated surface code. We extend the complete code-switching protocol to the d=5d=5d=5 doubled color code and perform the corresponding end-to-end simulations. Finally, we propose two fault-tolerant magic-state preparation protocols that combine phase-kickback checks with a transversal non-Clifford gate.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4)210.2p^3+O(p^4)210.2p3+O(p4). At p=10−3p=10^{-3}p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−72.2\times10^{-7}2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.