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arXiv:2503.18657 · PRX Quantum

Efficient Magic State Cultivation on R P 2

Zi-Han Chen, Ming-Cheng Chen, Chao-Yang Lu, Jian-Wei Pan

Published Nov 19, 2025• 3 citations
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Abstract

Preparation of high-fidelity logical magic states is crucial for fault-tolerant quantum computation. Among previous attempts to reduce the substantial cost of magic state preparation, magic state cultivation (MSC) [Gidney , Magic state cultivation: growing T states as cheap as CNOT gates, arXiv:2409.17595], a recently proposed protocol for logical T state preparation without magic state distillation, achieves state-of-the-art efficiency. Inspired by this work [Gidney , Magic state cultivation: growing T states as cheap as CNOT gates, arXiv:2409.17595], we propose a MSC procedure that can produce logical T states on the rotated surface code at a further reduced cost. To maintain high efficiency throughout our protocol, we design structured codes along with compact circuits bridging between them. More specifically, we construct a code family, the RP code, by putting the rotated surface code on R P 2 (a two-dimensional manifold), as well as two self-dual Calderbank-Shor-Steane codes, named SRP-3 and SRP-5, respectively. In our MSC protocol, we start with a cultivation process, in which a high-fidelity T state is prepared on a small RP code with distance 3 or 5. Then, to preserve the logical T state, we use an efficient and easy-to-decode expansion stage to grow a small RP code to a larger rotated surface code in one syndrome extraction (SE) round. The RP code serves as an efficient transfer station with efficient SE circuits and compact interfaces between the SRP-3 (or SRP-5) code—used in the cultivation process to efficiently verify the correctness of the logical T state—and larger rotated surface codes for preserving the prepared logical T state. Our MSC protocol utilizes nonlocal connectivity, available on both neutral atom array and ion trap platforms. According to our Monte Carlo sampling results, our MSC protocol requires about an order of magnitude smaller space-time volume to reach a target logical error rate of around 10 − 9 compared to the original MSC protocol.

DOI
10.1103/9kys-3whh

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Preparation of high-fidelity logical magic states is crucial for fault-tolerant quantum computation. Among previous attempts to reduce the substantial cost of magic state preparation, magic state cultivation (MSC) [Gidney , Magic state cultivation: growing T states as cheap as CNOT gates, arXiv:2409.17595], a recently proposed protocol for logical T state preparation without magic state distillation, achieves state-of-the-art efficiency. Inspired by this work [Gidney , Magic state cultivation: growing T states as cheap as CNOT gates, arXiv:2409.17595], we propose a MSC procedure that can produce logical T states on the rotated surface code at a further reduced cost. To maintain high efficiency throughout our protocol, we design structured codes along with compact circuits bridging between them. More specifically, we construct a code family, the RP code, by putting the rotated surface code on R P 2 (a two-dimensional manifold), as well as two self-dual Calderbank-Shor-Steane codes, named SRP-3 and SRP-5, respectively. In our MSC protocol, we start with a cultivation process, in which a high-fidelity T state is prepared on a small RP code with distance 3 or 5. Then, to preserve the logical T state, we use an efficient and easy-to-decode expansion stage to grow a small RP code to a larger rotated surface code in one syndrome extraction (SE) round. The RP code serves as an efficient transfer station with efficient SE circuits and compact interfaces between the SRP-3 (or SRP-5) code—used in the cultivation process to efficiently verify the correctness of the logical T state—and larger rotated surface codes for preserving the prepared logical T state. Our MSC protocol utilizes nonlocal connectivity, available on both neutral atom array and ion trap platforms. According to our Monte Carlo sampling results, our MSC protocol requires about an order of magnitude smaller space-time volume to reach a target logical error rate of around 10 − 9 compared to the original MSC protocol.
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.
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.
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 cultivation is a state-of-the-art protocol to prepare ultra-high fidelity non-Clifford resource states for universal quantum computation. It offers a significant reduction in spacetime overhead compared to traditional magic state distillation techniques. Cultivation protocols involve measuring a transversal logical Clifford operator on an initial small-distance code and then rapidly growing to a larger-distance code. In this work, we present a new cultivation scheme in which we measure the fold-transversal Hadamard of the unrotated surface code, and leverage unitary techniques to grow within the surface code family. Using both stabilizer and state vector simulations we find that this approach achieves the lowest known spacetime overhead for magic state cultivation. Practical implementation of our protocol is best suited to architectures with nonlocal connectivity, showing the strength of architectures where such connectivity is readily available.
Fault-tolerant quantum computing requires high-fidelity logical magic states for implementing non-Clifford operations. Magic-state cultivation provides a lower-overhead route to logical magic-state preparation, but its efficiency is limited by postselection loss during the early injection-and-cultivation stages. In this work, we propose an in-patch multiplexing scheme that uses early-stage idle resources within a single logical patch to create multiple local cultivation opportunities. A candidate that passes the early stages is forwarded to the standard escape pathway, while the escape stage and the decoder-based acceptance procedure are kept identical to those of the single-site baseline. Under a uniform depolarizing noise model with idle noise, the proposed protocol substantially reduces the injection-and-cultivation discard rate and the expected number of attempts required to obtain an accepted early-stage candidate. At a physical error rate of p=2×10−3p=2\times10^{-3}p=2×10−3, the injection-and-cultivation expected attempts are reduced by 45.46%45.46\%45.46% for d1=3d_1=3d1​=3 and by 72.91%72.91\%72.91% for d1=5d_1=5d1​=5, relative to the single-site MSC baseline. In the direct full-cycle evaluation including escape, the expected attempts per kept logical output are further reduced by 49.04%49.04\%49.04% for d1=3d_1=3d1​=3 and by 78.69%78.69\%78.69% for d1=5d_1=5d1​=5 at the same physical error rate. The full-cycle cost curves are shifted toward smaller expected attempts, while the final logical-error behavior remains governed by the escape-stage gap threshold. These results show that in-patch multiplexing can reduce postselection overhead while preserving the standard magic-state cultivation framework.
We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of O(θL2(1−Θ(1/d))pph)\mathcal{O}(θ_L^{2(1-Θ(1/d))}p_{\text{ph}})O(θL2(1−Θ(1/d))​pph​), while requiring only the ancillary space of a single surface code patch. Here, θLθ_LθL​ is the logical rotation angle, pphp_{\text{ph}}pph​ is the physical error rate, and ddd is the code distance. This scaling marks a significant improvement over the previous state-of-the-art, O(θLpph)\mathcal{O}(θ_L p_{\text{ph}})O(θL​pph​), making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for θL≲10−5θ_L \lesssim 10^{-5}θL​≲10−5, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard TTT-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+TTT+φφφ gate set, rather than the conventional Clifford+TTT or Clifford+φφφ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of pph=10−3p_{\text{ph}}=10^{-3}pph​=10−3.