We refine ideas from Knill 1996, Jones 2016, Chamberland 2020, Gidney 2023+2024, Bombin 2024, and Hirano 2024 to efficiently prepare good $|T\rangle$ 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 $2 \cdot 10^{-9}$ when subjected to $10^{-3}$ uniform depolarizing circuit noise. Halving the circuit noise to $5 \cdot 10^{-4}$ improves the achievable logical error rate to $4 \cdot 10^{-11}$. Cultivation's efficiency and strong response to improvements in physical noise suggest that further magic state distillation may never be needed in practice.
We refine ideas from Knill 1996, Jones 2016, Chamberland 2020, Gidney 2023+2024, Bombin 2024, and Hirano 2024 to efficiently prepare good ∣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⋅10−9 when subjected to 10−3 uniform depolarizing circuit noise. Halving the circuit noise to 5⋅10−4 improves the achievable logical error rate to 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.
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.
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=2×10−3, the injection-and-cultivation expected attempts are reduced by 45.46% for d1=3 and by 72.91% for d1=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% for d1=3 and by 78.69% for d1=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.
Abstract Error-corrected quantum computers limited to small code distances trade physical qubits against the number of logical operations they can run. Two tools set that trade-off in the early fault-tolerant era: logical-level error mitigation, which cancels residual logical errors at a sampling cost exponential in the total error, and magic-state cultivation, which prepares non-Clifford resource states far more cheaply than distillation. Their resource estimates use different accountings of time, idle error, and consumption, so their combination has not been costed consistently. Here we build one cost model in which data block, factory, and sampling overhead share one clock and precision target, and show that whether cultivation pays is decided by two architectural numbers: the T-state demand per time step, ρ=nLufT, which sets the factory area, and the exposure N/u, which sets the idle error, where u is the fraction of the nL logical qubits acted on per step and fT the T fraction. At ρ≈1 adding cultivation to error mitigation saves 6--9% of physical qubits; at ρ≈30 it saves 71--85% because factories dominate the footprint. The feasible circuit size grows by at most 1.8--2.7× and never reaches 1010 operations at d≤13. We also identify a storage-limited error floor for 15-to-1 distillation confined to patch distance dX≤11 (3.7×10−6 at error rate 10−3), compute the exact floor of the d1=3 cultivation stage (6.8×10−7), show that escaping to d2=7--11 costs 25--56% of the published volume, and find that the error of a cultivated state given its decoder gap follows one curve per d1 to within 2×. On reported logical error suppression factors, today's hardware supports only tens of mitigated logical operations; the regime opens below a physical error rate of 2×10−3.
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 logical error rates for the d=5 color code, with substantially lower space-time overhead than magic state distillation. However, due to the O(d) depth of the Clifford measurement circuit, magic state cultivation becomes impractical for d>5. Here, we perform logical XS† 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.
Direct Cultivation of Entangled ∣CS⟩ Magic States
Magic-state cultivation has largely targeted single-qubit non-Clifford resources. We show that the entangled state ∣CS⟩=CS∣++⟩ can be cultivated directly rather than synthesized from three T-type resources. Two commuting Hermitian Clifford involutions project onto four Pauli-frame branches, so no ideal branch is discarded, while a minimal six-bit [6,2,4] record protects their history. The small-code state cannot itself be certified: late two-fault patterns evade its record, placing the certified boundary after surface-code growth. Under a fixed decoder that sees 136 operational detector bits per patch and withholds 168, exact audits find no accepted closed-boundary logical failure through fault order two, while order-three mechanisms exist. Escaping from the small code directly into a distance-3 patch and growing it to distance 13 in a single round keeps this certificate. Compared with a re-decoded three-T comparator matched to the same binary logical-error target, the direct route uses 2% fewer active locations at p=5×10−4 and 16--28% more at 8×10−4--10−3, without charging the comparator for T-state storage. Counted in qubit-ticks under one rule for both routes, it uses 21--39% less spacetime. Adding idle-memory noise to both routes under one rule and one optimized schedule raises the direct route's cost more, to 8% and 46% above the comparator at 5×10−4 and 10−3, because that cost is limited by acceptance. The result is architectural: an entangled non-Clifford state can itself be cultivated with a certified output interface.
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=5, the ∣T⟩ output reaches similar logical error rates to the ∣S⟩ output, traditionally used as a cheap full Clifford proxy. This contrasts with the ∼10× discrepancy reported for the d=5 colour-code scheme of Gidney et al. We also find that our new d=5 scheme has ∼1.3× lower expected space-time cost while still reaching 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=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.