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 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 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.
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.
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). At p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 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.
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.
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.
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=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.