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 \(p=2\times10^{-3}\), the injection-and-cultivation expected attempts are reduced by \(45.46\%\) for \(d_1=3\) and by \(72.91\%\) for \(d_1=5\), 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 \(d_1=3\) and by \(78.69\%\) for \(d_1=5\) 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.
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\times10^{-3}), the injection-and-cultivation expected attempts are reduced by (45.46%) for (d_1=3) and by (72.91%) for (d_1=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 (d_1=3) and by (78.69%) for (d_1=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 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 offers a promising alternative for lowering the cost of non-Clifford operations in fault-tolerant quantum computing (FTQC). However, realizing cultivation in practice exposes two challenges: (i) the lack of an open-source classical runtime layer between logical software and physical control, and (ii) the latency constraints on protocol-specific decisions that determine magic-state readiness. To address these challenges, we adopt an algorithm--hardware co-design approach. At the algorithm level, we develop a two-stage early-escape scheme that identifies an informative subset of detectors offline, constructs a compact decoding problem, and performs partial decoding in parallel with complete decoding at runtime, allowing high-confidence attempts to advance before full decoding completes. At the hardware level, we present MagiCFirm, a configurable runtime that combines offline-compiled microprograms with dedicated datapaths for detector construction, event processing, and protocol control, enabling end-to-end execution of magic-state cultivation. Across evaluated configurations, MagiCFirm reduces wall-clock magic-state preparation time by up to 39.3% at matched logical error rate. For a representative magic-state-bound workload, this translates to an estimated 11% reduction in overall application runtime.
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.
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.
Erasure qubits constitute a promising approach for tackling the daunting resources required for fault-tolerant quantum computing. By heralding of erasure errors, both the error-correction threshold and the sub-threshold scaling of the logical error rate are significantly improved. While previous research has focused primarily on fault-tolerant quantum memories, we extend this investigation to magic state injection—a critical yet resource-intensive component of fault-tolerant quantum computation. We show that, after postselection on erasures, the logical error rate of the injected magic state is set by the residual Pauli error, while the space-time overhead is only marginally increased as compared with the case of non-erasure qubits with a similar noise strength. These conclusions hold both for injection into the surface code and for injection and cultivation on the color code. For the former, we show that most of the gains can be achieved by the use of just three strategically placed erasure qubits in the surface code patch, independent of the patch size. For the latter, in contrast, it is beneficial to have all the qubits in the cultivation patch be erasure qubits. Our results for cultivation suggest that algorithmically relevant logical error rates may be within reach magic state distillation for erasure rates ≲ 4 × 10 − 3 and residual Pauli error rates of ∼ 10 − 4 .
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.