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arXiv:2609.09993

Direct Cultivation of Entangled $|CS\rangle$ Magic States

Gunsik Min, Jun Heo

quant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
v1 · Sep 9, 2026v2 · Sep 10, 2026v3 · Sep 11, 2026v4 · Oct 27, 2026
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Abstract

Magic-state cultivation has so far focused mainly on single-qubit non-Clifford resources. We develop a direct cultivation architecture for the entangled state $|CS\rangle=CS|++\rangle$. Two commuting Clifford involutions project onto four usable branches related by Pauli-frame updates. A minimal six-bit $[6,2,4]$ record protects the branch label against readout errors that map one valid record to another. Verified CAT$_7$ gadgets, Steane error detection, CZZ-based controlled checks, and immediate Steane-to-surface expansion form the complete factory. The decoder uses 263 operational detector bits, while 168 additional bits are withheld for later validation. Decoding proceeds through exact low-order resolution, a precomputed higher-order catalogue, and four-coset BP+OSD. Under the stated active-location stochastic-Pauli model, exact enumeration finds no accepted closed-boundary logical-failure mechanism through fault order two, while explicit order-three failure mechanisms exist. Finite-$p$ simulations quantify acceptance, residual syndromes, and decoder workload, and targeted sampling of order-three faults estimates the leading logical-error channels. Optimizing the direct $d=5\rightarrow13$ expansion reduces accepted-output operation count by approximately 35--37\%. We compare the two routes at the same binary logical-error rate. Direct CS remains cheaper in operation count at the two lower-noise benchmark points even when the three-$T$ route is given pre-existing output patches. The ordering reverses between $8\times10^{-4}$ and $9\times10^{-4}$. These results show that an entangled non-Clifford state can be cultivated directly with a protected branch record and an explicitly certified fault-order-three output channel.

Related papers

Coded Clifford Measurements for Multiqubit Magic-State Cultivation

Gunsik Min, Jun Heo

Magic-state cultivation suppresses errors by measuring logical Clifford checks and discarding inconsistent outcomes. For an entangled resource, several measurement branches are usable, so their outcomes form a multibit classical record whose corruption can produce a logical-frame error. We show that this measurement record can be protected as a binary linear code. For any third-level Clifford-hierarchy unitary $U\in\mathcal C_3$, every parity of the branch bits can be measured by a commuting Hermitian Clifford check $C(v)=UX(v)U^\dagger$. Choosing which parities to measure therefore defines a binary code $y(a)=aG$. If the valid records have minimum distance $d$, at least $d$ readout-bit flips are required to confuse one valid branch with another, giving $P_{\rm wv}=O(q^d)$. A Plotkin bound limits the length of any binary branch record, and the Clifford schedules attain this limit: at distance four, six logical measurements suffice for $|CS\rangle$ and seven for $|CCZ\rangle$, compared with eight and twelve under independent repetition. Thus restricting the logical schedule to Clifford parities requires no additional measurements. In a native-CZZ-assisted Steane implementation, the $[6,2,4]$ CS schedule is also the unique minimum-cost distance-four solution within the compiled Clifford-check family, reducing the cultivation core by $26.6\%$ in active locations. State-vector simulations without a final ideal code-space projection further show higher acceptance and approximately half the residual error weight on the two Steane blocks. Coding the logical measurement record therefore reduces both measurement redundancy and compiled fault-tolerant overhead.

quant-pharXiv:2609.09994•Sep 10, 2026•0 citations

Reducing Postselection Overhead in Magic-State Cultivation by In-Patch Multiplexing

Dongmin Kim, Jeonggeun Seo, Aniket Patra +1 more

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.

quant-pharXiv:2605.03616•May 6, 2026•0 citations

Where magic-state cultivation pays in early fault-tolerant quantum computing: a single-clock cost model of logical error mitigation and magic-state supply

Hikaru Wakaura, Taiki Tanimae

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, $\rho=n_L u f_T$, which sets the factory area, and the exposure $N/u$, which sets the idle error, where $u$ is the fraction of the $n_L$ logical qubits acted on per step and $f_T$ the $T$ fraction. At $\rho\approx1$ adding cultivation to error mitigation saves $6$--$9\%$ of physical qubits; at $\rho\approx30$ it saves $71$--$85\%$ because factories dominate the footprint. The feasible circuit size grows by at most $1.8$--$2.7\times$ and never reaches $10^{10}$ operations at $d\le13$. We also identify a storage-limited error floor for 15-to-1 distillation confined to patch distance $d_X\le11$ ($3.7\times10^{-6}$ at error rate $10^{-3}$), compute the exact floor of the $d_1=3$ cultivation stage ($6.8\times10^{-7}$), show that escaping to $d_2=7$--$11$ costs $25$--$56\%$ of the published volume, and find that the error of a cultivated state given its decoder gap follows one curve per $d_1$ to within $2\times$. 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 $2\times10^{-3}$.

•Oct 5, 2026•0 citations

Diagnosing and Restoring the Degraded Fault Distance of Magic State Cultivation

Tim Chan, Armands Strikis, Zhu Sun +1 more

T-state cultivation is a resource-efficient protocol producing logical T states but recent benchmarks show that its logical error rate is considerably higher than intended i.e. than S-state cultivation, which is the analogous protocol for producing logical S states. In this paper, we explain this T-S discrepancy by analytically showing, under circuit-level depolarising noise, that distance-3 (-5) T-state cultivation has fault distance 2 (3) due to Pauli hook errors that propagate to coherent Clifford errors after its final double-check circuit; such errors remain Pauli in S-state cultivation, which consequently retains fault distance 3 (5). As part of our analysis we derive a general formula, and an $\mathcal O(n^3)$-time algorithm for fixed logical-qubit count, for the acceptance probability of a logical mixed state afflicted with a Clifford error, where $n$ is the physical qubit count. We then design flags that detect the malignant hook errors in cultivation, improving the pre-escape logical error rate from $Θ(p^3)$ to $Θ(p^5)$. At noise level $p =10^{-3}$, this is a 4.9$\times$ improvement, costing a 1.36$\times$ increase in attempts per accepted shot.

quant-pharXiv:2609.17706•Sep 17, 2026•0 citations

The practical cost of magic state cultivation

Rohan Mehta, Varun Menon, Hengyun Zhou +2 more

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.

quant-pharXiv:2610.06828•Oct 6, 2026•0 citations

Comparing magic state cultivation methods using matrix product states

Tom Hartweg, Asier Piñeiro Orioli

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 $|T\rangle$ output reaches similar logical error rates to the $|S\rangle$ output, traditionally used as a cheap full Clifford proxy. This contrasts with the $\sim10\times$ discrepancy reported for the $d=5$ colour-code scheme of Gidney et al. We also find that our new $d=5$ scheme has $\sim1.3\times$ 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.

quant-pharXiv:2609.19116•Sep 17, 2026•0 citations
Direct Cultivation of Entangled ∣CS⟩|CS\rangle∣CS⟩ Magic States
Magic-state cultivation has so far focused mainly on single-qubit non-Clifford resources. We develop a direct cultivation architecture for the entangled state ∣CS⟩=CS∣++⟩|CS\rangle=CS|++\rangle∣CS⟩=CS∣++⟩. Two commuting Clifford involutions project onto four usable branches related by Pauli-frame updates. A minimal six-bit [6,2,4][6,2,4][6,2,4] record protects the branch label against readout errors that map one valid record to another. Verified CAT7_77​ gadgets, Steane error detection, CZZ-based controlled checks, and immediate Steane-to-surface expansion form the complete factory. The decoder uses 263 operational detector bits, while 168 additional bits are withheld for later validation. Decoding proceeds through exact low-order resolution, a precomputed higher-order catalogue, and four-coset BP+OSD. Under the stated active-location stochastic-Pauli model, exact enumeration finds no accepted closed-boundary logical-failure mechanism through fault order two, while explicit order-three failure mechanisms exist. Finite-ppp simulations quantify acceptance, residual syndromes, and decoder workload, and targeted sampling of order-three faults estimates the leading logical-error channels. Optimizing the direct d=5→13d=5\rightarrow13d=5→13 expansion reduces accepted-output operation count by approximately 35--37%. We compare the two routes at the same binary logical-error rate. Direct CS remains cheaper in operation count at the two lower-noise benchmark points even when the three-TTT route is given pre-existing output patches. The ordering reverses between 8×10−48\times10^{-4}8×10−4 and 9×10−49\times10^{-4}9×10−4. These results show that an entangled non-Clifford state can be cultivated directly with a protected branch record and an explicitly certified fault-order-three output channel.
Magic-state cultivation suppresses errors by measuring logical Clifford checks and discarding inconsistent outcomes. For an entangled resource, several measurement branches are usable, so their outcomes form a multibit classical record whose corruption can produce a logical-frame error. We show that this measurement record can be protected as a binary linear code. For any third-level Clifford-hierarchy unitary U∈C3U\in\mathcal C_3U∈C3​, every parity of the branch bits can be measured by a commuting Hermitian Clifford check C(v)=UX(v)U†C(v)=UX(v)U^\daggerC(v)=UX(v)U†. Choosing which parities to measure therefore defines a binary code y(a)=aGy(a)=aGy(a)=aG. If the valid records have minimum distance ddd, at least ddd readout-bit flips are required to confuse one valid branch with another, giving Pwv=O(qd)P_{\rm wv}=O(q^d)Pwv​=O(qd). A Plotkin bound limits the length of any binary branch record, and the Clifford schedules attain this limit: at distance four, six logical measurements suffice for ∣CS⟩|CS\rangle∣CS⟩ and seven for ∣CCZ⟩|CCZ\rangle∣CCZ⟩, compared with eight and twelve under independent repetition. Thus restricting the logical schedule to Clifford parities requires no additional measurements. In a native-CZZ-assisted Steane implementation, the [6,2,4][6,2,4][6,2,4] CS schedule is also the unique minimum-cost distance-four solution within the compiled Clifford-check family, reducing the cultivation core by 26.6%26.6\%26.6% in active locations. State-vector simulations without a final ideal code-space projection further show higher acceptance and approximately half the residual error weight on the two Steane blocks. Coding the logical measurement record therefore reduces both measurement redundancy and compiled fault-tolerant overhead.
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.
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 TTT-state demand per time step, ρ=nLufT\rho=n_L u f_Tρ=nL​ufT​, which sets the factory area, and the exposure N/uN/uN/u, which sets the idle error, where uuu is the fraction of the nLn_LnL​ logical qubits acted on per step and fTf_TfT​ the TTT fraction. At ρ≈1\rho\approx1ρ≈1 adding cultivation to error mitigation saves 666--9%9\%9% of physical qubits; at ρ≈30\rho\approx30ρ≈30 it saves 717171--85%85\%85% because factories dominate the footprint. The feasible circuit size grows by at most 1.81.81.8--2.7×2.7\times2.7× and never reaches 101010^{10}1010 operations at d≤13d\le13d≤13. We also identify a storage-limited error floor for 15-to-1 distillation confined to patch distance dX≤11d_X\le11dX​≤11 (3.7×10−63.7\times10^{-6}3.7×10−6 at error rate 10−310^{-3}10−3), compute the exact floor of the d1=3d_1=3d1​=3 cultivation stage (6.8×10−76.8\times10^{-7}6.8×10−7), show that escaping to d2=7d_2=7d2​=7--111111 costs 252525--56%56\%56% of the published volume, and find that the error of a cultivated state given its decoder gap follows one curve per d1d_1d1​ to within 2×2\times2×. 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 2×10−32\times10^{-3}2×10−3.
T-state cultivation is a resource-efficient protocol producing logical T states but recent benchmarks show that its logical error rate is considerably higher than intended i.e. than S-state cultivation, which is the analogous protocol for producing logical S states. In this paper, we explain this T-S discrepancy by analytically showing, under circuit-level depolarising noise, that distance-3 (-5) T-state cultivation has fault distance 2 (3) due to Pauli hook errors that propagate to coherent Clifford errors after its final double-check circuit; such errors remain Pauli in S-state cultivation, which consequently retains fault distance 3 (5). As part of our analysis we derive a general formula, and an O(n3)\mathcal O(n^3)O(n3)-time algorithm for fixed logical-qubit count, for the acceptance probability of a logical mixed state afflicted with a Clifford error, where nnn is the physical qubit count. We then design flags that detect the malignant hook errors in cultivation, improving the pre-escape logical error rate from Θ(p3)Θ(p^3)Θ(p3) to Θ(p5)Θ(p^5)Θ(p5). At noise level p=10−3p =10^{-3}p=10−3, this is a 4.9×\times× improvement, costing a 1.36×\times× increase in attempts per accepted shot.
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.