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arXiv:2609.09994

Coded Clifford Measurements for Multiqubit Magic-State Cultivation

Gunsik Min, Jun Heo

quant-phSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
v1 · Sep 9, 2026v2 · Sep 10, 2026
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Abstract

Magic-state cultivation suppresses errors by measuring logical Clifford symmetries and discarding inconsistent outcomes. For an entangled resource several branches are accepted, so the retained branch carries a multibit classical record whose corruption produces a logical-frame error. We show that this record layer is exactly a binary linear code. For any third-level Clifford-hierarchy unitary $U \in \mathcal{C}_3$, the branches $UZ(a)|+\rangle^{\otimes k}$ are resolved by commuting Hermitian Clifford observables $C(v) = UX(v)U^{\dagger}$, so a measurement schedule is a generator matrix and postselection on valid records gives $P_{\mathrm{wv}} = O(q^{d})$ at record distance $d$. A Plotkin bound then limits every binary record, linear or not. The physically realizable Clifford family already saturates it: at distance four, six logical measurements suffice for $|CS\rangle$ and seven for $|CCZ\rangle$, against eight and twelve for independent repetition. Restricting to fault-tolerantly measurable parities therefore costs nothing in record length. The saving survives compilation: in a native-CZZ-assisted Steane realization the $[6,2,4]$ CS schedule is the unique minimum-cost distance-four solution, reducing the cultivation core by 26.6\% in active locations, and exact state-vector simulation without a final ideal code-space projection confirms higher acceptance and roughly half the residual Steane-boundary weight. Coding thus reduces logical redundancy and compiled overhead together.

Related papers

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

Gunsik Min, Jun Heo

Magic-state cultivation has largely targeted single-qubit non-Clifford resources. We show that the entangled state $|CS\rangle=CS|++\rangle$ 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 $p=5\times10^{-4}$ and $16$--$28\%$ more at $8\times10^{-4}$--$10^{-3}$, 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 $5\times10^{-4}$ 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.

quant-pharXiv:2609.09993•Sep 10, 2026•0 citations

Constant-depth magic state cultivation with Clifford measurements by gauging

Bence Hetényi, Benjamin J. Brown, Dominic J. Williamson

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 $\sim 10^{-9}$ logical error rates for the $d=5$ color code, with substantially lower space-time overhead than magic state distillation. However, due to the $\mathcal{O}(d)$ depth of the Clifford measurement circuit, magic state cultivation becomes impractical for $d>5$. Here, we perform logical $XS^\dagger$ 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.

quant-pharXiv:2603.05429•Physical Review Letters•Mar 6, 2026•0 citations

Diagnosing and Restoring the Degraded Fault Distance of Magic State Cultivation

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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

Magic Quantum Code Surgery

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Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.

quant-pharXiv:2610.03920•Oct 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

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
Magic-state cultivation suppresses errors by measuring logical Clifford symmetries and discarding inconsistent outcomes. For an entangled resource several branches are accepted, so the retained branch carries a multibit classical record whose corruption produces a logical-frame error. We show that this record layer is exactly a binary linear code. For any third-level Clifford-hierarchy unitary U∈C3U \in \mathcal{C}_3U∈C3​, the branches UZ(a)∣+⟩⊗kUZ(a)|+\rangle^{\otimes k}UZ(a)∣+⟩⊗k are resolved by commuting Hermitian Clifford observables C(v)=UX(v)U†C(v) = UX(v)U^{\dagger}C(v)=UX(v)U†, so a measurement schedule is a generator matrix and postselection on valid records gives Pwv=O(qd)P_{\mathrm{wv}} = O(q^{d})Pwv​=O(qd) at record distance ddd. A Plotkin bound then limits every binary record, linear or not. The physically realizable Clifford family already saturates it: at distance four, six logical measurements suffice for ∣CS⟩|CS\rangle∣CS⟩ and seven for ∣CCZ⟩|CCZ\rangle∣CCZ⟩, against eight and twelve for independent repetition. Restricting to fault-tolerantly measurable parities therefore costs nothing in record length. The saving survives compilation: in a native-CZZ-assisted Steane realization the [6,2,4][6,2,4][6,2,4] CS schedule is the unique minimum-cost distance-four solution, reducing the cultivation core by 26.6% in active locations, and exact state-vector simulation without a final ideal code-space projection confirms higher acceptance and roughly half the residual Steane-boundary weight. Coding thus reduces logical redundancy and compiled overhead together.
Direct Cultivation of Entangled ∣CS⟩|CS\rangle∣CS⟩ Magic States
Magic-state cultivation has largely targeted single-qubit non-Clifford resources. We show that the entangled state ∣CS⟩=CS∣++⟩|CS\rangle=CS|++\rangle∣CS⟩=CS∣++⟩ can be cultivated directly rather than synthesized from three TTT-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][6,2,4][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-TTT comparator matched to the same binary logical-error target, the direct route uses 2%2\%2% fewer active locations at p=5×10−4p=5\times10^{-4}p=5×10−4 and 161616--28%28\%28% more at 8×10−48\times10^{-4}8×10−4--10−310^{-3}10−3, without charging the comparator for TTT-state storage. Counted in qubit-ticks under one rule for both routes, it uses 212121--39%39\%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%8\%8% and 46%46\%46% above the comparator at 5×10−45\times10^{-4}5×10−4 and 10−310^{-3}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 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\sim 10^{-9}∼10−9 logical error rates for the d=5d=5d=5 color code, with substantially lower space-time overhead than magic state distillation. However, due to the O(d)\mathcal{O}(d)O(d) depth of the Clifford measurement circuit, magic state cultivation becomes impractical for d>5d>5d>5. Here, we perform logical XS†XS^\daggerXS† 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=7d=7d=7 version of our protocol gives access to the 10−1210^{-12}10−12 logical error rate regime at 0.05%0.05\%0.05% physical error rate while retaining more than 1%1\%1% of the shots after the equivalent of the cultivation stage.
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.
Non-Clifford operations required for universal quantum computation can be implemented by measuring logical Clifford operators. We present a generalized framework to fault-tolerantly measure transversal Clifford logical operators in quantum low-density parity check (QLDPC) codes. Our protocol takes any QLDPC code and deforms the code across additional ancillary qubits such that a product of stabilizers measures the desired logical Clifford operator. We prove that the deformed, Clifford-stabilized code remains LDPC and that its distance is preserved up to a constant factor independent of the number of physical qubits. We further prove that the Clifford measurement has a fault distance against data qubit and measurement errors that grows linearly with the distance of the original code. We apply our methods to implement logical non-Clifford gates in algorithmically relevant subroutines and to prepare magic states in quantum stabilizer codes.
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.
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.