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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• Revised Sep 10, 2026 (v2)
v1 · Sep 9, 2026v2 · Sep 10, 2026
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Abstract

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.

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We introduce a surface-code cultivation protocol for reusable logical catalyst states that implement exact fine dyadic phase gates $Z^{2^{-b}}$ by phase kickback. The catalyst is an eigenstate of a high-period Clifford circuit $U$, with a direct construction supported on $O(2^b)$ logical qubits. Once cultivated, each invocation implements the target phase through a controlled-$U$ gadget, removing Clifford+$T$ synthesis approximation error from the online gate and making the online non-Clifford depth independent of the target logical accuracy. As a concrete demonstration, we construct a catalyst for $\sqrt{T}=Z^{1/8}$, where $U$ is a nine-qubit brickwork Clifford circuit and controlled-$U$ consists of eight controlled-CNOTs. Starting from nine distance-three rotated-surface-code blocks, we cultivate the catalyst through logical-$U$ checks, syndrome extraction and postselection, code growth, and complementary-gap decoding. Due to the intrinsic fault tolerance of the phase read-out, a \emph{single} verification round already reaches the leading error-corrected scaling, in contrast to the repeated logical checks required when cultivating single-qubit magic states. A hybrid tensor-network and stabilizer simulation shows that, at physical error rate $p=10^{-3}$, the postselected catalyst can be grown to distance-seven rotated-surface-code blocks with logical leakage rate $\sim 10^{-6}$ using around seven expected attempts, and can be suppressed further with stronger postselection. Compared with existing protocols, our approach trades offline, phase-specific catalyst cultivation for exactness, reusability, and constant-depth online implementation of fixed fine dyadic phases in codes with restricted transversal gate sets.

quant-pharXiv:2606.27358•Jun 26, 2026•0 citations
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.
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.
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.
We introduce a surface-code cultivation protocol for reusable logical catalyst states that implement exact fine dyadic phase gates Z2−bZ^{2^{-b}}Z2−b by phase kickback. The catalyst is an eigenstate of a high-period Clifford circuit UUU, with a direct construction supported on O(2b)O(2^b)O(2b) logical qubits. Once cultivated, each invocation implements the target phase through a controlled-UUU gadget, removing Clifford+TTT synthesis approximation error from the online gate and making the online non-Clifford depth independent of the target logical accuracy. As a concrete demonstration, we construct a catalyst for T=Z1/8\sqrt{T}=Z^{1/8}T​=Z1/8, where UUU is a nine-qubit brickwork Clifford circuit and controlled-UUU consists of eight controlled-CNOTs. Starting from nine distance-three rotated-surface-code blocks, we cultivate the catalyst through logical-UUU checks, syndrome extraction and postselection, code growth, and complementary-gap decoding. Due to the intrinsic fault tolerance of the phase read-out, a \emph{single} verification round already reaches the leading error-corrected scaling, in contrast to the repeated logical checks required when cultivating single-qubit magic states. A hybrid tensor-network and stabilizer simulation shows that, at physical error rate p=10−3p=10^{-3}p=10−3, the postselected catalyst can be grown to distance-seven rotated-surface-code blocks with logical leakage rate ∼10−6\sim 10^{-6}∼10−6 using around seven expected attempts, and can be suppressed further with stronger postselection. Compared with existing protocols, our approach trades offline, phase-specific catalyst cultivation for exactness, reusability, and constant-depth online implementation of fixed fine dyadic phases in codes with restricted transversal gate sets.