Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

Published Oct 5, 2026• 0 citations
Publisher

Abstract

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 $2\times10^{-3}$.

DOI
10.21203/rs.3.rs-11231700/v1

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

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

Price and Payoff: Non-Determinism in Fault Tolerant Quantum Computation

Aditi Awasthi, Sayam Sethi, Sahil Khan +2 more

A promising approach to achieving scalable fault-tolerant quantum computation is the use of quantum error correction (QEC) codes augmented with magic states i.e. resource states produced via distillation, cultivation, or $R_z$ synthesis and teleported into the circuit as needed. Because magic-state production dominates the space-time volume of fault-tolerant programs, system architects must decide how many production units to allocate. Current approaches rely on deterministic analysis that either provisions for worst-case peak demand (wasting valuable qubit resources on factories that are never simultaneously utilized) or assumes average demand, which increases execution time. In this work, we build a simulation framework that couples circuit scheduling with different stochastic magic state production models, and use it to quantify the impact of non-determinism on circuit execution. We show that non-determinism has a dual effect that deterministic models cannot capture: it inflates total execution time (the price), while deflating peak per-cycle resource demand (the payoff). For distillation-based architectures, this demand smoothing shifts the space-time-optimal provisioning point: fewer factories are needed to minimize space-time volume than deterministic analysis predicts. Across benchmarks, stochastic-aware provisioning reduces space-time volume by up to 27% compared to the deterministic optimum for distillation, while requiring up to 30% fewer factories. We characterize these effects across each preparation mechanism, map the resulting design-space tradeoffs, and demonstrate that static resource estimation systematically mis-characterizes the cost of fault-tolerant execution. Our results establish that stochastic-aware analysis is necessary for right-sizing the factory allocations and should replace deterministic heuristics as the standard methodology for FTQC resource planning.

quant-pharXiv:2605.07983•May 11, 2026•0 citations

Magic state cultivation: growing T states as cheap as CNOT gates

Craig Gidney, Noah Shutty, Cody Jones

We refine ideas from Knill 1996, Jones 2016, Chamberland 2020, Gidney 2023+2024, Bombin 2024, and Hirano 2024 to efficiently prepare good $|T\rangle$ states. We call our construction "magic state cultivation" because it gradually grows the size and reliability of one state. Cultivation fits inside a surface code patch and uses roughly the same number of physical gates as a lattice surgery CNOT gate of equivalent reliability. We estimate the infidelity of cultivation (from injection to idling at distance 15) using a mix of state vector simulation, stabilizer simulation, error enumeration, and Monte Carlo sampling. Compared to prior work, cultivation uses an order of magnitude fewer qubit-rounds to reach logical error rates as low as $2 \cdot 10^{-9}$ when subjected to $10^{-3}$ uniform depolarizing circuit noise. Halving the circuit noise to $5 \cdot 10^{-4}$ improves the achievable logical error rate to $4 \cdot 10^{-11}$. Cultivation's efficiency and strong response to improvements in physical noise suggest that further magic state distillation may never be needed in practice.

arXiv:2409.17595•Sep 26, 2024•9 citations

STAR-Magic Mutation: Even More Efficient Analog Rotation Gates for Early Fault-Tolerant Quantum Computer

Riki Toshio, Shota Kanasugi, Jun Fujisaki +3 more

We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of $\mathcal{O}(θ_L^{2(1-Θ(1/d))}p_{\text{ph}})$, while requiring only the ancillary space of a single surface code patch. Here, $θ_L$ is the logical rotation angle, $p_{\text{ph}}$ is the physical error rate, and $d$ is the code distance. This scaling marks a significant improvement over the previous state-of-the-art, $\mathcal{O}(θ_L p_{\text{ph}})$, making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for $θ_L \lesssim 10^{-5}$, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard $T$-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+$T$+$φ$ gate set, rather than the conventional Clifford+$T$ or Clifford+$φ$ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of $p_{\text{ph}}=10^{-3}$.

quant-pharXiv:2603.22891•Mar 25, 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
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.
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.
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.
A promising approach to achieving scalable fault-tolerant quantum computation is the use of quantum error correction (QEC) codes augmented with magic states i.e. resource states produced via distillation, cultivation, or RzR_zRz​ synthesis and teleported into the circuit as needed. Because magic-state production dominates the space-time volume of fault-tolerant programs, system architects must decide how many production units to allocate. Current approaches rely on deterministic analysis that either provisions for worst-case peak demand (wasting valuable qubit resources on factories that are never simultaneously utilized) or assumes average demand, which increases execution time. In this work, we build a simulation framework that couples circuit scheduling with different stochastic magic state production models, and use it to quantify the impact of non-determinism on circuit execution. We show that non-determinism has a dual effect that deterministic models cannot capture: it inflates total execution time (the price), while deflating peak per-cycle resource demand (the payoff). For distillation-based architectures, this demand smoothing shifts the space-time-optimal provisioning point: fewer factories are needed to minimize space-time volume than deterministic analysis predicts. Across benchmarks, stochastic-aware provisioning reduces space-time volume by up to 27% compared to the deterministic optimum for distillation, while requiring up to 30% fewer factories. We characterize these effects across each preparation mechanism, map the resulting design-space tradeoffs, and demonstrate that static resource estimation systematically mis-characterizes the cost of fault-tolerant execution. Our results establish that stochastic-aware analysis is necessary for right-sizing the factory allocations and should replace deterministic heuristics as the standard methodology for FTQC resource planning.
We refine ideas from Knill 1996, Jones 2016, Chamberland 2020, Gidney 2023+2024, Bombin 2024, and Hirano 2024 to efficiently prepare good ∣T⟩|T\rangle∣T⟩ states. We call our construction "magic state cultivation" because it gradually grows the size and reliability of one state. Cultivation fits inside a surface code patch and uses roughly the same number of physical gates as a lattice surgery CNOT gate of equivalent reliability. We estimate the infidelity of cultivation (from injection to idling at distance 15) using a mix of state vector simulation, stabilizer simulation, error enumeration, and Monte Carlo sampling. Compared to prior work, cultivation uses an order of magnitude fewer qubit-rounds to reach logical error rates as low as 2⋅10−92 \cdot 10^{-9}2⋅10−9 when subjected to 10−310^{-3}10−3 uniform depolarizing circuit noise. Halving the circuit noise to 5⋅10−45 \cdot 10^{-4}5⋅10−4 improves the achievable logical error rate to 4⋅10−114 \cdot 10^{-11}4⋅10−11. Cultivation's efficiency and strong response to improvements in physical noise suggest that further magic state distillation may never be needed in practice.
We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of O(θL2(1−Θ(1/d))pph)\mathcal{O}(θ_L^{2(1-Θ(1/d))}p_{\text{ph}})O(θL2(1−Θ(1/d))​pph​), while requiring only the ancillary space of a single surface code patch. Here, θLθ_LθL​ is the logical rotation angle, pphp_{\text{ph}}pph​ is the physical error rate, and ddd is the code distance. This scaling marks a significant improvement over the previous state-of-the-art, O(θLpph)\mathcal{O}(θ_L p_{\text{ph}})O(θL​pph​), making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for θL≲10−5θ_L \lesssim 10^{-5}θL​≲10−5, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard TTT-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+TTT+φφφ gate set, rather than the conventional Clifford+TTT or Clifford+φφφ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of pph=10−3p_{\text{ph}}=10^{-3}pph​=10−3.
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.