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

arXiv:2605.28936

Hardware-Tailored Resource Estimation for Magic-State Distillation on Silicon Spin Qubits

Songqinghao Yang, Christopher K. Long, Rubén M. Otxoa, Prakash Murali, Crispin H. W. Barnes, David R. M. Arvidsson-Shukur

quant-phSubmitted May 27, 2026• Announced May 29, 2026• 0 citations
arXivHTMLPDF

Abstract

We present a resource analysis for generating high-fidelity logical magic states on silicon spin-qubit platforms. We consider a range of architectures, including a shuttling-based SpinBus design, a dense nearest-neighbor layout, and a hybrid scheme with shuttling-connected patches. We compare surface, color, and biased error-correcting codes, and analyze the $5\to1$ and $15\to1$ magic-state distillation protocols. Our approach combines bottom-up and top-down methodologies. We construct a hardware-level noise model based on a silicon-processor Hamiltonian with realistic parameters and $1/f$ non-Markovian noise, enabling estimation of physical resources required to reach target logical error rates. These results are propagated to system-level overheads for applications including spin dynamics, integer factorization, and quantum chemistry. Conversely, we fix target logical fidelities and derive corresponding constraints on hardware performance. Our framework enables systematic evaluation of resource-reduction strategies. We find that optimized control pulses reduce magic-state distillation overhead by 42\% compared to standard gate implementations. In addition, silicon-tailored biased error-correcting codes achieve an approximately threefold reduction in physical footprint relative to the surface code, even without physical-bias-preserving operations.

Comments
22 pages, 15 figures

Related papers

High-fidelity magic-state preparation with a biased-noise architecture

Shraddha Singh, Andrew S. Darmawan, Benjamin J. Brown +1 more

Magic state distillation is a resource intensive subroutine that consumes noisy input states to produce high-fidelity resource states that are used to perform logical operations in practical quantum-computing architectures. The resource cost of magic state distillation can be reduced by improving the fidelity of the raw input states. To this end, we propose an initialization protocol that offers a quadratic improvement in the error rate of the input magic states in architectures with biased noise. This is achieved by preparing an error-detecting code which detects the dominant errors that occur during state preparation. We obtain this advantage by exploiting the native gate operations of an underlying qubit architecture that experiences biases in its noise profile. We perform simulations to analyze the performance of our protocol with the XZZX surface code. Even at modest physical parameters with a two-qubit gate error rate of $0.7%$ and total probability of dominant errors in the gate $O({10}^{3})$ larger compared to that of nondominant errors, we find that our preparation scheme delivers magic states with logical error rate $O({10}^{\ensuremath{-}8})$ after a single round of the standard 15-to-1 distillation protocol, two orders of magnitude lower than using conventional state preparation. Our approach therefore promises considerable savings in overheads with near-term technology.

arXiv:2109.02677•Physical Review A•May 5, 2022•19 citations

Distilling Magic States in the Bicycle Architecture

Shifan Xu, Kun Liu, Patrick Rall +2 more

Magic State Distillation is considered to be one of the promising methods for supplying the non-Clifford resources required to achieve universal fault tolerance. Conventional MSD protocols implemented in surface codes often require multiple code blocks and lattice surgery rounds, resulting in substantial qubit overhead, especially at low target error rates. In this work, we present practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block. We formulate distillation circuit design as a joint optimization of logical qubit mapping, gate scheduling, measurement nativization, and protocol compression via qubit recycling. Based on detailed resource analysis and simulations, our BB factories have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint, and are particularly compelling as second-round distillers following magic state cultivations.

quant-pharXiv:2602.20546•Feb 25, 2026•0 citations

Zero-level $CCZ$ Distillation

Tomohiro Itogawa, Yutaka Hirano, Yutaro Akahoshi +1 more

Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the $T$ gate and the $CCZ$ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical $CCZ$ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal $T/T^\dagger$ operation of the $[[ 8,3,2 ]]$ code to fault-tolerantly encode the state $\overline{CCZ}|+++\rangle$, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as $p_L \simeq 300 \times p^2$ with respect to the physical error rate $p$. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at $p = 10^{-3}$ and $p = 10^{-4}$, respectively, compared to conventional seven-$T$-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of $CCZ$-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional $T$-state generation.

quant-pharXiv:2605.21867•May 22, 2026•0 citations

Unfolded distillation: very low-cost magic state preparation for biased-noise qubits

Diego Ruiz, Jérémie Guillaud, Christophe Vuillot +1 more

Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of 3 × 10 −7 , using only 53 qubits and 5.5 error correction rounds, under a noise bias of η ≳ 5 × 10 6 and a phase-flip noise rate of 0.1%. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of η ≳ 80, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the X stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as unfolded distillation.

arXiv:2507.12511•npj Quantum Information•Feb 20, 2026•2 citations

Low-Overhead Magic State Distillation with Color Codes

Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio +2 more

Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.

arXiv:2409.07707•PRX Quantum•May 30, 2025•11 citations

The Magic Scroll: Leveraging biased noise to improve magic state cultivation in register-based architectures

Ian D. Thorvaldson, Jeffrey Marshall, Jack R. Craig +3 more

Multiple quantum computing platforms across neutral atoms [1], nitrogen vacancy centres [2, 3], gate-defined dots [4-6] and 14|15 phosphorus atom qubits [7, 8] in silicon are experimentally exploring the use of high connectivity qubits, beyond that of nearest-neighbour planar lattices. Theoretical works consider modifications to fault-tolerant codes to leverage this higher qubit connectivity, such as non-local LDPC codes [9], inspiring superconducting [10] and photonic [11] platforms to also seek non-planar connectivity. In addition, separate theoretical works consider biased noise, where bit- and phase-flip errors are not equally likely. In this work, we present efficient methods for implementing 6.6.6 and 4.8.8 colour codes, as well as bilayer and folded surface codes, using two-qubit registers. We also leverage noise bias to avoid hook errors, demonstrating comparable performance to the surface code. By combining colour and bilayer codes, we show how magic state cultivation procedures [12-14] can be improved, in a procedure we refer to as the Magic Scroll. Not only does the Magic Scroll escape to a standard surface code, it also improves cultivation volumes by 3x and supports magic $|T\rangle$ state fidelities as low as $10^{-9}$. We show that this technique can be further leveraged to improve distillation performance, showing a 3x improvement to distillation volumes for error rates of $\sim 10^{-15}$. Through these constructions, we demonstrate techniques to leverage noise bias and high qubit connectivity, showing how register-based architectures can improve error rates and reduce quantum volumes in fault-tolerant quantum computing.

quant-pharXiv:2608.09018•Aug 11, 2026•0 citations
We present a resource analysis for generating high-fidelity logical magic states on silicon spin-qubit platforms. We consider a range of architectures, including a shuttling-based SpinBus design, a dense nearest-neighbor layout, and a hybrid scheme with shuttling-connected patches. We compare surface, color, and biased error-correcting codes, and analyze the 5→15\to15→1 and 15→115\to115→1 magic-state distillation protocols. Our approach combines bottom-up and top-down methodologies. We construct a hardware-level noise model based on a silicon-processor Hamiltonian with realistic parameters and 1/f1/f1/f non-Markovian noise, enabling estimation of physical resources required to reach target logical error rates. These results are propagated to system-level overheads for applications including spin dynamics, integer factorization, and quantum chemistry. Conversely, we fix target logical fidelities and derive corresponding constraints on hardware performance. Our framework enables systematic evaluation of resource-reduction strategies. We find that optimized control pulses reduce magic-state distillation overhead by 42% compared to standard gate implementations. In addition, silicon-tailored biased error-correcting codes achieve an approximately threefold reduction in physical footprint relative to the surface code, even without physical-bias-preserving operations.
Magic state distillation is a resource intensive subroutine that consumes noisy input states to produce high-fidelity resource states that are used to perform logical operations in practical quantum-computing architectures. The resource cost of magic state distillation can be reduced by improving the fidelity of the raw input states. To this end, we propose an initialization protocol that offers a quadratic improvement in the error rate of the input magic states in architectures with biased noise. This is achieved by preparing an error-detecting code which detects the dominant errors that occur during state preparation. We obtain this advantage by exploiting the native gate operations of an underlying qubit architecture that experiences biases in its noise profile. We perform simulations to analyze the performance of our protocol with the XZZX surface code. Even at modest physical parameters with a two-qubit gate error rate of 0.70.7%0.7 and total probability of dominant errors in the gate O(103)O({10}^{3})O(103) larger compared to that of nondominant errors, we find that our preparation scheme delivers magic states with logical error rate $O({10}^{\ensuremath{-}8})$ after a single round of the standard 15-to-1 distillation protocol, two orders of magnitude lower than using conventional state preparation. Our approach therefore promises considerable savings in overheads with near-term technology.
Magic State Distillation is considered to be one of the promising methods for supplying the non-Clifford resources required to achieve universal fault tolerance. Conventional MSD protocols implemented in surface codes often require multiple code blocks and lattice surgery rounds, resulting in substantial qubit overhead, especially at low target error rates. In this work, we present practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block. We formulate distillation circuit design as a joint optimization of logical qubit mapping, gate scheduling, measurement nativization, and protocol compression via qubit recycling. Based on detailed resource analysis and simulations, our BB factories have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint, and are particularly compelling as second-round distillers following magic state cultivations.
Zero-level CCZCCZCCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the TTT gate and the CCZCCZCCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZCCZCCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T†T/T^\daggerT/T† operation of the [[8,3,2]][[ 8,3,2 ]][[8,3,2]] code to fault-tolerantly encode the state CCZ‾∣+++⟩\overline{CCZ}|+++\rangleCCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2p_L \simeq 300 \times p^2pL​≃300×p2 with respect to the physical error rate ppp. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3p = 10^{-3}p=10−3 and p=10−4p = 10^{-4}p=10−4, respectively, compared to conventional seven-TTT-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZCCZCCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional TTT-state generation.
Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of 3 × 10 −7 , using only 53 qubits and 5.5 error correction rounds, under a noise bias of η ≳ 5 × 10 6 and a phase-flip noise rate of 0.1%. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of η ≳ 80, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the X stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as unfolded distillation.
Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.
Multiple quantum computing platforms across neutral atoms [1], nitrogen vacancy centres [2, 3], gate-defined dots [4-6] and 14|15 phosphorus atom qubits [7, 8] in silicon are experimentally exploring the use of high connectivity qubits, beyond that of nearest-neighbour planar lattices. Theoretical works consider modifications to fault-tolerant codes to leverage this higher qubit connectivity, such as non-local LDPC codes [9], inspiring superconducting [10] and photonic [11] platforms to also seek non-planar connectivity. In addition, separate theoretical works consider biased noise, where bit- and phase-flip errors are not equally likely. In this work, we present efficient methods for implementing 6.6.6 and 4.8.8 colour codes, as well as bilayer and folded surface codes, using two-qubit registers. We also leverage noise bias to avoid hook errors, demonstrating comparable performance to the surface code. By combining colour and bilayer codes, we show how magic state cultivation procedures [12-14] can be improved, in a procedure we refer to as the Magic Scroll. Not only does the Magic Scroll escape to a standard surface code, it also improves cultivation volumes by 3x and supports magic ∣T⟩|T\rangle∣T⟩ state fidelities as low as 10−910^{-9}10−9. We show that this technique can be further leveraged to improve distillation performance, showing a 3x improvement to distillation volumes for error rates of ∼10−15\sim 10^{-15}∼10−15. Through these constructions, we demonstrate techniques to leverage noise bias and high qubit connectivity, showing how register-based architectures can improve error rates and reduce quantum volumes in fault-tolerant quantum computing.