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arXiv:2608.09018

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

Ian D. Thorvaldson, Jeffrey Marshall, Jack R. Craig, Samuel K. Gorman, Charles D. Hill, Michelle Y. Simmons

quant-phSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

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.

Comments
Main text: 22 pages, 10 figures, 2 tables. Supplementary: 24 pages, 24 figures, 1 table

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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.
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Magic state cultivation is a leading approach for generating the resource states required for fault-tolerant quantum computation. Here we present a cultivation protocol that increases the success probability of magic state generation in platforms with flexible and nonlocal connectivity. Our method implements cultivation directly on the surface code, avoiding the detour through alternative, less efficient error-correcting codes used in prior approaches. This both improves the acceptance rate and preserves compatibility with the geometry of the code and the hardware. Numerical simulations show that our protocols improve success probabilities and reduce output error rates compared with protocols tailored to locally connected platforms. Under realistic noise models for cold-atom and trapped-ion systems, we improve the rate of magic state generation by more than a factor of 20. Finally, we study qubit loss and erasure and show that very low error rates can be achieved with minimal overhead, reaching below 10 − 6 infidelity using only nine physical erasure qubits.
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.
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.
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.