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arXiv:2610.32267

Two-Layer Verification for Fault-Tolerant Magic-State Preparation in High-Rate Quantum Codes

Naoyuki Kanomata, Hayato Goto

quant-phSubmitted Oct 23, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

Encoded magic states are essential for universal fault-tolerant quantum computation, but their resource-efficient preparation in high-rate quantum codes remains challenging. To address this limitation, we propose two-layer verification, which combines local verification of independently prepared subblocks with logical verification after their embedding into a target code. This hierarchy detects harmful error patterns, including even-parity logical-eigenvalue flips that are invisible to a collective product check alone. We demonstrate this scheme for a $[[21,3,5]]$ code by constructing a circuit that embeds three verified seven-qubit Steane-code magic states into the target code. Circuit-level simulations of $T$- and $S$-state preparation show logical-error suppression consistent with the code distance. These results demonstrate the potential of two-layer verification as a design principle for reliable magic-state preparation in quantum codes encoding multiple logical qubits.

AI Summary

Preparing multiple logical magic states in a high-rate code using a transversal product check leaves an ambiguity: an even number of logical verification-eigenvalue flips passes the check. This paper investigates whether local verification before embedding, followed by collective verification and error detection, can suppress these failures. The construction decomposes an encoder for the self-dual CSS $[[21,3,5]]$ code into three independently prepared and verified $[[7,1,3]]$ Steane-code states, a qubit permutation, and a residual entangling circuit. After embedding, a product of logical verification operators is measured, followed by syndrome-based error detection implemented through error-detecting teleportation (EDT). Circuit-level Monte Carlo simulations find approximately cubic conditional logical-error scaling, $p_L\propto p_{\mathrm{circ}}^\alpha$: $\alpha=2.93(2)$ for $T$ states with ideal final error detection, $3.04(2)$ for $S$ states with ideal final detection, and $3.16(1)$ for $S$ states with noisy EDT. These provide numerical evidence of distance-5-compatible suppression. The Clifford-only $S$ circuits serve as proxies for the more expensive non-Clifford simulations. At $p_{\mathrm{circ}}\simeq0.1\%$, the reported average physical-qubit requirement for EDT-based $S$ preparation is below that of the studied fault-tolerant logical-CNOT implementation. The noise model includes preparation, measurement, and CNOT faults but neglects single-qubit-gate and memory errors. Fully noisy EDT-based $T$ preparation is inferred from the proxy results rather than directly simulated. The construction relies on a specific Steane-subblock decomposition; extending it to other high-rate codes requires finding analogous decompositions and embedding circuits.

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Error found: The abstract does not disclose that the T-state simulations use ideal final error detection, whereas noisy final error detection is tested only with the Clifford S-state proxy. The numerical evidence assumes error-free single-qubit gates and memory, so it does not test robustness to faults on the non-Clifford verification gates. The decoder appendix incorrectly states that eight syndrome bits and 256 table entries exhaust the syndromes: the supplied parity-check matrix has nine independent rows, giving 512 possible single-basis syndromes.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Comments
13 pages, 13 figures

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Encoded magic states are essential for universal fault-tolerant quantum computation, but their resource-efficient preparation in high-rate quantum codes remains challenging. To address this limitation, we propose two-layer verification, which combines local verification of independently prepared subblocks with logical verification after their embedding into a target code. This hierarchy detects harmful error patterns, including even-parity logical-eigenvalue flips that are invisible to a collective product check alone. We demonstrate this scheme for a [[21,3,5]][[21,3,5]][[21,3,5]] code by constructing a circuit that embeds three verified seven-qubit Steane-code magic states into the target code. Circuit-level simulations of TTT- and SSS-state preparation show logical-error suppression consistent with the code distance. These results demonstrate the potential of two-layer verification as a design principle for reliable magic-state preparation in quantum codes encoding multiple logical qubits.
Preparing multiple logical magic states in a high-rate code using a transversal product check leaves an ambiguity: an even number of logical verification-eigenvalue flips passes the check. This paper investigates whether local verification before embedding, followed by collective verification and error detection, can suppress these failures.
The construction decomposes an encoder for the self-dual CSS [[21,3,5]][[21,3,5]][[21,3,5]] code into three independently prepared and verified [[7,1,3]][[7,1,3]][[7,1,3]] Steane-code states, a qubit permutation, and a residual entangling circuit. After embedding, a product of logical verification operators is measured, followed by syndrome-based error detection implemented through error-detecting teleportation (EDT).
Circuit-level Monte Carlo simulations find approximately cubic conditional logical-error scaling, pL∝pcircαp_L\propto p_{\mathrm{circ}}^\alphapL​∝pcircα​: α=2.93(2)\alpha=2.93(2)α=2.93(2) for TTT states with ideal final error detection, 3.04(2)3.04(2)3.04(2) for SSS states with ideal final detection, and 3.16(1)3.16(1)3.16(1) for SSS states with noisy EDT. These provide numerical evidence of distance-5-compatible suppression. The Clifford-only SSS circuits serve as proxies for the more expensive non-Clifford simulations. At pcirc≃0.1%p_{\mathrm{circ}}\simeq0.1\%pcirc​≃0.1%, the reported average physical-qubit requirement for EDT-based SSS preparation is below that of the studied fault-tolerant logical-CNOT implementation.
The noise model includes preparation, measurement, and CNOT faults but neglects single-qubit-gate and memory errors. Fully noisy EDT-based TTT preparation is inferred from the proxy results rather than directly simulated. The construction relies on a specific Steane-subblock decomposition; extending it to other high-rate codes requires finding analogous decompositions and embedding circuits.
Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with 82.58% probability, with an infidelity of at most 5.1 ( 2.7 ) × 10 − 4 . The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least 2.7, indicating the quantum processor is below the pseudothreshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.
The seven-qubit quantum error-correcting code originally proposed by Steane is one of the best known quantum codes. The Steane code has a desirable property that most basic operations can be performed easily in a fault-tolerant manner. A major obstacle to fault-tolerant quantum computation with the Steane code is fault-tolerant preparation of encoded states, which requires large computational resources. Here we propose efficient state preparation methods for zero and magic states encoded with the Steane code, where the zero state is one of the computational basis states and the magic state allows us to achieve universality in fault-tolerant quantum computation. The methods minimize resource overheads for the fault-tolerant state preparation, and therefore reduce necessary resources for quantum computation with the Steane code. Thus, the present results will open a new possibility for efficient fault-tolerant quantum computation.
As quantum hardware enters the fault-tolerant regime, algorithms are beginning to be executed on encoded qubits, yet scalable certification of how well they run remains an open problem. Here we introduce magic-free certification, a certification protocol for universal fault-tolerant circuits that consumes no magic, the resource that generally places them beyond efficient classical simulation and is by far the most expensive element of a fault-tolerant computation. The protocol is code- and decoder-agnostic, runs interleaved with the target circuit on the same device, and has a sample complexity that grows linearly with the number of logical qubits and non-Clifford gates while remaining decoupled from the logical error rate itself. Given a predefined error tolerance pL∗p_L^*pL∗​, the protocol determines at a chosen confidence level whether the device executes the target circuit at a logical error rate below pL∗p_L^*pL∗​. The method relies on trap circuits, copies of the target with every magic state replaced by a stabilizer state, whose outcomes are deterministic and which, being indistinguishable from the target to stochastic Pauli noise, fail at the same rate. We illustrate the protocol in Steane-code simulations of a magic state injection gadget and a quantum Fourier transform.
The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance quantum computing requires error-corrected nonClifford operations, in addition to Clifford gates, and for the former, it is important to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavyhexagonal surface code for IBM quantum processors (ibm_fez) and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of≈0.37%\approx 0.37 \%≈0.37%and≈0.31%\approx 0.31 \%≈0.31%, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states∣HL⟩\vert H_{L}\rangle∣HL​⟩and∣TL⟩\vert T_{L}\rangle∣TL​⟩achieves fidelities of0.8806±0.00020.8806 \pm 0.00020.8806±0.0002and0.8665±0.00030.8665 \pm 0.00030.8665±0.0003, respectively, which are both above the magic state distillation threshold. Our work demonstrates the potential for realising nonClifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
Magic states are essential yet resource-intensive components for realizing universal fault-tolerant quantum computation. Preparing magic states within emerging quantum low-density parity-check (qLDPC) codes poses additional challenges, due to the complex encoding structures. Here, we introduce a generic and scalable method for magic state injection into arbitrarily selected logical qubits encoded using qLDPC codes. Our approach, based on parallelized code surgery, supports the injection from either physical qubits or low-distance logical qubits. For qLDPC code families with asymptotically constant encoding rates, the method achieves injection into Θ(k)Θ(k)Θ(k) logical qubits -- where kkk denotes the logical qubit number of the code -- with only constant qubit overhead and a time complexity of O~(d2)\tilde{O}(d^2)O~(d2), where ddd is the code distance. A central contribution of this work is a rigorous proof that errors affecting the injected magic states remain independent throughout the procedure. This independence ensures the resilience of logical qubits against interactions with noisy ancillae and preserves the presumption of subsequent magic state distillation protocols. We further support our theoretical results with numerical validation through circuit-level simulations. These findings advance the feasibility of scalable, fault-tolerant universal quantum computing using qLDPC codes, offering a pathway to significantly reduced qubit resource requirements in magic state injection.