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arXiv:2508.14200

Flag at origin: a modular fault-tolerant preparation for CSS codes

Diego Forlivesi, David Amaro

Published Aug 19, 2025• 0 citations
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Abstract

Fault-tolerant (FT) preparation of diverse logical stabilizer states in quantum error-correcting (QEC) codes is essential for FT computation. Existing constructions of these FT circuits are often constrained by classical computational resources or result in unnecessarily large quantum circuits. This work introduces a modular construction for FT preparation circuits in CSS codes of arbitrary distance, yielding significantly more resource-efficient circuits than previous approaches, especially for the largest codes studied. The key insight is that in bipartite CX circuits used to prepare CSS states, $X$ errors propagate in one direction across the qubit partition, while $Z$ errors propagate in the opposite direction. By appending $X$-detecting flag gadgets to the first partition and $Z$-detecting flag gadgets to the second, the circuit becomes FT. To manage the associated overhead, we propose an algorithm that discovers optimal (or near-optimal) flag gadgets at any distance. These gadgets are reusable across different QEC codes and FT subroutines, such as flag-based QEC. We estimate the logical state preparation error using subset-sampling Monte Carlo simulations at the circuit level, combined with approximate maximum-likelihood look-up table decoding. On Quantinuum's H2-1 device, preparation of the $\lvert\bar{0}\rangle$ state in the [[23,1,7]] Golay code achieves a logical SPAM error rate of $3.3_{-2.4}^{+8.6} \times 10^{-4}$ with an acceptance rate of $47.23(86)\%$. This surpasses (within $95\%$ confidence intervals) the minimum SPAM error rate of $6.0(1.6) \times 10^{-4}$ for a physical $\lvert 0\rangle$, as well as the best previously demonstrated logical state preparations.

DOI
10.48550/arxiv.2508.14200

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Fault-tolerant (FT) preparation of diverse logical stabilizer states in quantum error-correcting (QEC) codes is essential for FT computation. Existing constructions of these FT circuits are often constrained by classical computational resources or result in unnecessarily large quantum circuits. This work introduces a modular construction for FT preparation circuits in CSS codes of arbitrary distance, yielding significantly more resource-efficient circuits than previous approaches, especially for the largest codes studied. The key insight is that in bipartite CX circuits used to prepare CSS states, XXX errors propagate in one direction across the qubit partition, while ZZZ errors propagate in the opposite direction. By appending XXX-detecting flag gadgets to the first partition and ZZZ-detecting flag gadgets to the second, the circuit becomes FT. To manage the associated overhead, we propose an algorithm that discovers optimal (or near-optimal) flag gadgets at any distance. These gadgets are reusable across different QEC codes and FT subroutines, such as flag-based QEC. We estimate the logical state preparation error using subset-sampling Monte Carlo simulations at the circuit level, combined with approximate maximum-likelihood look-up table decoding. On Quantinuum's H2-1 device, preparation of the ∣0ˉ⟩\lvert\bar{0}\rangle∣0ˉ⟩ state in the [[23,1,7]] Golay code achieves a logical SPAM error rate of 3.3−2.4+8.6×10−43.3_{-2.4}^{+8.6} \times 10^{-4}3.3−2.4+8.6​×10−4 with an acceptance rate of 47.23(86)%47.23(86)\%47.23(86)%. This surpasses (within 95%95\%95% confidence intervals) the minimum SPAM error rate of 6.0(1.6)×10−46.0(1.6) \times 10^{-4}6.0(1.6)×10−4 for a physical ∣0⟩\lvert 0\rangle∣0⟩, as well as the best previously demonstrated logical state preparations.
Post-selected stabilizer state preparation is a necessary subroutine in fault-tolerant quantum computation, both for initialization of logical qubits, and for logical-ancilla-based error correction gadgets (e.g. Steane and Knill). Therefore, reducing the number of gates needed to prepare a stabilizer state fault-tolerantly can simultaneously reduce time-to-solution and increase reliability. For small, low-distance codes such as the [[7, 1, 3]] Steane code, circuits with low gate counts can be found by inspection. This becomes impractical for larger codes, necessitating automation. There are two state-of-the-art methods for automated fault-tolerant state preparation, SAT-based stabilizer measurement and flag-at-origin. In this work, we optimize state preparation circuits using the circuit gauge operator formalism to express the construction of flag circuits as an integer linear program. This allows the construction of circuits with equal or lower gate count than the state of the art, while detecting up to three errors. We use this technique to derive a Steane error correction gadget for the [[24, 10, 4]] two-block group algebra code, and test it on Quantinuum's System Model H2 quantum computer with 10,000 shots, resulting in a logical block error rate ~0.00014 (~0.000014 per logical qubit), with ~1.6% of the shots post-selected due to weight-two errors.
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Modularization promises to break down the design and implementation complexity of large scale quantum processors into smaller manageable subtasks. In this approach, quantum channels, realized for instance through physical rerouting of qubits or quantum teleportation, connect multiple modules. Each of those modules hosts a subset of qubits, e.g. multiple logical qubits, and provides quantum operations on them. In this work, we implement for the first time all logical operations required for modular fault-tolerant universal quantum computing with a non-Calderbank-Shor-Steane (CSS) code, the perfect [[5,1,3]][[5, 1, 3]][[5,1,3]] code, on a trapped-ion quantum computer. This code is the smallest quantum error-correcting (QEC) code capable of correcting any single-qubit error, making it a compact alternative to larger CSS codes. We demonstrate logical state teleportation and a full suite of fault-tolerant operations required for universal logical control, including logical state preparation, QEC with real-time feedback, logical measurements, magic-state preparation, logical entangling operations, and magic-state injection. Moreover, we characterize the logical spectator error picked up by idling logical qubits during quantum operations on distinct qubit registers and demonstrate a logical Pauli quantum process tomography that minimizes required sampling resources for logical tomography.
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