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arXiv:2605.19471

Efficient Fault-Tolerant Ancilla Preparation for Quantum BCH codes via Cyclic Symmetry

Kohei Yamamoto, Keisuke Fujii

quant-phSubmitted May 19, 2026• Announced May 20, 2026• 0 citations
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Abstract

One of the major challenges in realizing fault-tolerant quantum computers (FTQCs) is the requirement for a large number of physical qubits. To address this issue, high-rate quantum error correcting codes, which efficiently embed logical qubits into physical qubits, have recently attracted considerable attention. Among such codes, quantum BCH codes, which offer both high rates and large code distances, are promising yet underexplored candidates. However, no fault-tolerant ancilla preparation method specialized for this class had been established. We employ a two-stage approach (non-fault-tolerant preparation + entanglement distillation) for ancilla preparation. We then propose a framework for designing low-overhead distillation method that strategically leverages the cyclic symmetry of quantum BCH codes to determine which non-fault-tolerant circuits can successfully produce a fault-tolerant state. Numerical simulations on several high-performance quantum BCH codes up to 127 qubits demonstrate that our method achieves lower spatial overhead and logical error rates than conventional distillation circuits. Furthermore, we evaluated the logical error rates under a circuit-level noise model, and obtained performance benchmarks in realistic settings. This efficient state preparation technique is expected to contribute to the early realization of practical FTQCs, particularly on highly connected quantum platforms such as neutral atom systems.

Comments
19 pages, 15 figures

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One of the major challenges in realizing fault-tolerant quantum computers (FTQCs) is the requirement for a large number of physical qubits. To address this issue, high-rate quantum error correcting codes, which efficiently embed logical qubits into physical qubits, have recently attracted considerable attention. Among such codes, quantum BCH codes, which offer both high rates and large code distances, are promising yet underexplored candidates. However, no fault-tolerant ancilla preparation method specialized for this class had been established. We employ a two-stage approach (non-fault-tolerant preparation + entanglement distillation) for ancilla preparation. We then propose a framework for designing low-overhead distillation method that strategically leverages the cyclic symmetry of quantum BCH codes to determine which non-fault-tolerant circuits can successfully produce a fault-tolerant state. Numerical simulations on several high-performance quantum BCH codes up to 127 qubits demonstrate that our method achieves lower spatial overhead and logical error rates than conventional distillation circuits. Furthermore, we evaluated the logical error rates under a circuit-level noise model, and obtained performance benchmarks in realistic settings. This efficient state preparation technique is expected to contribute to the early realization of practical FTQCs, particularly on highly connected quantum platforms such as neutral atom systems.
Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.
Fault-tolerant quantum computation (FTQC) schemes that use multiqubit large block codes can potentially reduce the resource overhead to a great extent. A major obstacle is the requirement for a large number of clean ancilla states of different types without correlated errors inside each block. These ancilla states are usually logical stabilizer states of the data-code blocks, which are generally difficult to prepare if the code size is large. Previously, we have proposed an ancilla distillation protocol for Calderbank-Shor-Steane (CSS) codes by classical error-correcting codes. It was assumed that the quantum gates in the distillation circuit were perfect; however, in reality, noisy quantum gates may introduce correlated errors that are not treatable by the protocol. In this paper, we show that additional postselection by another classical error-detecting code can be applied to remove almost all correlated errors. Consequently, the revised protocol is fully fault tolerant and capable of preparing a large set of stabilizer states sufficient for FTQC using large block codes. At the same time, the yield rate can be boosted from $O({t}^{\ensuremath{-}2})$ to O(1)O(1)O(1) in practice for an [[n,k,d=2t+1]][[n,k,d=2t+1]][[n,k,d=2t+1]] CSS code. Ancilla preparation for the [[23,1,7]] quantum Golay code is numerically studied in detail through Monte Carlo simulation. The results support the validity of the protocol when the gate failure rate is reasonably low. To the best of our knowledge, this approach is one of the first attempts to prepare general large block stabilizer states free of correlated errors for FTQC in a fault-tolerant and efficient manner.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [ ⁣[7,1,3] ⁣][\![7,1,3]\!][[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors - a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×\times× compared to standard approaches. Our implementations achieve logical error rates as low as 5.1×10−55.1 \times 10^{-5}5.1×10−5 under realistic noise conditions with physical error rates of 10−310^{-3}10−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
Abstract Fault-tolerant quantum computing promises significant computational speedup over classical computing for a variety of important problems. One of the biggest challenges for realizing fault-tolerant quantum computing is preparing magic states with sufficiently low error rates. Magic state distillation is one of the most efficient schemes for preparing high-quality magic states. However, since magic state distillation circuits are not fault-tolerant, all the operations in the distillation circuits must be encoded in a large distance error-correcting code, resulting in a significant resource overhead. Here, we propose a fault-tolerant scheme for directly preparing high-quality magic states, which makes magic state distillation unnecessary. In particular, we introduce a concept that we call redundant ancilla encoding. The latter combined with flag qubits allows for circuits to both measure stabilizer generators of some code, while also being able to measure global operators to fault-tolerantly prepare magic states, all using nearest neighbor interactions. We apply such schemes to a planar architecture of the triangular color code family and demonstrate that our scheme requires at least an order of magnitude fewer qubits and space–time overhead compared to the most competitive magic state distillation schemes. Since our scheme requires only nearest-neighbor interactions in a planar architecture, it is suitable for various quantum computing platforms currently under development.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors—a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we’re trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor’s cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane’s encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM’s Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×compared to standard approaches. Our implementations achieve logical error rates as low as 5.1 ×10^−5 under realistic noise conditions with physical error rates of 10^−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
Abstract Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [7,1,3] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors—a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4x compared to standard approaches. Our implementations achieve logical error rates as low as 5.1 x 10 -5 under realistic noise conditions with physical error rates of 10 -3 , while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.