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KSII Transactions on Internet and Information Systems

Parallel Low-Depth Flagged Syndrome Extraction for Quantum Stabilizer Codes

Yunxing Zhang, Lan Luo, Chaofeng Guan

Published May 31, 2026• 0 citations
Publisher

Abstract

In this work, we propose a parallel low-depth flagged syndrome extraction protocol for faulttolerant quantum error correction.The protocol combines parallel measurement of different stabilizers with dynamic selection of a minimal stabilizer set, guided by flag outcomes.Circuit-level optimizations, including CNOT reordering, are integrated while rigorously preserving fault-tolerance guarantees.Implementations for several distance-3 and distance-5 codes achieve substantial reductions in circuit depth and idle locations.Numerical simulations under a circuit-level depolarizing noise model demonstrate consistent pseudo-threshold improvements, such as 12.56% and 4.43% over prior schemes for [[5, 1, 3]], 16.2% for [[7, 1, 3]], and 14.55% to 30.13% for distance-5 codes, confirming the effectiveness of the cooptimized approach.This co-optimized framework offers a practical pathway towards more resource-efficient fault-tolerant quantum computation.

DOI
10.3837/tiis.2026.05.025

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In this work, we propose a parallel low-depth flagged syndrome extraction protocol for faulttolerant quantum error correction.The protocol combines parallel measurement of different stabilizers with dynamic selection of a minimal stabilizer set, guided by flag outcomes.Circuit-level optimizations, including CNOT reordering, are integrated while rigorously preserving fault-tolerance guarantees.Implementations for several distance-3 and distance-5 codes achieve substantial reductions in circuit depth and idle locations.Numerical simulations under a circuit-level depolarizing noise model demonstrate consistent pseudo-threshold improvements, such as 12.56% and 4.43% over prior schemes for [[5, 1, 3]], 16.2% for [[7, 1, 3]], and 14.55% to 30.13% for distance-5 codes, confirming the effectiveness of the cooptimized approach.This co-optimized framework offers a practical pathway towards more resource-efficient fault-tolerant quantum computation.
Given that quantum error correction processes are unreliable, an efficient error syndrome extraction circuit should use fewer ancillary qubits, quantum gates, and measurements, while maintaining low circuit depth, to minimizing the circuit area, roughly defined as the product of circuit depth and the number of physical qubits. We propose to design parallel flagged syndrome extraction with shared flag qubits for quantum stabilizer codes. Versatile parallelization techniques are employed to minimize the required circuit area, thereby improving the error threshold and overall performance. Specifically, measurement outcomes across multiple rounds of syndrome extraction are integrated into a lookup table decoder, enabling parallelization of multiple stabilizer measurements with shared flag qubits. Additionally, we introduce an adaptive technique to reduce the overhead from excessive syndrome extraction. We present flag-sharing and fully parallel schemes for the[ ⁣[17,1,5] ⁣][\![17,1,5]\!][[17,1,5]],[ ⁣[19,1,5] ⁣][\![19,1,5]\!][[19,1,5]]Calderbank-Shor-Steane (CSS) codes and the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]non-CSS code, where the[ ⁣[5,1,3] ⁣][\![5,1,3]\!][[5,1,3]]implementation achieves the minimum known circuit area. Numerical simulations have demonstrated improved pseudo-thresholds for these codes by up to an order of magnitude compared to previous schemes in the literature.
Flag-style fault-tolerance has become a linchpin in the realization of small fault-tolerant quantum-error correction experiments. The flag protocol's utility hinges on low qubit over-head, which is typically much smaller than in other approaches. However, as in most fault-tolerance protocols, the advantages of flag-style error correction come with a tradeoff: fault tolerance can be guaranteed, but such protocols involve high-depth circuits, due to the need for repeated stabilizer measurements. Here, we demonstrate that a dynamic choice of stabilizer measurements, based on past syndromes, and the utilization of elements from the full stabilizer group, leads to flag protocols with lower-depth syndrome-extraction circuits for the [[5], [1], [3]] code, as well as for the Steane code when compared to the standard methods in flag fault tolerance. We methodically prove that our new protocols yield fault-tolerant lookup tables, and demonstrate them with a pseudothreshold simulation, showcasing large improvements for all protocols when compared to previously-established methods. This work opens the dialogue on exploiting the properties of the full stabilizer group for reducing circuit overhead in fault-tolerant quantum-error correction.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4)210.2p^3+O(p^4)210.2p3+O(p4). At p=10−3p=10^{-3}p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−72.2\times10^{-7}2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.
Abstract Flagged fault‐tolerant error correction schemes have attracted a lot of attention for their low qubit overhead, but usually require a high circuit depth. One way to reduce the depth is to develop parallel flagged syndrome extraction circuits. This study investigates parallel syndrome extraction circuits for Calderbank‐Shor‐Steane codes with distance 5. A simple algorithm for finding parallel flagged syndrome extraction circuits is provided. Using this algorithm, the parallel flagged circuits of the [[17,1,5]] and [[19,1,5]] color codes are constructed precisely. The error pseudo‐thresholds for code schemes are also precisely simulated and calculated. The parallel scheme in this study, based on the [[17,1,5]] and [[19,1,5]] color codes, outperforms traditional flagged schemes, particularly when the idle location error rate is high.
To perform achieve fault-tolerant quantum computation, one can use flagged syndrome extraction with fewer ancilla qubits. However, it suffers from long circuit depth if one stabilizer is measured at a time. Previously, Reichardt showed that it is possible to measure multiple stabilizers with at most one shared flag qubit for certain small quantum codes. In this paper, we propose a procedure for general Calderbank-Shor-Steane codes of distance three so that multiple ZZZ stabilizers (XXX stabilizers) can be fault tolerantly measured in parallel with one shared flag qubit. We simulate the memory and computation pseudothresholds for various code schemes. In particular, our parallel scheme based on Shor's nine-qubit code performs better than known seven- and nine-qubit schemes in the literature.
Fault-tolerant syndrome extraction is a key ingredient in implementing fault-tolerant quantum computation. While conventional methods use a number of extra qubits that are linear in the weight of the syndrome, several improvements have been introduced using flag gadgets. In this work, we develop a framework to design flag gadgets using classical codes. Using this framework, we show how to perform fault-tolerant syndrome extraction for any stabilizer code with arbitrary distance using exponentially fewer qubits than conventional methods when qubit measurement and reset are relatively slow compared to a round of error correction. In particular, our method requires only ( 2 t + 1 ) t ⌈ log 2  ( w ) ⌉ flag qubits to fault-tolerantly measure a weight- w stabilizer. We further take advantage of the saving provided by our construction to fault-tolerantly measure multiple stabilizers using a single gadget and show that it maintains the same exponential advantage when it is used to fault-tolerantly extract the syndromes of quantum low-density parity-check codes. Using the developed framework, we perform computer-assisted search to find several small examples where our constructions reduce the number of qubits required. These small examples may be relevant to near-term experiments on small-scale quantum computers. Published by the American Physical Society 2024