Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2305.00784 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes

Dhruv Bhatnagar, Matthew Steinberg, David Elkouss, Carmen G. Almudéver, Sebastian Feld

Published Sep 17, 2023• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/qce57702.2023.00016

Related papers

Reducing Quantum Error Correction Overhead With Versatile Flag-Sharing Syndrome Extraction Circuits

Pei-Hao Liou, Ching–Yi Lai

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$[\![17,1,5]\!]$,$[\![19,1,5]\!]$Calderbank-Shor-Steane (CSS) codes and the$[\![5,1,3]\!]$non-CSS code, where the$[\![5,1,3]\!]$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.

arXiv:2407.00607•IEEE Transactions on Quantum Engineering•Jan 1, 2025•5 citations

Ultra Low Overhead Syndrome Extraction for the Steane code

Boldizsár Poór, Benjamin Rodatz, Aleks Kissinger

We establish a new performance benchmark for the fault-tolerant syndrome extraction of the [[7, 1, 3]] Steane code with a dynamic protocol. Our method is built on two highly optimized circuits derived using fault-equivalent ZX-rewrites: a primary fault-tolerant circuit with 14 CNOTs and an efficient non-fault-tolerant recovery circuit with 11 CNOTs. The protocol uses an adaptive response to internal faults, discarding flagged measurements and falling back to the recovery circuit to correct potentially detrimental errors. Monte Carlo simulations confirm the efficiency of our protocol, reducing the logical error rate per cycle by an average of ~14.3% relative to the optimized Steane method [arXiv:2506.17181] and ~17.3% compared to Reichardt's three-qubit method [arXiv:1804.06995], the leading prior techniques.

quant-pharXiv:2511.13700•Quantum Science and Technology•Nov 18, 2025•0 citations

Low-Depth Flagged Syndrome Extraction for Calderbank-Shor-Steane Codes of Distance 3

Chao Du, Zhi Ma, Yiting Liu +2 more

In this work, we present a low-depth flagged syndrome extraction strategy for Calderbank-Shor-Steane (CSS) codes of distance 3. We demonstrate that if one stabilizer generator of a distance-3 CSS code measured using the flagged circuit has and only has two distinct qubit supports with certain stabilizers of opposite types, then the number of stabilizers required for the second round of measurements can be reduced. The number of these stabilizers depends on the weight of the measured generator. This novel strategy utilizes past syndrome information and elements from the stabilizer group, enabling the implementation of a flag-FTEC protocol for distance-3 CSS codes with lower-depth syndrome extraction circuits compared to previously flag fault-tolerant protocol if the total weight of these stabilizers is less than that of stabilizer generators. Based on our strategy, we also improve the flag-FTEC protocol for cyclic CSS codes of distance 3, which requires fewer stabilizer measurements. We utilize the$[[{14,2,3}]]$CSS code,$[[{15,1,3}]]$quantum Reed-Muller code and$[[{15,7,3}]]$cyclic CSS code to demonstrate how the strategy operates. Furthermore, we also provide a low-depth flagged syndrome extraction strategy for a$[[{19,1,5}]]$CSS-type code.

IEEE Transactions on Information Theory•Jul 10, 2024•2 citations

Parallel Low-Depth Flagged Syndrome Extraction for Quantum Stabilizer Codes

Yunxing Zhang, Lan Luo, Chaofeng Guan

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.

KSII Transactions on Internet and Information Systems•May 31, 2026•0 citations

Flag Gadgets Based on Classical Codes

Benjamin Anker, Milad Marvian

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

arXiv:2212.10738•PRX Quantum•Dec 10, 2024•3 citations

Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits

Ben W Reichardt

Abstract Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.

arXiv:1804.06995•Quantum Science and Technology•Nov 24, 2020•70 citations
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.
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.
We establish a new performance benchmark for the fault-tolerant syndrome extraction of the [[7, 1, 3]] Steane code with a dynamic protocol. Our method is built on two highly optimized circuits derived using fault-equivalent ZX-rewrites: a primary fault-tolerant circuit with 14 CNOTs and an efficient non-fault-tolerant recovery circuit with 11 CNOTs. The protocol uses an adaptive response to internal faults, discarding flagged measurements and falling back to the recovery circuit to correct potentially detrimental errors. Monte Carlo simulations confirm the efficiency of our protocol, reducing the logical error rate per cycle by an average of ~14.3% relative to the optimized Steane method [arXiv:2506.17181] and ~17.3% compared to Reichardt's three-qubit method [arXiv:1804.06995], the leading prior techniques.
In this work, we present a low-depth flagged syndrome extraction strategy for Calderbank-Shor-Steane (CSS) codes of distance 3. We demonstrate that if one stabilizer generator of a distance-3 CSS code measured using the flagged circuit has and only has two distinct qubit supports with certain stabilizers of opposite types, then the number of stabilizers required for the second round of measurements can be reduced. The number of these stabilizers depends on the weight of the measured generator. This novel strategy utilizes past syndrome information and elements from the stabilizer group, enabling the implementation of a flag-FTEC protocol for distance-3 CSS codes with lower-depth syndrome extraction circuits compared to previously flag fault-tolerant protocol if the total weight of these stabilizers is less than that of stabilizer generators. Based on our strategy, we also improve the flag-FTEC protocol for cyclic CSS codes of distance 3, which requires fewer stabilizer measurements. We utilize the[[14,2,3]][[{14,2,3}]][[14,2,3]]CSS code,[[15,1,3]][[{15,1,3}]][[15,1,3]]quantum Reed-Muller code and[[15,7,3]][[{15,7,3}]][[15,7,3]]cyclic CSS code to demonstrate how the strategy operates. Furthermore, we also provide a low-depth flagged syndrome extraction strategy for a[[19,1,5]][[{19,1,5}]][[19,1,5]]CSS-type code.
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.
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
Abstract Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.