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:1707.09788

Performance of five-qubit code for arbitrary error channels

Long Yang Huang, Xiaohua Wu, Tao Zhou

Published Jul 31, 2017• 0 citations
arXivHTMLPDF

Abstract

In quantum error correction, it is an important assumption that errors on different qubits are independent. In our previous work [Phys. Rev. A {\bf 92}, 052320 (2015)], the generality of the concatenated five-qubit code has been investgated when noises of the principal system and the auxiliary environment are assumed to be the same. In the error correction with concatenated code, tiny differences (in fidelity or independent errors) between initial quantum channels may introduce different effective channels in the next level, and therefore, it is necessary to study a meaningful question: Does five-qubit code still work efficiently when errors on different qubits are different? In the present work, it is discovered that even errors for different qubits are arbitrary, the five-qubit code still works efficiently. Since it is much easier and more accurate to measure the fidelity, one can construct quantum error correction with five-qubit code according to the initial channel fidelity, and it is not necessary to know the complete information of the initial channel. Moreover, when the initial channel fidelity is below $0.992$, the fidelity threshold for five-qubit code is the fidelity of the effective channel after error correction in bit-flip channels.

DOI
10.48550/arxiv.1707.09788

Related papers

Generality of the concatenated five-qubit code

Long Yang Huang, Bo You, Xiaohua Wu +1 more

In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used to construct a near-perfect effective qubit channel (with a error below ${10}^{\ensuremath{-}5}$) from arbitrary noise channels. The exact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicit protocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary size of the concatenated five-qubit code depends only on the entanglement fidelity of the initial noise channels.

arXiv:1511.00059•Physical Review A•Nov 18, 2015•5 citations

The advantage of the concatenated three-qubit codes

Long Huang

In this work, the efficient quantum error-correction protocol against the general independent noise is constructed with the three-qubit codes. The rules of concatenation are summarized according to the error-correcting capability of the codes. The codes not only play the role of correcting errors, but the role of polarizing the effective channel. For any independent noise, the most suitable error-correction protocol can be constructed based on the rules of concatenation. The most significant aspect of using the concatenated three-qubit codes is to realize quantum error-correction with the non-ideal gate operations, because the error-correction schemes of the three-qubit codes are simple and modular. For example, for the amplitude damping noise with the initial channel fidelity 0.9, the effective channel fidelity can reach 0.94731 (or 0.961634 with the ideal quantum gate operations) when using the 4 levels concatenated quantum error-correction with the three-qubit codes. The protocol with the three-qubit codes needs 81 qubits and 366 quantum gate operations when the accuracy rate is 0.9995. Meanwhile, when using the 3 levels concatenated quantum error-correction with the five-qubit code, the effective channel fidelity can only reach 0.922798 (or 0.975488 with the ideal quantum gate operations). The protocol with the five-qubit code needs 125 qubits and 1147 quantum gate operations when the accuracy rate is 0.9995. The physical resources costed is multiple of the physical resources costed by realizing the three-qubit quantum error-correction, the protocol has a higher fault tolerance threshold, and no increase in complexity. So, we believe it will be helpful for realizing quantum error-correction in the physical system.

arXiv:2209.15435•Sep 28, 2022•0 citations

Performance of quantum error correction with coherent errors

Eric Huang, Andrew C. Doherty, Steven T. Flammia

We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where $d$ is the distance of the code and $c$ is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance $d$ will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.

arXiv:1805.08227•Physical Review A•Feb 12, 2019•76 citations

CONCATENATION OF ERROR AVOIDING WITH ERROR CORRECTING QUANTUM CODES FOR CORRELATED NOISE MODELS

Carlo Cafaro, Stefano Mancini

We study the performance of simple error correcting and error avoiding quantum codes together with their concatenation for correlated noise models. Specifically, we consider two error models: (i) a bit-flip (phase-flip) noisy Markovian memory channel (model I); (ii) a memory channel defined as a memory degree dependent linear combination of memoryless channels with Kraus decompositions expressed solely in terms of tensor products of X-Pauli (Z-Pauli) operators (model II). The performance of both the three-qubit bit flip (phase flip) and the error avoiding codes suitable for the considered error models is quantified in terms of the entanglement fidelity. We explicitly show that while none of the two codes is effective in the extreme limit when the other is, the three-qubit bit flip (phase flip) code still works for high enough correlations in the errors, whereas the error avoiding code does not work for small correlations. Finally, we consider the concatenation of such codes for both error models and show that it is particularly advantageous for model II in the regime of partial correlations.

arXiv:1006.2051•International Journal of Quantum Information•Jan 1, 2011•8 citations

Channel-adapted quantum error correction via code concatenation

Bei Zeng

Quantum error correction (QEC) is crucial for faithful transmission of quantum information and successful realization of quantum computers. Typically, QEC is designed with minimal assumptions about the noise process, i.e. depolarizing noise that happens independently on each individual particle. In real physical systems, errors are unlikely to be arbitrary. Rather we have reasonable knowledge about the system to model its quantum decoherence process. In this talk, we examine QEC that are adapted to the physical noise models. We start from an overview of the concepts and general methods for channel-adapted QEC. We then examine a technique, called code concatenation, for constructing channel-adapted quantum codes. Applying to the amplitude damping channels, we demonstrate the power of the code concatenation method for constructing good codes for channel-adapted QEC.

•Jan 1, 2016•0 citations

Learning to Concatenate Quantum Codes

Nico Meyer, Christopher Mutschler, Dominik Seuß +2 more

Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.

quant-pharXiv:2604.14931•Apr 17, 2026•0 citations
In quantum error correction, it is an important assumption that errors on different qubits are independent. In our previous work [Phys. Rev. A {\bf 92}, 052320 (2015)], the generality of the concatenated five-qubit code has been investgated when noises of the principal system and the auxiliary environment are assumed to be the same. In the error correction with concatenated code, tiny differences (in fidelity or independent errors) between initial quantum channels may introduce different effective channels in the next level, and therefore, it is necessary to study a meaningful question: Does five-qubit code still work efficiently when errors on different qubits are different? In the present work, it is discovered that even errors for different qubits are arbitrary, the five-qubit code still works efficiently. Since it is much easier and more accurate to measure the fidelity, one can construct quantum error correction with five-qubit code according to the initial channel fidelity, and it is not necessary to know the complete information of the initial channel. Moreover, when the initial channel fidelity is below 0.9920.9920.992, the fidelity threshold for five-qubit code is the fidelity of the effective channel after error correction in bit-flip channels.
In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used to construct a near-perfect effective qubit channel (with a error below ${10}^{\ensuremath{-}5}$) from arbitrary noise channels. The exact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicit protocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary size of the concatenated five-qubit code depends only on the entanglement fidelity of the initial noise channels.
In this work, the efficient quantum error-correction protocol against the general independent noise is constructed with the three-qubit codes. The rules of concatenation are summarized according to the error-correcting capability of the codes. The codes not only play the role of correcting errors, but the role of polarizing the effective channel. For any independent noise, the most suitable error-correction protocol can be constructed based on the rules of concatenation. The most significant aspect of using the concatenated three-qubit codes is to realize quantum error-correction with the non-ideal gate operations, because the error-correction schemes of the three-qubit codes are simple and modular. For example, for the amplitude damping noise with the initial channel fidelity 0.9, the effective channel fidelity can reach 0.94731 (or 0.961634 with the ideal quantum gate operations) when using the 4 levels concatenated quantum error-correction with the three-qubit codes. The protocol with the three-qubit codes needs 81 qubits and 366 quantum gate operations when the accuracy rate is 0.9995. Meanwhile, when using the 3 levels concatenated quantum error-correction with the five-qubit code, the effective channel fidelity can only reach 0.922798 (or 0.975488 with the ideal quantum gate operations). The protocol with the five-qubit code needs 125 qubits and 1147 quantum gate operations when the accuracy rate is 0.9995. The physical resources costed is multiple of the physical resources costed by realizing the three-qubit quantum error-correction, the protocol has a higher fault tolerance threshold, and no increase in complexity. So, we believe it will be helpful for realizing quantum error-correction in the physical system.
We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where ddd is the distance of the code and ccc is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance ddd will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
We study the performance of simple error correcting and error avoiding quantum codes together with their concatenation for correlated noise models. Specifically, we consider two error models: (i) a bit-flip (phase-flip) noisy Markovian memory channel (model I); (ii) a memory channel defined as a memory degree dependent linear combination of memoryless channels with Kraus decompositions expressed solely in terms of tensor products of X-Pauli (Z-Pauli) operators (model II). The performance of both the three-qubit bit flip (phase flip) and the error avoiding codes suitable for the considered error models is quantified in terms of the entanglement fidelity. We explicitly show that while none of the two codes is effective in the extreme limit when the other is, the three-qubit bit flip (phase flip) code still works for high enough correlations in the errors, whereas the error avoiding code does not work for small correlations. Finally, we consider the concatenation of such codes for both error models and show that it is particularly advantageous for model II in the regime of partial correlations.
Quantum error correction (QEC) is crucial for faithful transmission of quantum information and successful realization of quantum computers. Typically, QEC is designed with minimal assumptions about the noise process, i.e. depolarizing noise that happens independently on each individual particle. In real physical systems, errors are unlikely to be arbitrary. Rather we have reasonable knowledge about the system to model its quantum decoherence process. In this talk, we examine QEC that are adapted to the physical noise models. We start from an overview of the concepts and general methods for channel-adapted QEC. We then examine a technique, called code concatenation, for constructing channel-adapted quantum codes. Applying to the amplitude damping channels, we demonstrate the power of the code concatenation method for constructing good codes for channel-adapted QEC.
Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.