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arXiv:1810.00390 · Physical Review A

Robustness of the concatenated quantum error-correction protocol against noise for channels affected by fluctuation

Long Yang Huang, Xiaohua Wu, Tao Zhou

Published Oct 21, 2019• 5 citations
arXivHTMLPDF

Abstract

In quantum error correction, the description of the noise channel cannot be completely accurate and fluctuation always appears in the noise channel. It is found that when fluctuation of the physical noise channel is considered, the average effective channel is dependent only on the average of the physical noise channel, and the average of the physical noise channel here plays the role of the independent error model in previous works. Now one may conclude that in the independent error model, the results in previous works are also valid for the average channel where fluctuation exists. In some typical cases, our numerical simulations in the concatenated quantum error-correction protocol with a five-qubit code, seven-qubit Steane code, and nine-qubit Shor confirm this conjecture. For a five-qubit code, the effective channels approximate to the depolarizing channel as the concatenated level increases. For the Steane code, the effective channels approximate to one Pauli channel as the concatenated level increases. For the Shor code, the effective channels approximate to either a Pauli-$X$ or Pauli-$Z$ channel in each level, and in the next concatenated level, the effective channels approximate to the other. The numerical results for these codes indicate that the degree of approximation increases as the concatenated level increases, and the fluctuation of the noise channel decays exponentially as concatenated quantum error correction is performed. On the error-correction threshold, the attenuation ratio of the standard deviation of channel fidelity has a roughly stable value. In contrast, standard deviations of off-diagonal elements of the quantum process matrix (Pauli form) decay more quickly than standard deviations of diagonal elements.

DOI
10.1103/physreva.100.042321

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In quantum error correction, the description of the noise channel cannot be completely accurate and fluctuation always appears in the noise channel. It is found that when fluctuation of the physical noise channel is considered, the average effective channel is dependent only on the average of the physical noise channel, and the average of the physical noise channel here plays the role of the independent error model in previous works. Now one may conclude that in the independent error model, the results in previous works are also valid for the average channel where fluctuation exists. In some typical cases, our numerical simulations in the concatenated quantum error-correction protocol with a five-qubit code, seven-qubit Steane code, and nine-qubit Shor confirm this conjecture. For a five-qubit code, the effective channels approximate to the depolarizing channel as the concatenated level increases. For the Steane code, the effective channels approximate to one Pauli channel as the concatenated level increases. For the Shor code, the effective channels approximate to either a Pauli-XXX or Pauli-ZZZ channel in each level, and in the next concatenated level, the effective channels approximate to the other. The numerical results for these codes indicate that the degree of approximation increases as the concatenated level increases, and the fluctuation of the noise channel decays exponentially as concatenated quantum error correction is performed. On the error-correction threshold, the attenuation ratio of the standard deviation of channel fidelity has a roughly stable value. In contrast, standard deviations of off-diagonal elements of the quantum process matrix (Pauli form) decay more quickly than standard deviations of diagonal elements.
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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 ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
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.
A universal, scalable quantum computer will require the use of quantum error correction in order to achieve fault tolerance. The assessment and comparison of error-correcting strategies is performed by classical simulation. However, due to the prohibitive exponential scaling of general quantum circuits, simulations are restrained to specific subsets of quantum operations. This creates a gap between accuracy and efficiency which is particularly problematic when modeling noise, because most realistic noise models are not efficiently simulable on a classical computer. We have introduced extensions to the Pauli channel, the traditional error channel employed to model noise in simulations of quantum circuits. These expanded error channels are still computationally tractable to simulate, but result in more accurate approximations to realistic error channels at the single qubit level. Using the Steane [[7,1,3]] code, we have also investigated the behavior of these expanded channels at the logical error-corrected level. We have found that it depends strongly on whether the error is incoherent or coherent. In general, the Pauli channel will be an excellent approximation to incoherent channels, but an unsatisfactory one for coherent channels, especially because it severely underestimates the magnitude of the error. Finally, we also studied the honesty and accuracy of the expanded channels at the logical level. Our results suggest that these measures can be employed to generate lower and upper bounds to a quantum code's threshold under the influence of a specific error channel.
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.