Advances in quantum error correction based on superconducting quantum systems

Chen Zi-Jie; Pan Xiao-Xuan; Hua Zi-Yue; Wang Wei-Ting; Ma Yu-Wei; Li Ming; Zou Xu-Bo; Sun Lu-Yan; Zou Chang-Ling

doi:10.7498/aps.71.20221824

Abstract

Information processing technology based on the basic principle of quantum mechanics shows great potential applications in computing, sensing and other fields, and is far superior to classical technology. With the advance of experimental technology, quantum control technology develops rapidly. Compared with other quantum information processing platforms, the superconducting system based on solid materials has the advantages of accurate quantum controllability, excellent quantum coherence and the potential for large-scale integration. Therefore, superconducting quantum system is one of the most promising platforms for quantum information processing. The existing superconducting circuits, which can integrate about one hundred qubits, have already demonstrated the advantages of quantum systems, but further development is limited by system noise. In order to break through this bottleneck, quantum error correction technology, which is developed from the classical error correction technology, has attracted extensive attention. Here, we mainly summarize the research progress of quantum error correction in superconducting quantum systems including the basic principles of superconducting quantum systems, the quantum error correction codes, the related control techniques and the recent applications. At the end of the article, we summarize seven key problems in this field.

Keywords:

Authors and contacts

1.
Key Laboratory of Quantum Information, Chinese Academy of Sciences, University of Science and Technology of China, Hefei 230026, China
2.
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China

Corresponding author: Sun Lu-Yan, luyansun@mail.tsinghua.edu.cn ; Zou Chang-Ling, clzou321@ustc.edu.cn

Funds: Project supported by the National Key R&D Program of China (Grant No. 2017YFA0304303), the National Natural Science Foundation of China (Grant Nos. 92165209, 11925404, 11874342, 11922411, 12061131011), and the China Postdoctoral Science Foundation (Grant No. BX2021167)

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References

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Cited By

图 1 (a)二维transmon量子比特结构示意图(图片来自文献[15]). 左右两边黄色电路分别表示一个transmon量子比特, 其中交叉表示约瑟夫森结, 其两端并联大旁路电容, 而中间的黑色电路表示辅助模式, 用于两个量子比特之间的可调耦合. (b)三维超导谐振腔与transmon量子比特耦合结构示意图(图片来自文献[45]). 橙色和绿色区域为高纯铝制备的三维超导谐振腔, 两腔通过铝块侧面开孔与辅助超导量子比特进行耦合. (c)基于二维transmon量子比特的架构(图片修改自文献[30]). 图中黄色线路为读取用微波传输线, 末端接隔离器和约瑟夫森参量放大器, 用于读取信号的单向传输和放大; 绿色区域为读取谐振腔; 浅蓝色区域为可调频transmon量子比特, 量子比特之间通过一个共同的辅助模式B耦合; 红色线路为transmon量子比特的微波驱动线, 蓝色线路为transmon量子比特的磁通驱动线. (d)三维谐振腔架构拓展示意图^[53]. 红色和绿色方块为transmon量子比特, 浅渌色部分为高Q超导谐振腔(用于存储量子信息), 灰色部分为低Q超导谐振腔(用于transmon量子比特的读取). 谐振腔与transmon量子比特通过电容直接耦合

Figure 1. (a) Structure of coupled two dimensional (2D) transmon qubits (Reprinted with permission from Ref. [15]). The left and right circuits (yellow) represent two transmon qubits, where each cross represents a Josephson junction and its two ends are coupled with a large capacitor. The middle black circuit represents the auxiliary mode for adjustable coupling between the two qubits. (b) Structure of 3D superconducting cavities coupled with transmon qubits (Reprinted with permission from Ref. [45]). The orange and green parts are the three dimensional (3D) superconducting cavities made of high-purity aluminum, and the two cavities are coupled to the auxiliary superconducting qubits through trenches on the sides of the aluminum block. (c) Architecture based on 2D transmon qubits (Reprinted with permission from Ref. [30]). The yellow lines in the figure are the microwave transmission lines for readout, and are connected with an isolator and a Josephson parametric amplifier, which are used for one-way transmission and amplification of the readout signal. The green parts are the readout resonators. The light blue parts are the tunable transmon qubits, which are coupled by a common auxiliary mode B. The red and the blue lines are the microwave drive and the flux drive lines for the transmon qubits respectively. (d) Architecture based on 3D superconducting resonators^[53]. The red and green squares are transmon qubits, the light green parts are the high-Q superconducting resonators (for storing quantum information), and the gray parts are the low-Q superconducting resonators (for readout of transmon qubits). The resonator is directly coupled to the transmon qubit through a capacitor

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图 2 纠错流程图. 蓝色的圆圈表示编码空间, 绿色的圆圈表示错误空间, 黄色的箭头表示要保护的量子态

Figure 2. Diagram of quantum error correction. The blue circles represent the code space, the green circles represent the error space, and the yellow arrows represent the quantum states to be protected

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图 3 (a) $ d=8 $的表面码及其逻辑算符$ {\boldsymbol{\hat{Z}}}_{{\rm{L}}} $与$ {\boldsymbol{\hat{X}}}_{{\rm{L}}} $的示意图. 图中网格的每条边上都有一个白点表示一个数据量子比特. 黄色菱形以及蓝色圆形分别表示其中一个A和B类型的稳定子. 蓝色和红色阴影线分别表示逻辑编码的$ {\boldsymbol{\hat{Z}}}_{{\rm{L}}} $和$ {\boldsymbol{\hat{X}}}_{{\rm{L}}} $算符. 绿色阴影线表示蓝色阴影线对应的算符乘上一个稳定算符, 其也是逻辑$ {\boldsymbol{\hat{Z}}}_{{\rm{L}}} $算符. (b)和(c)分别是A, B类型稳定子算符测量的量子线路图. 此图由文献[76]改编

Figure 3. (a) Schematic of $ d=8 $ surface code and the corresponding logical operators $ {\boldsymbol{\hat{Z}}}_{{\rm{L}}} $ and $ {\boldsymbol{\hat{X}}}_{{\rm{L}}} $. The white dots on each edge of the grid represent data qubits. The yellow diamonds and the blue circles represent A and B types of stabilizers, respectively. The blue and red shaded lines represent the logical $ {\boldsymbol{\hat{Z}}}_{{\rm{L}}} $ and $ {\boldsymbol{\hat{X}}}_{{\rm{L}}} $ operators, respectively. The green shaded line represents the operator corresponding to the blue shaded line multiplied by a stablizer operator, which is also the logical $ {\boldsymbol{\hat{Z}}}_{{\rm{L}}} $ operator. (b) and (c) are quantum circuits that measure the stabilizer operators of type A and B, respectively. This figure is adapted from Ref. [76]

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图 4 表面码的错误症状及纠错操作示意图. 图中红色交叉表示对应物理比特发生了位翻转噪声, 蓝色圆形表示测量结果异常的B类型稳定子的位置, 即缺陷的位置, 阴影线均表示对线上的物理比特执行的$ {\boldsymbol{\hat{X}}} $操作. 四个区域分别对应以下情况: (a)网格中单个比特发生噪声产生了两个相邻的缺陷; (b)网格中的连续的几个比特发生噪声产生了位于错误链两端的缺陷, 而蓝色和绿色阴影线对应的操作均能纠正逻辑错误; (c)含有网格左边界的比特的错误链导致单个缺陷的产生, 浅绿色和黄色的阴影线对应的操作均能纠正逻辑错误; (d)含有网格左边界的比特的错误链导致单个缺陷的产生, 红色阴影线对应的操作能纠正逻辑错误, 但是紫色阴影线的操作会导致无法被错误诊断识别的逻辑错误

Figure 4. Schematic of error syndromes and error correction operations of surface codes. The red cross in the figure indicates a bit-flip error occuring in the corresponding physical qubit, while the blue circle indicates the position of the B type stabilizer with abnormal measurement results, i.e., the position of the defect, and the shaded lines indicate an $ {\boldsymbol{\hat{X}}} $ operation performed on each of the physical qubit on the line. The four areas correspond to the following situations: (a) A single bit-flip error in the grid results in two adjacent defects. (b) Several consecutive bit-flip errors in the grid result in two defects located at the two ends of the error chain. Both the blue and green shaded lines can correct these errors. (c) The error chain containing a qubit on the left boundary of the grid results in a single defect, and the operations corresponding to either the light green or the yellow shaded line can orrect these errors. (d) The error chain containing a qubit on the left boundary of the grid results in a single defect. The operation corresponding to the red shaded line can correct these errors, but the operation corresponding to the purple shaded line causes a logic error that cannot be identified by the error syndrome

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图 5 (a)谷歌公司实验^[84]中制备表面码初态所用超导芯片的架构以及对应稳定子测量结果. 图中白色的十字为数据比特, 紫色和蓝色的图形分别代表$ {\boldsymbol{B}_{p}} $类型和$ {\boldsymbol{A}_{s}} $类型的稳定子测量, 图中的数值是初态对应的稳定子测量结果的平均值. (b)表面码初态制备对应的门操作线路

Figure 5. (a) Architecture of the superconducting circuit used to prepare the initial state of the surface code in Google’s experiment^[84] and the corresponding stabilizer measurement results. The white crosses are data qubits, the purple and blue regions represent stabilizer measurement results of type $ {\boldsymbol{B}_{p}} $ and $ {\boldsymbol{A}_{s}} $, respectively. The values in the figure are the average values of the stabilizer measurement results corresponding to the initial state. (b) Operation circuit to prepare the initial state of the surface code

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图 6 一种二项式编码逻辑态的维格纳函数图像. 左上: 逻辑态$ |0_{{\rm{L}}}\rangle=( | 0\rangle +\sqrt{3}|4\rangle)/2 $; 右上: $ |1_{{\rm{L}}}\rangle=(\sqrt{3}|2\rangle +|6\rangle)/2 $; 左下: $ |+_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle+|1_{{\rm{L}}}\rangle)/\sqrt{2} $; 右下: $ |-_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle-|1_{{\rm{L}}}\rangle)/\sqrt{2} $. 这是平均光子数最小的且能纠正错误集合$ \{{\boldsymbol{\hat{I}}},\; {\boldsymbol{\hat{a}}},\; {\boldsymbol{\hat{a}}}^{\dagger}{\boldsymbol{\hat{a}}}\} $的二项式编码, 可以看到其$ |0/1_{{\rm{L}}}\rangle $本身具有$ {\text{π}}/4 $的旋转对称性, 而$ |+/-_{{\rm{L}}}\rangle $在相空间中接近正交

Figure 6. Wigner functions of the logical states of one binomial code. Upper left: Logical state $ |0_{{\rm{L}}}\rangle=( | 0\rangle +\sqrt{3}|4\rangle)/2 $; Upper right: Logical state $ |1_{{\rm{L}}}\rangle=(\sqrt{3}|2\rangle +|6\rangle)/2 $; Lower left: $ |+_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle+|1_{{\rm{L}}}\rangle)/\sqrt{2} $; Lower right: $ |-_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle-|1_{{\rm{L}}}\rangle)/\sqrt{2} $. This is the binomial code that has the minimal average photon number and can correct the error set $ \{{\boldsymbol{\hat{I}}}, \;{\boldsymbol{\hat{a}}},\; {\boldsymbol{\hat{a}}}^{\dagger}{\boldsymbol{\hat{a}}}\} $. The logical state $ |0/1_{{\rm{L}}}\rangle $ has a rotational symmetry of $ {\text{π}}/4 $ and $ |+/-_{{\rm{L}}}\rangle $ are almost orthogonal in phase space

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图 7 猫态编码逻辑态的维格纳函数图像, 其中$ \alpha=2.5,\; n=4 $. 左上: 逻辑态$ |0_{{\rm{L}}}\rangle $; 右上: $ |1_{{\rm{L}}}\rangle $; 左下: $ |+_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle+|1_{{\rm{L}}}\rangle)/\sqrt{2} $; 右下: $ |-_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle-|1_{{\rm{L}}}\rangle)/\sqrt{2} $

Figure 7. Wigner functions of the logical states of the cat code with $ \alpha=2.5,\; n=4 $. Upper left: Logical state $ |0_{{\rm{L}}}\rangle $; Upper right: Logical state $ |1_{{\rm{L}}}\rangle $; Lower left:$ |+_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle+|1_{{\rm{L}}}\rangle)/\sqrt{2} $; Lower right:$ |-_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle-|1_{{\rm{L}}}\rangle)/\sqrt{2} $

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图 8 近似GKP编码逻辑态的维格纳函数图像, 其中$\varDelta=0.3$. 左上: 逻辑态$ |0_{{\rm{L}}}\rangle $; 右上: $ |1_{{\rm{L}}}\rangle $; 左下: $ |+_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle+|1_{{\rm{L}}}\rangle)/\sqrt{2} $; 右下: $ |-_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle-|1_{{\rm{L}}}\rangle)/\sqrt{2} $

Figure 8. Wigner functions of the logical states of an approximate GKP code with $\varDelta=0.3$. Upper left: Logical state $ |0_{{\rm{L}}}\rangle $; Upper right: $ |1_{{\rm{L}}}\rangle $; Lower left: $ |+_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle+|1_{{\rm{L}}}\rangle)/\sqrt{2} $; Lower right: $ |-_{{\rm{L}}}\rangle=(|0_{{\rm{L}}}\rangle-|1_{{\rm{L}}}\rangle)/\sqrt{2} $

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图 9 (a)横向逻辑门操作示意图; (b)单比特态传输的方式实现逻辑${{\hat{{\boldsymbol{T}}}}}$门. 图中斜线表示此逻辑比特包括多个物理比特, 而图中红色阴影区域的线路等价于将辅助比特初态制备成$\left|\varTheta\right\rangle$, 红色区域外的门操作全是横向逻辑门

Figure 9. (a) Schematic of the transversal logical operation; (b) logical ${{\hat{{\boldsymbol{T}}}}}$ gate implemented through one-bit teleportation. The slash in the figure indicates that the logical qubit is encoded with several physical qubits, and the red shaded area is equivalent to initializing the auxiliary qubit as $\left|\varTheta\right\rangle$. All logical gates outside the red shaded area are transversal

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表 1 超导系统中二能级编码的纠错实验进展. 带*号的是物理比特中最优的数据, /表示文献中没有直接给出相关数据. 其中, 文献[72, 90, 91]的实验均在国际商用机器公司的IBMQ平台上进行

Table 1. Experimental progress on quantum error correction with two-level codes in superconducting systems. Datas with * refer to the optimal datas among the physical qubits. / indicate that the relevant datas are not directly given in the documents. The experiments in the Refs. [72, 90, 91] were implemented on IBMQ

时间	文献	编码	逻辑$ T_{1} $(物理$ T_{1} $)$/\text{µ} {\rm{s}}$	逻辑 $ T_{2} $(物理$ T_{2} $)$/\text{µ} {\rm{s}}$
2012	耶鲁大学^[68]	$ \left[3, 1, 3\right] $重复码	/	/
2014	加州大学圣芭芭拉分校^[69]	$ \left[3, 1, 3\right] $, $ \left[5, 1, 5\right] $重复码	/	/
2015	代尔夫特理工大学^[25]	$ \left[3, 1, 3\right] $重复码	/	/
2017	IBM^[71]	4比特颜色码	/	/
2018	巴塞尔大学(University of Basel)^[72]	$ \left[8, 1, 8\right] $重复码	/	/
2018	德国于利希研究中心(Forschungszentrum Jülich)^[90]	4比特颜色码	/	/
2019	悉尼大学^[91]	$ [4, 2, 2] $错误检测码	/	/
2019	中国科学技术大学^[92]	$ [[5, 1, 3]] $编码	/	/
2020	苏黎世联邦理工学院^[93]	$ [[4, 1, 2]] $表面码	$ 62.7\pm9.4 $($ 16.8^{*} $)	$ 72.5\pm32.9 $($ 21.5^{*} $)
2021	谷歌公司^[84]	实现表面码的基态	/	/
2021	中国科学技术大学^[31]	$ [[9, 1, 3]] $表面码	$ 137.8 $($ 36.6^{*} $)	/
2021	苏黎世联邦理工学院^[94]	$ [[9, 1, 3]] $表面码	$ 16.4 $($ 32.5 $)	$ 18.2 $($ 37.5 $)
2021	谷歌公司^[73]	$ [11, 1, 11] $重复码以及$ [[4, 1, 2]] $表面码	/	/
2022	苏黎世联邦理工学院^[24]	$ [[4, 1, 2]] $表面码	/	/
2022	IBM^[95]	$ [[9, 1, 3]] $Heavy-hexagon编码	/	/
2022	谷歌公司^[8]	$ [[25, 1, 5]] $表面码以及$ [25, 1, 25] $重复码	/	/

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