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多逻辑比特表面码结构设计及其逻辑CNOT门实现

权东晓 ,  吕晓杰 ,  张雯菲

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多逻辑比特表面码结构设计及其逻辑CNOT门实现

权东晓, 吕晓杰, 张雯菲
, 2024, 73(4): 040304.
doi: 10.7498/aps.73.20231138

Structure design and logical CNOT implementation of multi-logical-qubits surface code

Quan Dong-Xiao, Lü Xiao-Jie, Zhang Wen-Fei
Acta Phys. Sin., 2024, 73(4): 040304.
doi: 10.7498/aps.73.20231138
科大讯飞翻译 (iFLYTEK Translation)
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  • 摘要

    量子计算因具有并行处理能力, 相比于经典计算有着指数级的加速, 但量子系统具有脆弱性, 极易受到噪声的影响, 量子纠错码是克服量子噪声的有效手段. 量子表面码是一种拓扑稳定子码, 由于其结构上的最近邻居特点和较高的容错阈值, 表面码在大规模容错量子计算方面具有巨大的潜力. 目前已有的基于边界的表面码均为编码一个逻辑比特的表面码, 本文主要研究基于边界如何实现多逻辑量子比特的编码, 包括设计表面码的结构, 根据结构找出对应的稳定子和逻辑操作, 进一步根据稳定子设计出基于稳定子实现的编码线路; 在研究基于测量和纠正的单量子比特间CNOT实现原理和基于融合操作和分割操作的单逻辑量子比特表面码间CNOT门实现原理的基础上, 优化了基于融合操作和分割操作的单逻辑量子比特表面码间CNOT门实现方案, 将其扩展到所设计的多逻辑量子比特表面码上实现了多逻辑量子比特表面码之间的CNOT操作, 并通过仿真验证量子线路的正确性. 本文设计的多逻辑比特表面码克服了单比特表面码不能密铺于量子芯片的缺点且提高了某些逻辑操作的长度, 提高了容错能力. 基于联合测量的思想降低了对辅助比特的要求且减小了实现过程中对量子资源的需求.

    Abstract

    As its parallel processing ability, quantum computing has an exponential acceleration over classical computing. However, quantum systems are fragile and susceptible to noise. Quantum error correction code is an effective means to overcome quantum noise. Quantum surface codes are topologically stable subcodes that have great potential for large-scale fault-tolerant quantum computing because of their structural nearest neighbor characteristics and high fault-tolerance thresholds. The existing boundary-based surface codes can encode one logical qubit. This paper mainly studies how to implement multi-logical-qubits encoding based on the boundary, including designing the structure of the surface code, finding out the corresponding stabilizers and logical operations according to the structure, and further designing the coding circuit based on the stabilizers. After research on the single qubit CNOT implementation principle based on measurement and correcting and the logic CNOT implementation based on fusion and segmentation, we further optimized implementation scheme of the logic CNOT implementation based on fusion and segmentation. The scheme is extended to the designed multi-logical-qubits surface code to realize the CNOT operation between the multi-logical-qubits surface codes, and the correctness of the quantum circuit is verified by simulation. The multi-logical-qubits surface code designed in this paper overcomes the disadvantage that the single-logical-qubit surface code can not be densely embedded in the quantum chip, improves the length of some logical operations, and increases the fault tolerance ability. The idea of joint measurement reduces the requirement for ancilla qubits and reduces the demand for quantum resources in the implementation process.

    作者及机构信息

    • 1.

      西安电子科技大学通信工程学院, 西安　710071

    • 2.

      西安电子科技大学, 量子信息协同创新中心, 西安　710071

      • 通信作者: 权东晓, dxquan@xidian.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 62001351)和陕西省重点研发计划(批准号: 2019ZDLGY09-02)资助的课题.

    Authors and contacts

    • 1.

      School of Telecommunications Engineering, Xidian University, Xi’an 710071, China

    • 2.

      Collaborative Innovation Center of Quantum Information, Xidian University, Xi’an 710071, China

      • Corresponding author: Quan Dong-Xiao, dxquan@xidian.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 62001351) and the Key Research and Development Program of Shaanxi Province, China (Grant No. 2019ZDLGY09-02).

    参考文献

    [1]

    Feynman R P 1982 Int. J. Theor. Phys. 21 467Google Scholar

    [2]

    Shor P W 1999 SIREV 41 303Google Scholar

    [3]

    Preskill J 2012 arXiv: 1203.5813v3 [quant-ph
    [4]

    张诗豪, 张向东, 李绿周 2021 70 210301Google Scholar

    Zhang S H, Zhang X D, Li L Z 2021 Acta Phys. Sin. 70 210301Google Scholar

    [5]

    周文豪, 王耀, 翁文康, 金贤敏 2022 71 240302Google Scholar

    Zhou W H, Wang Y, Weng W K, Jin X M 2022 Acta Phys. Sin. 71 240302Google Scholar

    [6]

    宋克慧 2005 54 4730Google Scholar

    Song K H 2005 Acta Phys. Sin. 54 4730Google Scholar

    [7]

    Grover L 1996 Proc. 28th ACM Symp. Theo. Comp. 212
    [8]

    范桁 2023 72 070303Google Scholar

    Fan H 2023 Acta Phys. Sin. 72 070303Google Scholar

    [9]

    Shor P W 1995 Phys. Rev. A 52 2493Google Scholar

    [10]

    Steane A M 1996 Phys. Rev. Lett. 77 793Google Scholar

    [11]

    Frank A, Kunal A, Ryan B, et al. 2019 Nature 574 505Google Scholar

    [12]

    Davide C 2023 Nature 618 656
    [13]

    Deng Y H, Gu Y C, Liu H L, Gong S Q, Su H, Zhang Z J, Tang H Y, Jia M H, Xu J M, Chen M C, Qin J, Peng L C, Yan J R, Hu Y, Huang J, Li H, Li Y X, Chen Y J, Jiang X, Gan L, Yang G W, You L X, Li L, Zhong H S, Wang H, Liu N L, Renema J J, Lu C Y, Pan J W 2023 Phys. Rev. Lett. 131 150601Google Scholar

    [14]

    Huang J S, Chen X J, Li X D, Wang J W 2023 AAPPS Bull. 14 33
    [15]

    Fowler A G, Mariantoni M, Martinis J M, Cleland A N 2012 Phys. Rev. A 86 032324Google Scholar

    [16]

    Horsman C, Fowler A G, Devitt S, van Meter R 2012 New J. Phys. 14 123011Google Scholar

    [17]

    Kitaev A Y 1997 Quantum Communication, Computing, and Measurement (New York: Plenum Press) pp181–188
    [18]

    Kitaev A Y 1997 Russ. Math. Surv. 52 1191Google Scholar

    [19]

    Kitaev A Y 2003 Ann. Phys. 303 2Google Scholar

    [20]

    Bravyi S B, Kitaev A Y 1998 arXiv: 9811052 v1 [quant-ph
    [21]

    Freedman M H, Meyer D A 2001 Found Comput. Math. 1 325Google Scholar

    [22]

    Wang C Y, Harrington J, Preskill J 2003 Ann. Phys. 303 31Google Scholar

    [23]

    Raussendorf R, Harrington J, Goyal K 2006 Ann. Phys. 321 2242Google Scholar

    [24]

    邢莉娟, 李卓, 白宝明, 王新梅 2008 57 4695Google Scholar

    Xing L J, Li Z, Bai B M, Wang X M 2008 Acta Phys. Sin. 57 4695Google Scholar

    [25]

    Fowler A G, Stephens A M, Groszkowski P 2009 Phys. Rev. A 80 052312Google Scholar

    [26]

    DiVincenzo D P 2009 Phys. Scr. 137 014020
    [27]

    Tomita Y, Svore K M 2014 Phys. Rev. A 90 062320Google Scholar

    [28]

    Brown B J, Laubscher K, Kesselring M S, Wootton J R 2017 Phys. Rev. X 7 021029Google Scholar

    [29]

    Litinski D, von Oppen F 2018 Quantum 2 62Google Scholar

    [30]

    Krylov G, Lukac M 2018 arXiv: 1809.11134v1 [quant-ph
    [31]

    Beaudrap de N, Horsman D 2020 Quantum 4 218Google Scholar

    [32]

    Camps D, van Beeumen R 2020 Phys. Rev. A 102 052411Google Scholar

    [33]

    Shirakawa T, Ueda H, Yunoki S 2021 arXiv: 2112.14524v1 [quant-ph
    [34]

    Wang H W, Xue Y J, Ma Y L, Hua N, Ma H Y 2022 Chin. Phys. B 31 010303Google Scholar

    [35]

    Marques J F, Varbanov B M, Moreira M S, Ali H, Muthusubramanian N, Zachariadis C, Battistel F, Beekman M, Haider N, Vlothuizen W, Bruno A, Terhal B M, DiCarlo L 2022 Nat. Phys. 18 80Google Scholar

    [36]

    Kumari K, Rajpoot G, Ranjan Jain S 2022 arXiv: 2211. 12695v4 [quant-ph
    [37]

    Chen P H, Yan B W, Cui S X 2022 arXiv: 2210.01682v2 [cond-mat.str-el
    [38]

    Chen X B, Zhao L Y, Xu G, Pan X B, Chen S Y, Cheng Z W, Yang Y X 2022 Chin. Phys. B 31 040305Google Scholar

    [39]

    Xue Y J, Wang H W, Tian Y B, Wang Y N, Wang Y X, Wang S M 2022 Quantum Eng. 2022 9
    [40]

    Ding L, Wang H W, Wang Y N, Wang S M 2022 Quantum Eng. 2022 8
    [41]

    Siegel A, Strikis A, Flatters T, Benjamin S 2023 Quantum 7 1065Google Scholar

    [42]

    Quan D X, Liu C S, Lü X J, Pei C X 2022 Entropy 24 1107Google Scholar

  • 图 1  3 × 3表面码的结构图

    Fig. 1.  Structure diagram of 3 × 3 surface code

    下载: 全尺寸图片 幻灯片

    图 2  基于边界编码三位逻辑比特的表面码结构

    Fig. 2.  Structure of surface code based on boundary encoding three logical qubits

    下载: 全尺寸图片 幻灯片

    图 3  图2所示编码三位逻辑比特表面码的编码线路图

    Fig. 3.  Quantum encoding circuit for the surface code shown in Fig. 2

    下载: 全尺寸图片 幻灯片

    图 4  基于联合测量的单量子比特间CNOT门的实现原理

    Fig. 4.  CNOT gate implementation for single qubit based on joint measurement

    下载: 全尺寸图片 幻灯片

    图 5  表面码的粗糙融合

    Fig. 5.  Rough fusion of surface codes

    下载: 全尺寸图片 幻灯片

    图 6  表面码的粗糙分割

    Fig. 6.  Rough segmentation of surface codes

    下载: 全尺寸图片 幻灯片

    图 7  表面码的光滑融合

    Fig. 7.  Smooth fusion of surface codes

    下载: 全尺寸图片 幻灯片

    图 8  表面码的光滑分割

    Fig. 8.  Smooth segmentation of surface codes

    下载: 全尺寸图片 幻灯片

    图 9  单逻辑比特量子表面码逻辑CNOT门的实现

    Fig. 9.  Implementation of logic CNOT gate for the single logical qubit surface code

    下载: 全尺寸图片 幻灯片

    图 10  不同输入状态下的仿真输出　(a)$ \left|{\mathrm{CQ}}\right\rangle $=$ \left|0\right\rangle $, $ \left|{\mathrm{TQ}}\right\rangle $=$ \left|0\right\rangle $; (b)$ \left|{\mathrm{CQ}}\right\rangle $=$ \left|0\right\rangle $, $ \left|{\mathrm{TQ}}\right\rangle $=$ \left|1\right\rangle $; (c)$ \left|{\mathrm{CQ}}\right\rangle $=$ \left|1\right\rangle $, $ \left|{\mathrm{TQ}}\right\rangle $=$ \left|0\right\rangle $; (d)$ \left|{\mathrm{CQ}}\right\rangle $ = $ \left|1\right\rangle $, $ \left|{\mathrm{TQ}}\right\rangle $=$ \left|1\right\rangle $

    Fig. 10.  Simulation output under different input states: (a)$ \left|{\mathrm{CQ}}\right\rangle $=$ \left|0\right\rangle $, $ \left|\mathrm{TQ}\right\rangle $=$ \left|0\right\rangle $; (b)$ \left|\mathrm{CQ}\right\rangle $=$ \left|0\right\rangle $, $ \left|\mathrm{TQ}\right\rangle $=$ \left|1\right\rangle $; (c)$ \left|\mathrm{CQ}\right\rangle $=$ \left|1\right\rangle $, $ \left|\mathrm{TQ}\right\rangle $=$ \left|0\right\rangle $; (d)$ \left|\mathrm{CQ}\right\rangle $=$ \left|1\right\rangle $, $ \left|\mathrm{TQ}\right\rangle $=$ \left|1\right\rangle $

    下载: 全尺寸图片 幻灯片

    图 11  基于联合测量和逻辑测量的多逻辑量子比特表面码逻辑CNOT门的实现

    Fig. 11.  Implementation of logic CNOT gate for multiple logical qubits surface code based on joint measuremnt and logical measurement

    下载: 全尺寸图片 幻灯片

    图 12  基于联合测量和逻辑测量的多逻辑量子比特表面码逻辑CNOT门的优化

    Fig. 12.  Optimization of logic CNOT gate for multiple logical qubits surface code based on joint measuremnt and logical measurement

    下载: 全尺寸图片 幻灯片

    图 13  控制比特和目标比特都为逻辑比特3时, 逻辑CNOT门的优化

    Fig. 13.  Optimization of logic CNOT gate when both control qubit and target qubit are the 3rd logical qubits

    下载: 全尺寸图片 幻灯片

    表 1  图2所示表面码的逻辑操作

    Table 1.  Logical operation of the surface code shown in Fig. 2

    $X_{{\mathrm{L}}}$ $Z_{{\mathrm{L}}}$
    $X_{{\mathrm{L}}1}=X_{1}X_{3}$ $Z_{{\mathrm{L}}1}=Z_{1}Z_{2} $
    $X_{{\mathrm{L}}2}=X_{10}X_{12}$ $Z_{{\mathrm{L}}2}=Z_{5}Z_{10}$
    $X_{{\mathrm{L}}3}=X_{8}X_{11}$ $Z_{{\mathrm{L}}3}=Z_{8}Z_{6} Z_{4} Z_{2}$
    下载: 导出CSV

    表 2  对辅助比特M在Z基测量后的输出结果

    Table 2.  Output states after the measurements of ancilla qubit M in the Z basis.

    测量结果 输出态
    $M_{1}$=0, $M_{2}$=0, $M_{3}$=0 $\alpha \left |00 \right \rangle (m\left | 0 \right \rangle+n\left | 1 \right \rangle )+\beta \left | 10 \right \rangle(m\left | 1 \right \rangle+n\left | 0 \right \rangle)$
    $M_{1}$=0, $M_{2}$=0, $M_{3}$=1 $\alpha \left |01 \right \rangle (m\left |1 \right \rangle+n\left |0 \right \rangle )+\beta \left | 11 \right \rangle(m\left | 0 \right \rangle+n\left | 1 \right \rangle)$
    $M_{1}$=0, $M_{2}$=1, $M_{3}$=0 $\alpha \left |00 \right \rangle (m\left | 0 \right \rangle+n\left | 1 \right \rangle )-\beta \left | 10 \right \rangle(m\left | 1 \right \rangle+n\left | 0 \right \rangle)$
    $M_{1}$=0, $M_{2}$=1, $M_{3}$=1 $-\alpha \left |01 \right \rangle (m\left |1 \right \rangle+n\left |0 \right \rangle )+\beta \left | 11 \right \rangle(m\left | 0 \right \rangle+n\left | 1 \right \rangle)$
    $M_{1}$=1, $M_{2}$=0, $M_{3}$=0 $\alpha \left |00 \right \rangle (m\left | 1 \right \rangle+n\left | 0 \right \rangle )+\beta \left | 10 \right \rangle(m\left | 0 \right \rangle+n\left | 1 \right \rangle)$
    $M_{1}$=1, $M_{2}$=0, $M_{3}$=1 $\alpha \left |01 \right \rangle (m\left |0 \right \rangle+n\left |1 \right \rangle )+\beta \left | 11 \right \rangle(m\left | 1 \right \rangle+n\left | 0 \right \rangle)$
    $M_{1}$=1, $M_{2}$=1, $M_{3}$=0 $-\alpha \left |00 \right \rangle (m\left |1 \right \rangle+n\left | 0 \right \rangle )+\beta \left | 10 \right \rangle(m\left | 0 \right \rangle+n\left |1 \right \rangle)$
    $M_{1}$=1, $M_{2}$=1, $M_{3}$=1 $\alpha \left |01 \right \rangle (m\left |0 \right \rangle+n\left |1 \right \rangle )-\beta \left | 11 \right \rangle(m\left | 1 \right \rangle+n\left | 0 \right \rangle)$
    下载: 导出CSV

    表 3  $ \left|\mathrm{CQ}\right\rangle $=$\left|AB0\right\rangle $, $ \left|\mathrm{TQ}\right\rangle $=$\left|CD0\right\rangle $, $\left ( A, B, C, D\in {(0, 1)}\right )$ 时的输出

    Table 3.  Output when the input is $ \left|\mathrm{CQ}\right\rangle $=$\left|AB0\right\rangle $, $ \left|\mathrm{TQ}\right\rangle $=$\left|CD0\right\rangle $, $\left( A, B, C, D\in {(0, 1)}\right )$

    测量结果($M_{1}$$M_{2}$$M_{3}$) 000 001 010 011
    $ \left|\mathrm{CQ}\right\rangle\otimes\left|\mathrm{INT}\right\rangle\otimes\left|\mathrm{TQ}\right\rangle $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD0\right\rangle $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD0\right\rangle $
    测量结果($M_{1}$$M_{2}$$M_{3}$) 100 101 110 111
    $ \left|\mathrm{CQ}\right\rangle\otimes\mathrm{\left|INT\right\rangle}\otimes\mathrm{\left|TQ\right\rangle} $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD0\right\rangle $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD0\right\rangle $
    下载: 导出CSV

    表 6  $ \mathrm{\left|CQ\right\rangle} $=$\left|AB0\right\rangle $, $ \mathrm{\left|TQ\right\rangle} $=$\left|CD1\right\rangle $, $\left (A, B, C, D\in {(0, 1)}\right) $时的输出

    Table 6.  Output when the input is $ \mathrm{\left|CQ\right\rangle} $=$\left|AB0\right\rangle $, $ \mathrm{\left|TQ\right\rangle} $=$\left|CD1\right\rangle $, $\left (A, B, C, D\in {(0, 1)}\right) $

    测量结果($M_{1}$$M_{2}$$M_{3}$) 000 001 010 011
    $ \mathrm{\left|CQ\right\rangle\otimes\left|INT\right\rangle\otimes\left|TQ\right\rangle} $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD1\right\rangle $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD1\right\rangle $
    测量结果($M_{1}$$M_{2}$$M_{3}$) 100 101 110 111
    $ \mathrm{\left|CQ\right\rangle\otimes\left|INT\right\rangle\otimes\left|TQ\right\rangle} $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD1\right\rangle $ $\left|AB0\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB0\right\rangle \left|1\right\rangle \left|CD1\right\rangle $
    下载: 导出CSV

    表 4  $ \mathrm{\left|CQ\right\rangle} $=$\left|AB1\right\rangle $, $ \mathrm{\left|TQ\right\rangle} $=$\left|CD1\right\rangle $, $\left ( A, B, C, D\in {(0, 1)}\right )$ 时的输出

    Table 4.  Output when the input is ${\mathrm{\left|{{CQ}}\right\rangle}} $=$\left|AB1\right\rangle $, $\left|{\mathrm{TQ}}\right\rangle $=$\left|CD1\right\rangle $, $\left( A, B, C, D\in {(0, 1)}\right )$

    测量结果($M_{1}$$M_{2}$$M_{3}$) 000 001 010 011
    $ \mathrm{\left|CQ\right\rangle\otimes\left|INT\right\rangle\otimes\left|TQ\right\rangle} $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD0\right\rangle $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD0\right\rangle $
    测量结果($M_{1}$$M_{2}$$M_{3}$) 100 101 110 111
    $ \mathrm{\left|CQ\right\rangle\otimes\left|INT\right\rangle\otimes\left|TQ\right\rangle} $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD0\right\rangle $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD0\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD0\right\rangle $
    下载: 导出CSV

    表 5  $ \mathrm{\left|CQ\right\rangle} $=$\left|AB1\right\rangle $, $ \mathrm{\left|TQ\right\rangle} $=$\left|CD0\right\rangle $, $\left ( A, B, C, D\in {(0, 1)}\right )$时的输出

    Table 5.  Output when the input is $ \mathrm{\left|CQ\right\rangle} $=$\left|AB1\right\rangle $, $ \mathrm{\left|TQ\right\rangle} $=$\left|CD0\right\rangle $, $\left ( A, B, C, D\in {(0, 1)}\right )$

    测量结果($M_{1}$$M_{2}$$M_{3}$) 000 001 010 011
    $ \mathrm{\left|CQ\right\rangle\otimes\left|INT\right\rangle\otimes\left|TQ\right\rangle} $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD1\right\rangle $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD1\right\rangle $
    测量结果($M_{1}$$M_{2}$$M_{3}$) 100 101 110 111
    $ \mathrm{\left|CQ\right\rangle\otimes\left|INT\right\rangle\otimes\left|TQ\right\rangle} $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD1\right\rangle $ $\left|AB1\right\rangle \left|0\right\rangle \left|CD1\right\rangle $ $\left|AB1\right\rangle \left|1\right\rangle \left|CD1\right\rangle $
    下载: 导出CSV

    表 7  两种逻辑CNOT门实现方法的资源消耗对比

    Table 7.  Comparison of the resource consumption of the two logic CNOT gate implementation methods

    基于联合测量和
    逻辑测量的方法    		 基于晶格融合
    与分割的方法
    辅助表面码的码距 3 4
    辅助表面码的数据量子
    比特数目    		 13 25
    量子门数目
    (不含纠正操作)    		 19 40
    测量次数 3 15
    最大纠正次数 2 15
    下载: 导出CSV

    表 A1  图3所示的编码线路的8种输出码字

    Table A1.  Eight output codewords for the encoding circuit shown in Fig. 3

    000 001 010 011 100 101 110 111
    $|000000000000\rangle$ $|010000110011\rangle$ $|000000000101\rangle$ $|010000110110\rangle$ $|000000010010\rangle$ $|010000100001\rangle$ $|000000010111\rangle$ $|010000100100\rangle$
    $|000000001011\rangle$ $|010000111000\rangle$ $|000000001110\rangle$ $|010000111101\rangle$ $|000000011001\rangle$ $|010000101010\rangle$ $|000000011100\rangle$ $|010000101111\rangle$
    $|000010100100\rangle$ $|010010010111\rangle$ $|000010100001\rangle$ $|010010010010\rangle$ $|000010110110\rangle$ $|010010000101\rangle$ $|000010110011\rangle$ $|010010000000\rangle$
    $|000010101111\rangle$ $|010010011100\rangle$ $|000010101010\rangle$ $|010010011001\rangle$ $|000010111101\rangle$ $|010010001110\rangle$ $|000010111000\rangle$ $|010010001011\rangle$
    $|000101100011\rangle$ $|010101010000\rangle$ $|000101100110\rangle$ $|010101010101\rangle$ $|000101110001\rangle$ $|010101000010\rangle$ $|000101110100\rangle$ $|010101000111\rangle$
    $|000101101000\rangle$ $|010101011011\rangle$ $|000101101101\rangle$ $|010101011110\rangle$ $|000101111010\rangle$ $|010101001001\rangle$ $|000101111111\rangle$ $|010101001100\rangle$
    $|000111000111\rangle$ $|010111110100\rangle$ $|000111000010\rangle$ $|010111110001\rangle$ $|000111010101\rangle$ $|010111100110\rangle$ $|000111010000\rangle$ $|010111100011\rangle$
    $|000111001100\rangle$ $|010111111111\rangle$ $|000111001001\rangle$ $|010111111010\rangle$ $|000111011110\rangle$ $|010111101101\rangle$ $|000111011011\rangle$ $|010111101000\rangle$
    $|001001010000\rangle$ $|011001100011\rangle$ $|001001010101\rangle$ $|011001100110\rangle$ $|001001000010\rangle$ $|011001110001\rangle$ $|001001000111\rangle$ $|011001110100\rangle$
    $|001001011011\rangle$ $|011001101000\rangle$ $|001001011110\rangle$ $|011001101101\rangle$ $|001001001001\rangle$ $|011001111010\rangle$ $|001001001100\rangle$ $|011001111111\rangle$
    $|001011110100\rangle$ $|011011000111\rangle$ $|001011110001\rangle$ $|011011000010\rangle$ $|001011100110\rangle$ $|011011010101\rangle$ $|001011100011\rangle$ $|011011010000\rangle$
    $|001011111111\rangle$ $|011011001100\rangle$ $|001011111010\rangle$ $|011011001001\rangle$ $|001011101101\rangle$ $|011011011110\rangle$ $|001011101000\rangle$ $|011011011011\rangle$
    $|001100110011\rangle$ $|011100000000\rangle$ $|001100110110\rangle$ $|011100000101\rangle$ $|001100100001\rangle$ $|011100010010\rangle$ $|001100100100\rangle$ $|011100010111\rangle$
    $|001100111000\rangle$ $|011100001011\rangle$ $|001100111101\rangle$ $|011100001110\rangle$ $|001100101010\rangle$ $|011100011001\rangle$ $|001100101111\rangle$ $|011100011100\rangle$
    $|001110010111\rangle$ $|011110100100\rangle$ $|001110010010\rangle$ $|011110100001\rangle$ $|001110000101\rangle$ $|011110110110\rangle$ $|001110000000\rangle$ $|011110110011\rangle$
    $|001110011100\rangle$ $|011110101111\rangle$ $|001110011001\rangle$ $|011110101010\rangle$ $|001110001110\rangle$ $|011110111101\rangle$ $|001110001011\rangle$ $|011110111000\rangle$
    $|110001100011\rangle$ $|100001010000\rangle$ $|110001100110\rangle$ $|100001010101\rangle$ $|110001110001\rangle$ $|100001000010\rangle$ $|110001110100\rangle$ $|100001000111\rangle$
    $|110001101000\rangle$ $|100001011011\rangle$ $|110001101101\rangle$ $|100001011110\rangle$ $|110001111010\rangle$ $|100001001001\rangle$ $|110001111111\rangle$ $|100001001100\rangle$
    $|110011000111\rangle$ $|100011110100\rangle$ $|110011000010\rangle$ $|100011110001\rangle$ $|110011010101\rangle$ $|100011100110\rangle$ $|110011010000\rangle$ $|100011100011\rangle$
    $|110011001100\rangle$ $|100011111111\rangle$ $|110011001001\rangle$ $|100011111010\rangle$ $|110011011110\rangle$ $|100011101101\rangle$ $|110011011011\rangle$ $|100011101000\rangle$
    $|110100000000\rangle$ $|100100110011\rangle$ $|110100000101\rangle$ $|100100110110\rangle$ $|110100010010\rangle$ $|100100100001\rangle$ $|110100010111\rangle$ $|100100100100\rangle$
    $|110100001011\rangle$ $|100100111000\rangle$ $|110100001110\rangle$ $|100100111101\rangle$ $|110100011001\rangle$ $|100100101010\rangle$ $|110100011100\rangle$ $|100100101111\rangle$
    $|110110100100\rangle$ $|100110010111\rangle$ $|110110100001\rangle$ $|100110010010\rangle$ $|110110110110\rangle$ $|100110000101\rangle$ $|110110110011\rangle$ $|100110000000\rangle$
    $|110110101111\rangle$ $|100110011100\rangle$ $|110110101010\rangle$ $|100110011001\rangle$ $|110110111101\rangle$ $|100110001110\rangle$ $|110110111000\rangle$ $|100110001011\rangle$
    $|111000110011\rangle$ $|101000000000\rangle$ $|111000110110\rangle$ $|101000000101\rangle$ $|111000100001\rangle$ $|101000010010\rangle$ $|111000100100\rangle$ $|101000010111\rangle$
    $|111000111000\rangle$ $|101000001011\rangle$ $|111000111101\rangle$ $|101000001110\rangle$ $|111000101010\rangle$ $|101000011001\rangle$ $|111000101111\rangle$ $|101000011100\rangle$
    $|111010010111\rangle$ $|101010100100\rangle$ $|111010010010\rangle$ $|101010100001\rangle$ $|111010000101\rangle$ $|101010110110\rangle$ $|111010000000\rangle$ $|101010110011\rangle$
    $|111010011100\rangle$ $|101010101111\rangle$ $|111010011001\rangle$ $|101010101010\rangle$ $|111010001110\rangle$ $|101010111101\rangle$ $|111010001011\rangle$ $|101010111000\rangle$
    $|111101010000\rangle$ $|101101100011\rangle$ $|111101010101\rangle$ $|101101100110\rangle$ $|111101000010\rangle$ $|101101110001\rangle$ $|111101000111\rangle$ $|101101110100\rangle$
    $|111101011011\rangle$ $|101101101000\rangle$ $|111101011110\rangle$ $|101101101101\rangle$ $|111101001001\rangle$ $|101101111010\rangle$ $|111101001100\rangle$ $|101101111111\rangle$
    $|111111110100\rangle$ $|101111000111\rangle$ $|111111110001\rangle$ $|101111000010\rangle$ $|111111100110\rangle$ $|101111010101\rangle$ $|111111100011\rangle$ $|101111010000\rangle$
    $|111111111111\rangle$ $|101111001100\rangle$ $|111111111010\rangle$ $|101111001001\rangle$ $|111111101101\rangle$ $|101111011110\rangle$ $|111111101000\rangle$ $|101111011011\rangle$
    下载: 导出CSV
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  • [1]

    Feynman R P 1982 Int. J. Theor. Phys. 21 467Google Scholar

    [2]

    Shor P W 1999 SIREV 41 303Google Scholar

    [3]

    Preskill J 2012 arXiv: 1203.5813v3 [quant-ph
    [4]

    张诗豪, 张向东, 李绿周 2021 70 210301Google Scholar

    Zhang S H, Zhang X D, Li L Z 2021 Acta Phys. Sin. 70 210301Google Scholar

    [5]

    周文豪, 王耀, 翁文康, 金贤敏 2022 71 240302Google Scholar

    Zhou W H, Wang Y, Weng W K, Jin X M 2022 Acta Phys. Sin. 71 240302Google Scholar

    [6]

    宋克慧 2005 54 4730Google Scholar

    Song K H 2005 Acta Phys. Sin. 54 4730Google Scholar

    [7]

    Grover L 1996 Proc. 28th ACM Symp. Theo. Comp. 212
    [8]

    范桁 2023 72 070303Google Scholar

    Fan H 2023 Acta Phys. Sin. 72 070303Google Scholar

    [9]

    Shor P W 1995 Phys. Rev. A 52 2493Google Scholar

    [10]

    Steane A M 1996 Phys. Rev. Lett. 77 793Google Scholar

    [11]

    Frank A, Kunal A, Ryan B, et al. 2019 Nature 574 505Google Scholar

    [12]

    Davide C 2023 Nature 618 656
    [13]

    Deng Y H, Gu Y C, Liu H L, Gong S Q, Su H, Zhang Z J, Tang H Y, Jia M H, Xu J M, Chen M C, Qin J, Peng L C, Yan J R, Hu Y, Huang J, Li H, Li Y X, Chen Y J, Jiang X, Gan L, Yang G W, You L X, Li L, Zhong H S, Wang H, Liu N L, Renema J J, Lu C Y, Pan J W 2023 Phys. Rev. Lett. 131 150601Google Scholar

    [14]

    Huang J S, Chen X J, Li X D, Wang J W 2023 AAPPS Bull. 14 33
    [15]

    Fowler A G, Mariantoni M, Martinis J M, Cleland A N 2012 Phys. Rev. A 86 032324Google Scholar

    [16]

    Horsman C, Fowler A G, Devitt S, van Meter R 2012 New J. Phys. 14 123011Google Scholar

    [17]

    Kitaev A Y 1997 Quantum Communication, Computing, and Measurement (New York: Plenum Press) pp181–188
    [18]

    Kitaev A Y 1997 Russ. Math. Surv. 52 1191Google Scholar

    [19]

    Kitaev A Y 2003 Ann. Phys. 303 2Google Scholar

    [20]

    Bravyi S B, Kitaev A Y 1998 arXiv: 9811052 v1 [quant-ph
    [21]

    Freedman M H, Meyer D A 2001 Found Comput. Math. 1 325Google Scholar

    [22]

    Wang C Y, Harrington J, Preskill J 2003 Ann. Phys. 303 31Google Scholar

    [23]

    Raussendorf R, Harrington J, Goyal K 2006 Ann. Phys. 321 2242Google Scholar

    [24]

    邢莉娟, 李卓, 白宝明, 王新梅 2008 57 4695Google Scholar

    Xing L J, Li Z, Bai B M, Wang X M 2008 Acta Phys. Sin. 57 4695Google Scholar

    [25]

    Fowler A G, Stephens A M, Groszkowski P 2009 Phys. Rev. A 80 052312Google Scholar

    [26]

    DiVincenzo D P 2009 Phys. Scr. 137 014020
    [27]

    Tomita Y, Svore K M 2014 Phys. Rev. A 90 062320Google Scholar

    [28]

    Brown B J, Laubscher K, Kesselring M S, Wootton J R 2017 Phys. Rev. X 7 021029Google Scholar

    [29]

    Litinski D, von Oppen F 2018 Quantum 2 62Google Scholar

    [30]

    Krylov G, Lukac M 2018 arXiv: 1809.11134v1 [quant-ph
    [31]

    Beaudrap de N, Horsman D 2020 Quantum 4 218Google Scholar

    [32]

    Camps D, van Beeumen R 2020 Phys. Rev. A 102 052411Google Scholar

    [33]

    Shirakawa T, Ueda H, Yunoki S 2021 arXiv: 2112.14524v1 [quant-ph
    [34]

    Wang H W, Xue Y J, Ma Y L, Hua N, Ma H Y 2022 Chin. Phys. B 31 010303Google Scholar

    [35]

    Marques J F, Varbanov B M, Moreira M S, Ali H, Muthusubramanian N, Zachariadis C, Battistel F, Beekman M, Haider N, Vlothuizen W, Bruno A, Terhal B M, DiCarlo L 2022 Nat. Phys. 18 80Google Scholar

    [36]

    Kumari K, Rajpoot G, Ranjan Jain S 2022 arXiv: 2211. 12695v4 [quant-ph
    [37]

    Chen P H, Yan B W, Cui S X 2022 arXiv: 2210.01682v2 [cond-mat.str-el
    [38]

    Chen X B, Zhao L Y, Xu G, Pan X B, Chen S Y, Cheng Z W, Yang Y X 2022 Chin. Phys. B 31 040305Google Scholar

    [39]

    Xue Y J, Wang H W, Tian Y B, Wang Y N, Wang Y X, Wang S M 2022 Quantum Eng. 2022 9
    [40]

    Ding L, Wang H W, Wang Y N, Wang S M 2022 Quantum Eng. 2022 8
    [41]

    Siegel A, Strikis A, Flatters T, Benjamin S 2023 Quantum 7 1065Google Scholar

    [42]

    Quan D X, Liu C S, Lü X J, Pei C X 2022 Entropy 24 1107Google Scholar

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出版历程
  • 收稿日期:  2023-07-14
  • 修回日期:  2023-11-21
  • 上网日期:  2023-12-22
  • 刊出日期:  2024-02-20

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