Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Effects of quantum noise on Shor’s algorithm

Huang Tian-Long Wu Yong-Zheng Ni Ming Wang Shi Ye Yong-Jin

Citation:

Effects of quantum noise on Shor’s algorithm

Huang Tian-Long, Wu Yong-Zheng, Ni Ming, Wang Shi, Ye Yong-Jin
PDF
HTML
Get Citation
  • Shor’s quantum factoring algorithm (Shor’s algorithm) can solve factorization problem of large integers by using a fully-operational quantum computer with the complexity of polynomial-time level, thereby cracking a series of encryption algorithms (such as Rivest-Shamir-Adleman encryption algorithm, and Diffie-Hellman key exchange protocol) whose security is guaranteed by factorizing large integers, which is a difficult problem. We are currently in a noisy intermediate-scale quantum era, which means that we can only operate on quantum computers with a limited number of qubits and we have to take care of the effects of quantum noise. Quantum states on a quantum computer are prone to quantum noise caused by low-fidelity gates or interactions between qubits and the environment, which results in inaccurate measurements. We study the influence of quantum noise on Shor’s algorithm through 3 typical quantum noise channels: the depolarizing channel, the state preparation and measurement channel, and the thermal relaxation channel. We successfully simulate the factorization of the numbers 15, 21, and 35 into their corresponding prime factors by using the quantum circuit we have constructed on a classical computer. Then we simulate a running quantum circuit of Shor’s algorithm in a noisy environment with different level of noise for a certain type of noise channel and present numerical results. We can obtain precise measurements by calculating the state vector prior to measurement, instead of simulating and measuring expending much time, which contributes to higher efficiency. Each experiment is repeated 1000 times to reduce discrepancy. Our research indicates that Shor’s algorithm is easily affected by quantum noise. Successful rate of Shor’s algorithm decreases exponentially with the increase of noise level in the depolarizing channel, where the successful rate is an indicator we propose in this research to quantify the influence of noise on Shor’s algorithm, meanwhile the noise in the state preparation and measurement channel and the thermal relaxation channel can linearly affect the successful rate of Shor’s algorithm. There are $O(n^4) $ quantum gates in the circuit, each of which is disrupted by noise in depolarizing channel during running the circuit, meanwhile there are only O(n) interruptions caused by noise in state preparation and measurement channel since we repeat the measurements only O(n) times in the circuit where n is the number of bits of the integer about to be factored. Linear relationship in thermal relaxation channel is mainly due to the large gap between quantum gate time and relaxation time even if each gate in the circuit is disrupted by noise in thermal relaxation channel such as depolarizing channel. The present research results can be used for correcting the subsequent errors, improving Shor’s algorithm, and providing guidance for the fidelity required in engineering implementation of Shor’s algorithm.
      Corresponding author: Wu Yong-Zheng, yzwu15@fudan.edu.cn
    [1]

    Shor P W 1994 Proceedings of the 35th Annual Symposium on Foundations of Computer Science Washington DC, USA, November 20–22, 1994 p124

    [2]

    Shor P W 1999 SIAM Rev. Soc. Ind. Appl. Math 41 303

    [3]

    Lenstra A K, Hendrik Jr W 1993 The Development of the Number Field Sieve(Vol. 1554) (Heidelberg : Springer Science & Business Media) p5

    [4]

    Lenstra A K, Lenstra Jr H W, Manasse M S, Pollard J M 1990 Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing Baltimore Maryland, USA, May 13–17, 1990 p564

    [5]

    Buhler J P, Lenstra H W, Pomerance C 1993 The Development of the Number Field Sieve (Berlin Heidelberg: Springer) pp50–94

    [6]

    Kleinjung T, Aoki K, Franke J, Lenstra A K, Thomé E, Bos J W, Gaudry P, Kruppa A, Montgomery P L, Osvik D A, Riele H T, Timofeev A, Zimmermann P 2010 Advances in Cryptology–CRYPTO 2010: 30th Annual Cryptology Conference Santa Barbara, CA, USA, August 15–19, 2010 p333

    [7]

    Gidney C, Ekerå M 2021 Quantum 5 433Google Scholar

    [8]

    Preskill J 2018 Quantum 2 79Google Scholar

    [9]

    Harper R, Flammia S T, Wallman J J 2020 Nat. Phys. 16 1184Google Scholar

    [10]

    Georgopoulos K, Emary C, Zuliani P 2021 Phys. Rev. A 104 062432Google Scholar

    [11]

    Brown K R, Harrow A W, Chuang I L 2004 Phys. Rev. A 70 052318Google Scholar

    [12]

    Bassi A, Großardt A, Ulbricht H 2017 Classical Quantum Gravity 34 193002Google Scholar

    [13]

    Viola L, Knill E, Lloyd S 1999 Phys. Rev. Lett. 82 2417Google Scholar

    [14]

    Vedral V, Barenco A, Ekert A 1996 Phys. Rev. A 54 147Google Scholar

    [15]

    Draper T G 2000 arXiv: 0008033 v1[quant-ph

    [16]

    Beauregard S 2002 arXiv: 0205095 v3[quant-ph

    [17]

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

    [18]

    O’Gorman J, Campbell E T 2017 Phys. Rev. A 95 032338Google Scholar

    [19]

    Hwang Y, Kim T, Baek C, Choi B S 2020 Phys. Rev. Appl. 13 054033Google Scholar

    [20]

    Ha J, Lee J, Heo J 2022 Quantum Inf. Process. 21 60Google Scholar

    [21]

    Horsman D, Fowler A G, Devitt S, Van M R 2012 New J. Phys. 14 123011Google Scholar

    [22]

    Gidney C 2019 arXiv: 1905.07682 v1[quant-ph

    [23]

    Xiao L, Qiu D, Luo L, Mateus P 2022 arXiv: 2207.05976 v1[quant-ph

    [24]

    Rossi M, Asproni L, Caputo D, Rossi S, Cusinato A, Marini R, Agosti A, Magagnini M 2022 Quant. Mach. Intell. 4 18Google Scholar

    [25]

    Bogdanov Y I, Chernyavskiy A Y, Holevo A, Lukichev V F, Orlikovsky A A 2013 International Conference Micro-and Nano-Electronics Zvenigorod, Russian Federation, October 1–5, 2012 p404

    [26]

    Nachman B, Urbanek M, de Jong W A, Bauer C W 2020 NPJ Quantum Inf. 6 84Google Scholar

    [27]

    Xue C, Chen Z Y, Wu Y C, Guo G P 2021 Chin. Phys. Lett. 38 030302Google Scholar

    [28]

    Farhi E, Goldstone J, Gutmann S 2014 arXiv: 1411.4028 v1[quant-ph

    [29]

    Wallman J J, Emerson J 2016 Phys. Rev. A 94 052325Google Scholar

    [30]

    Nielsen M A, Chuang I L 2010 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) pp226–386

    [31]

    Wilde M M 2013 Quantum Information Theory (Cambridge: Cambridge University Press) pp175–176

    [32]

    Ji Z, Wang G, Duan R, Feng Y, Ying M 2008 IEEE Trans. Inf. Theory 54 5172Google Scholar

    [33]

    Ryan-Anderson C, Bohnet J G, Lee K, Gresh D, Hankin A, Gaebler J P, Stutz R P 2021 Phys. Rev. X 11 041058

    [34]

    Aliferis P, Preskill J 2008 Phys. Rev. A 78 052331Google Scholar

    [35]

    Tuckett D K, Bartlett S D, Flammia S T 2018 Phys. Rev. Lett. 120 050505Google Scholar

    [36]

    Coppersmith D 2002 arXiv: 0201067 v1[quant-ph

    [37]

    Ekert A, Jozsa R 1996 Rev. Mod. Phys. 68 733Google Scholar

    [38]

    Jozsa R 1998 Proc. R. Soc. London, Ser. A 454 323Google Scholar

    [39]

    Portugal R 2022 arXiv: 2201.10574 v5[quant-ph

    [40]

    King C 2003 IEEE Trans. Inf. Theory 49 221Google Scholar

    [41]

    Resch S, Karpuzcu U R 2021 ACM Comput. Surv. 54 1

    [42]

    Gottesman D 2009 arXiv: 0904.2557 v1[quant-ph

    [43]

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

    [44]

    McKay D C, Alexander T, Bello L, et al. 2018 arXiv: 1809.03452 v1[quant-ph

  • 图 1  Shor算法中的量子线路

    Figure 1.  Quantum circuit of Shor’s algorithm.

    图 2  Shor算法分解(N = 21, a = 2)实验中$ \left| {{\psi _4}} \right\rangle $的预期概率分布

    Figure 2.  Expected probability distribution of $ \left| {{\psi _4}} \right\rangle $ in factoring (N = 21, a = 2).

    图 3  去极化通道中单比特门噪声对(N = 15, a = 2), (N = 21, a = 2), (N = 35, a = 2)中$ \left| {{\psi _4}} \right\rangle $概率分布的影响

    Figure 3.  Effect of one-qubit gate noise in DC on probability distribution of $ \left| {{\psi _4}} \right\rangle $ in (N = 15, a = 2), (N = 21, a = 2), (N = 35, a = 2)

    图 4  去极化通道中双比特门噪声对(N = 15, a = 2), (N = 21, a = 2), (N = 35, a = 2)中$ \left| {{\psi _4}} \right\rangle $概率分布的影响

    Figure 4.  Effect of two-qubit gate noise in DC on probability distribution of $ \left| {{\psi _4}} \right\rangle $ in (N = 15, a = 2), (N = 21, a = 2), (N = 35, a = 2)

    图 5  (a) $ P_{\mathrm{s}} $与去极化通道中单比特门噪声的关系; (b) $ P_{\mathrm{s}} $与去极化通道中双比特门噪声的关系; (c) $ P_{\mathrm{s}} $与状态制备与测量通道中噪声的关系

    Figure 5.  (a) Effect of one-qubit gate noise in DC on $ P_{\mathrm{s}} $; (b) effect of two-qubit gate noise in DC on $ P_{\mathrm{s}} $; (c) effect of noise in SPAM channel on $ P_{\mathrm{s}} $.

    图 6  (a) MSE与去极化通道中单比特门噪声的关系; (b) MSE与去极化通道中双比特门噪声的关系; (c) MSE与状态制备与测量通道中噪声的关系

    Figure 6.  (a) Effect of one-qubit gate noise in DC on MSE; (b) effect of two-qubit gate noise in DC on MSE; (c) effect of noise in SPAM channel on MSE.

    图 7  状态制备与测量通道中噪声对(N = 15, a = 2), (N = 21, a = 2), (N = 35, a = 2)中$ \left| {{\psi _4}} \right\rangle $概率分布的影响

    Figure 7.  Effect of noise in SPAM channel on probability distribution of $ \left| {{\psi _4}} \right\rangle $ in (N = 15 a = 2), (N = 21, a = 2) and (N = 35, a = 2)

    图 8  热退相干通道中不同$ T_1, \ T_2 $对(N = 15, a = 2), (N = 21, a = 2), (N = 35, a = 2) 中的$ \left| {{\psi _4}} \right\rangle $概率分布的影响

    Figure 8.  Different $ T_1, \ T_2 $ of TRC on probability distribution of $ \left| {{\psi _4}} \right\rangle $ in (N = 15, a = 2), (N = 21, a = 2) and (N = 35, a = 2).

    图 9  $ P_{\mathrm{s }}$与热退相干通道的噪声的关系

    Figure 9.  Effect of noise in TRC on $ P_{\mathrm{s}} $.

    图 10  MSE与热退相干通道的噪声的关系

    Figure 10.  Effect of noise in TRC on MSE.

    图 B1  分解实验中$ P_{\mathrm{s}} $与去极化通道中单比特门噪声的关系 (a) N = 15; (b) N = 21; (c) N = 35

    Figure B1.  Effect of noise of one-qubit gate in DC on $ P_{\mathrm{s}} $ of factoring: (a) N = 15; (b) N = 21; (c) N = 35.

    图 B2  分解实验中MSE与去极化通道中单比特门噪声的关系 (a) N = 15; (b) N = 21; (c) N = 35

    Figure B2.  Effect of noise of one-qubit gate in DC on MSE of factoring: (a) N = 15; (b) N = 21; (c) N = 35.

    图 B3  分解实验中$ P_{\mathrm{s }}$与去极化通道中双比特门噪声的关系 (a) N = 15; (b) N = 21; (c) N = 35

    Figure B3.  Effect of noise of two-qubit gate in DC on $ P_{\mathrm{s}} $ of factoring: (a) N = 15; (b) N = 21; (c) N = 35.

    图 B4  分解实验中MSE与去极化通道中双比特门噪声的关系 (a) N = 15; (b) N = 21; (c) N = 35

    Figure B4.  Effect of noise of two-qubit gate in DC on MSE of factoring: (a) N = 15; (b) N = 21; (c) N = 35.

    图 B5  分解实验中$ P_{\mathrm{s }}$与状态制备与测量通道中噪声的关系 (a) N = 15; (b) N = 21; (c) N = 35

    Figure B5.  Effect of noise in SPAM channel on $ P_{\mathrm{s}} $ of factoring: (a) N = 15; (b) N = 21; (c) N = 35.

    图 B6  分解实验中MSE与状态制备与测量通道中噪声的关系 (a) N = 15; (b) N = 21; (c) N = 35

    Figure B6.  Effect of noise in SPAM channel on MSE of factoring: (a) N = 15; (b) N = 21; (c) N = 35.

    图 B7  分解N = 15, N = 21, N = 35实验中退相干噪声与$ P_{\mathrm{s}} $的关系

    Figure B7.  Effect of noise in TRC on $ P_{\rm s} $ of factoring N = 15, N = 21, N = 35.

    图 B8  分解N = 15, N = 21, N = 35实验中退相干噪声与MSE的关系

    Figure B8.  Effect of noise in TRC on MSE of factoring N = 15, N = 21, N = 35.

    表 1  Shor算法流程

    Table 1.  Pseudo of Shor’s algorithm

    算法 1 Shor算法
    输入: 待分解的整数N
    输出: N的非平凡素因子p, q
    1: if N is even then
    2:   return $2, {N}/{2}$
    3: end if
    4: if $N=p^k$, where $p$ is prime, $k > 0$ then
    5:   return $p$
    6: end if
    7: select a random number a, where $1 < a < N$
    8: if $\mathrm{gcd}(a, N)>1$ then
    9:   return $\mathrm{gcd}(a, N)$, $ {N}/{{\rm{gcd}}(a, N)}$
    10: end if
    11: excute quantum circuit, find minimal positive period of $f_a(x)=a^x\ {\rm{mod}}\ N$, denoted as r
    12: if r is odd or $a^{\frac{r}{2}}\equiv -1\ ({\rm{mod}}\ N)$ then
    13:   GOTO 7
    14: end if
    15: calculate $p = \mathrm{gcd}(a^{\frac{r}{2}} - 1, N)$ and $q = \mathrm{gcd}(a^{\frac{r}{2}} + 1, N) $
    16: return $p, q$
    DownLoad: CSV

    表 2  实验选择的待分解整数与随机数组合(N, a)

    Table 2.  Combination of integer about to be factored and random number (N, a).

    N a
    15 [2, 4, 7]
    21 [2, 8, 11]
    35 [2, 4, 9]
    DownLoad: CSV

    表 3  退相干通道中$T_1, T_2$的取值

    Table 3.  Choice of $T_1, T_2$ in TRC.

    $T_1/\text{μ} {\rm{s}}$ $T_2/\text{μ} {\rm{s}}$
    $[20, 30, \cdots, 90, 110]$ 30
    $[30, 40, \cdots, 100, 120]$ 50
    $[40, 50, \cdots, 110, 130]$ 70
    $[50, 60, \cdots, 120, 140]$ 90
    $[60, 70, \cdots, 130, 150]$ 110
    $[70, 80, \cdots, 140, 160]$ 130
    $[80, 90, \cdots, 150, 170]$ 150
    $[90, 100, \cdots, 160, 180]$ 170
    $[100, 110, \cdots, 170, 190]$ 190
    DownLoad: CSV

    表 B1  所有整数与随机数组合(N, a)在不同噪声的环境下的成功率

    Table B1.  Effect of different levels of noise in 3 channels on success rates of all (N, a) pairs.

    N 15 21 35
    a 2 4 7 2 8 11 2 4 9
    单比特门数量 1584 2860 4680
    双比特门数量 8476 17045 30594
    线路深度 6432 12059 20565
    线路宽度(量子比特数) 18 22 26
    平均运行一次所需时间/s 1.15 11.89 142.34
    理论成功率(无噪声) 0.75 0.5 0.75 0.4559 0.5 0.4559 0.4559 0.4559 0.4559
    噪声环境中
    的成功率
    $P_1=0.0025$ 0.2635 0.1593 0.2125 0.0408 0.0482 0.0327 0.0085 0.0058 0.0037
    $P_1=0.005$ 0.0964 0.0526 0.0776 0.0089 0.0078 0.0085 0.0029 0.0013 0.0014
    $P_1=0.0075$ 0.0455 0.0204 0.0358 0.0052 0.0031 0.0051 0.0021 0.0011 0.0011
    $P_1=0.01$ 0.0295 0.0142 0.0278 0.0046 0.0019 0.0044 0.0020 0.0010 0.0010
    $P_2=0.00075$ 0.1017 0.0512 0.0787 0.0073 0.0062 0.0082 0.0025 0.0012 0.0012
    $P_2=0.0015$ 0.0284 0.0124 0.0255 0.0042 0.0015 0.0042 0.0019 0.0009 0.0010
    $P_2=0.00225$ 0.0161 0.0064 0.0158 0.0039 0.0011 0.0039 0.0019 0.0008 0.0009
    $P_2=0.003$ 0.0137 0.0048 0.0138 0.0038 0.0010 0.0038 0.0018 0.0008 0.0009
    $P_{\rm{spam}}=0.05$ 0.5513 0.3441 0.5295 0.2721 0.3596 0.3373 0.3029 0.2739 0.3079
    $P_{\rm{spam}}=0.1$ 0.4058 0.2354 0.4155 0.1823 0.1603 0.1438 0.1335 0.1223 0.1645
    $P_{\rm{spam}}=0.15$ 0.2805 0.1812 0.2955 0.1066 0.1226 0.0758 0.0937 0.0427 0.0680
    $P_{\rm{spam}}=0.2$ 0.2131 0.1193 0.2025 0.0705 0.0816 0.0721 0.0763 0.0125 0.0572
    $T_1=T_2=10~{\text{μs}}$ 0.0792 0.0386 0.0646 0.0067 0.0041 0.0083 0.0029 0.0014 0.0016
    $T_1=T_2=70~{\text{μs}}$ 0.5455 0.3444 0.5497 0.1556 0.2740 0.1963 0.1004 0.0722 0.1019
    $T_1=T_2= 130~\text{μs}$ 0.6314 0.3992 0.6238 0.2714 0.3579 0.2811 0.2202 0.1972 0.2007
    $T_1=T_2= 190\text{μs}$ 0.6603 0.4770 0.6513 0.3690 0.4521 0.3960 0.2998 0.3226 0.3014
    DownLoad: CSV
    Baidu
  • [1]

    Shor P W 1994 Proceedings of the 35th Annual Symposium on Foundations of Computer Science Washington DC, USA, November 20–22, 1994 p124

    [2]

    Shor P W 1999 SIAM Rev. Soc. Ind. Appl. Math 41 303

    [3]

    Lenstra A K, Hendrik Jr W 1993 The Development of the Number Field Sieve(Vol. 1554) (Heidelberg : Springer Science & Business Media) p5

    [4]

    Lenstra A K, Lenstra Jr H W, Manasse M S, Pollard J M 1990 Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing Baltimore Maryland, USA, May 13–17, 1990 p564

    [5]

    Buhler J P, Lenstra H W, Pomerance C 1993 The Development of the Number Field Sieve (Berlin Heidelberg: Springer) pp50–94

    [6]

    Kleinjung T, Aoki K, Franke J, Lenstra A K, Thomé E, Bos J W, Gaudry P, Kruppa A, Montgomery P L, Osvik D A, Riele H T, Timofeev A, Zimmermann P 2010 Advances in Cryptology–CRYPTO 2010: 30th Annual Cryptology Conference Santa Barbara, CA, USA, August 15–19, 2010 p333

    [7]

    Gidney C, Ekerå M 2021 Quantum 5 433Google Scholar

    [8]

    Preskill J 2018 Quantum 2 79Google Scholar

    [9]

    Harper R, Flammia S T, Wallman J J 2020 Nat. Phys. 16 1184Google Scholar

    [10]

    Georgopoulos K, Emary C, Zuliani P 2021 Phys. Rev. A 104 062432Google Scholar

    [11]

    Brown K R, Harrow A W, Chuang I L 2004 Phys. Rev. A 70 052318Google Scholar

    [12]

    Bassi A, Großardt A, Ulbricht H 2017 Classical Quantum Gravity 34 193002Google Scholar

    [13]

    Viola L, Knill E, Lloyd S 1999 Phys. Rev. Lett. 82 2417Google Scholar

    [14]

    Vedral V, Barenco A, Ekert A 1996 Phys. Rev. A 54 147Google Scholar

    [15]

    Draper T G 2000 arXiv: 0008033 v1[quant-ph

    [16]

    Beauregard S 2002 arXiv: 0205095 v3[quant-ph

    [17]

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

    [18]

    O’Gorman J, Campbell E T 2017 Phys. Rev. A 95 032338Google Scholar

    [19]

    Hwang Y, Kim T, Baek C, Choi B S 2020 Phys. Rev. Appl. 13 054033Google Scholar

    [20]

    Ha J, Lee J, Heo J 2022 Quantum Inf. Process. 21 60Google Scholar

    [21]

    Horsman D, Fowler A G, Devitt S, Van M R 2012 New J. Phys. 14 123011Google Scholar

    [22]

    Gidney C 2019 arXiv: 1905.07682 v1[quant-ph

    [23]

    Xiao L, Qiu D, Luo L, Mateus P 2022 arXiv: 2207.05976 v1[quant-ph

    [24]

    Rossi M, Asproni L, Caputo D, Rossi S, Cusinato A, Marini R, Agosti A, Magagnini M 2022 Quant. Mach. Intell. 4 18Google Scholar

    [25]

    Bogdanov Y I, Chernyavskiy A Y, Holevo A, Lukichev V F, Orlikovsky A A 2013 International Conference Micro-and Nano-Electronics Zvenigorod, Russian Federation, October 1–5, 2012 p404

    [26]

    Nachman B, Urbanek M, de Jong W A, Bauer C W 2020 NPJ Quantum Inf. 6 84Google Scholar

    [27]

    Xue C, Chen Z Y, Wu Y C, Guo G P 2021 Chin. Phys. Lett. 38 030302Google Scholar

    [28]

    Farhi E, Goldstone J, Gutmann S 2014 arXiv: 1411.4028 v1[quant-ph

    [29]

    Wallman J J, Emerson J 2016 Phys. Rev. A 94 052325Google Scholar

    [30]

    Nielsen M A, Chuang I L 2010 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) pp226–386

    [31]

    Wilde M M 2013 Quantum Information Theory (Cambridge: Cambridge University Press) pp175–176

    [32]

    Ji Z, Wang G, Duan R, Feng Y, Ying M 2008 IEEE Trans. Inf. Theory 54 5172Google Scholar

    [33]

    Ryan-Anderson C, Bohnet J G, Lee K, Gresh D, Hankin A, Gaebler J P, Stutz R P 2021 Phys. Rev. X 11 041058

    [34]

    Aliferis P, Preskill J 2008 Phys. Rev. A 78 052331Google Scholar

    [35]

    Tuckett D K, Bartlett S D, Flammia S T 2018 Phys. Rev. Lett. 120 050505Google Scholar

    [36]

    Coppersmith D 2002 arXiv: 0201067 v1[quant-ph

    [37]

    Ekert A, Jozsa R 1996 Rev. Mod. Phys. 68 733Google Scholar

    [38]

    Jozsa R 1998 Proc. R. Soc. London, Ser. A 454 323Google Scholar

    [39]

    Portugal R 2022 arXiv: 2201.10574 v5[quant-ph

    [40]

    King C 2003 IEEE Trans. Inf. Theory 49 221Google Scholar

    [41]

    Resch S, Karpuzcu U R 2021 ACM Comput. Surv. 54 1

    [42]

    Gottesman D 2009 arXiv: 0904.2557 v1[quant-ph

    [43]

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

    [44]

    McKay D C, Alexander T, Bello L, et al. 2018 arXiv: 1809.03452 v1[quant-ph

  • [1] Li Tian-Yin, Xing Hong-Xi, Zhang Dan-Bo. Quantum computing based high-energy nuclear physics. Acta Physica Sinica, 2023, 72(20): 200303. doi: 10.7498/aps.72.20230907
    [2] Wang Chen-Xu, He Ran, Li Rui-Rui, Chen Yan, Fang Ding, Cui Jin-Ming, Huang Yun-Feng, Li Chuan-Feng, Guo Guang-Can. Advances in the study of ion trap structures in quantum computation and simulation. Acta Physica Sinica, 2022, 71(13): 133701. doi: 10.7498/aps.71.20220224
    [3] Wang Ning, Wang Bao-Chuan, Guo Guo-Ping. New progress of silicon-based semiconductor quantum computation. Acta Physica Sinica, 2022, 71(23): 230301. doi: 10.7498/aps.71.20221900
    [4] Zhou Zong-Quan. “Quantum memory” quantum computers and noiseless phton echoes. Acta Physica Sinica, 2022, 71(7): 070305. doi: 10.7498/aps.71.20212245
    [5] Fan Hong-Yi, Wu Ze. Classical correspondence of quantum entanglement in mesoscopic circuit. Acta Physica Sinica, 2022, 71(1): 010302. doi: 10.7498/aps.71.20210992
    [6] Zhou Wen-Hao, Wang Yao, Weng Wen-Kang, Jin Xian-Min. Research progress of integrated optical quantum computing. Acta Physica Sinica, 2022, 71(24): 240302. doi: 10.7498/aps.71.20221782
    [7] Zhang Jie-Yin, Gao Fei, Zhang Jian-Jun. Research progress of silicon and germanium quantum computing materials. Acta Physica Sinica, 2021, 70(21): 217802. doi: 10.7498/aps.70.20211492
    [8] Zhang Shi-Hao, Zhang Xiang-Dong, Li Lü-Zhou. Research progress of measurement-based quantum computation. Acta Physica Sinica, 2021, 70(21): 210301. doi: 10.7498/aps.70.20210923
    [9] Classical correspondence of quantum entanglement in mesoscopic circuit. Acta Physica Sinica, 2021, (): . doi: 10.7498/aps.70.20210992
    [10] Ding Chen, Li Tan, Zhang Shuo, Guo Chu, Huang He-Liang, Bao Wan-Su. A quantum state readout method based on a single ancilla qubit. Acta Physica Sinica, 2021, 70(21): 210303. doi: 10.7498/aps.70.20211066
    [11] Chen Ran-Yi-Liu, Zhao Ben-Chi, Song Zhi-Xin, Zhao Xuan-Qiang, Wang Kun, Wang Xin. Hybrid quantum-classical algorithms: Foundation, design and applications. Acta Physica Sinica, 2021, 70(21): 210302. doi: 10.7498/aps.70.20210985
    [12] Wang Fu-Rong, Yang Fan, Zhang Ya, Li Shi-Zhong, Wang He-Feng. Matrix low-rank approximate quantum algorithm based on singular value decomposition. Acta Physica Sinica, 2021, 70(15): 150201. doi: 10.7498/aps.70.20210411
    [13] Lin Jian, Ye Meng, Zhu Jia-Wei, Li Xiao-Peng. Machine learning assisted quantum adiabatic algorithm design. Acta Physica Sinica, 2021, 70(14): 140306. doi: 10.7498/aps.70.20210831
    [14] Yang Le, Li Kai, Dai Hong-Yi, Zhang Ming. A novel scheme of quantum state tomography based on quantum algorithms. Acta Physica Sinica, 2019, 68(14): 140301. doi: 10.7498/aps.68.20190157
    [15] Tian Yu-Ling, Feng Tian-Feng, Zhou Xiao-Qi. Collaborative quantum computation with redundant graph state. Acta Physica Sinica, 2019, 68(11): 110302. doi: 10.7498/aps.68.20190142
    [16] Fan Heng. Quantum computation and quantum simulation. Acta Physica Sinica, 2018, 67(12): 120301. doi: 10.7498/aps.67.20180710
    [17] Yang Guang, Lian Bao-Wang, Nie Min. Characteristics of multi-hop noisy quantum entanglement channel and optimal relay protocol. Acta Physica Sinica, 2015, 64(24): 240304. doi: 10.7498/aps.64.240304
    [18] Li Pan-Chi, Wang Hai-Ying, Dai Qing, Xiao Hong. Quantum process neural networks model algorithm and applications. Acta Physica Sinica, 2012, 61(16): 160303. doi: 10.7498/aps.61.160303
    [19] Li Pan-Chi, Wang Hai-Ying, Song Kao-Ping, Yang Er-Long. Research on the improvement of quantum potential well-based particle swarm optimization algorithm. Acta Physica Sinica, 2012, 61(6): 060302. doi: 10.7498/aps.61.060302
    [20] Lu Cui-Ping, Yuan Chun-Hua, Zhang Wei-Ping. The property of quantum noise in active Raman gain medium. Acta Physica Sinica, 2008, 57(11): 6976-6981. doi: 10.7498/aps.57.6976
Metrics
  • Abstract views:  2069
  • PDF Downloads:  90
  • Cited By: 0
Publishing process
  • Received Date:  03 September 2023
  • Accepted Date:  10 December 2023
  • Available Online:  04 January 2024
  • Published Online:  05 March 2024

/

返回文章
返回
Baidu
map