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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.
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Keywords:
- quantum surface code /
- multi-logical-qubits encoding /
- logic CNOT gate /
- fusion operation /
- segmentation operation
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图 1 3 × 3表面码的结构图
Figure 1. Structure diagram of 3 × 3 surface code
图 2 基于边界编码三位逻辑比特的表面码结构
Figure 2. Structure of surface code based on boundary encoding three logical qubits
图 4 基于联合测量的单量子比特间CNOT门的实现原理
Figure 4. CNOT gate implementation for single qubit based on joint measurement
图 5 表面码的粗糙融合
Figure 5. Rough fusion of surface codes
图 6 表面码的粗糙分割
Figure 6. Rough segmentation of surface codes
图 7 表面码的光滑融合
Figure 7. Smooth fusion of surface codes
图 8 表面码的光滑分割
Figure 8. Smooth segmentation of surface codes
图 9 单逻辑比特量子表面码逻辑CNOT门的实现
Figure 9. Implementation of logic CNOT gate for the single logical qubit surface code
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 $
Figure 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 $
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
Figure 11. Implementation of logic CNOT gate for multiple logical qubits surface code based on joint measuremnt and logical measurement
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
Figure 12. Optimization of logic CNOT gate for multiple logical qubits surface code based on joint measuremnt and logical measurement
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
Figure 13. Optimization of logic CNOT gate when both control qubit and target qubit are the 3rd logical qubits
$X_{{\mathrm{L}}}$ $Z_{{\mathrm{L}}}$ $X_{{\mathrm{L}}1}=X_{1}X_{3}$ $Z_{{\mathrm{L}}1}=Z_{1}Z_{2} $ $X_{{\mathrm{L}}2}=X_{10}X_{12}$ $Z_{{\mathrm{L}}2}=Z_{5}Z_{10}$ $X_{{\mathrm{L}}3}=X_{8}X_{11}$ $Z_{{\mathrm{L}}3}=Z_{8}Z_{6} Z_{4} Z_{2}$ 
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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 $\alpha \left |01 \right \rangle (m\left |0 \right \rangle+n\left |1 \right \rangle )-\beta \left | 11 \right \rangle(m\left | 1 \right \rangle+n\left | 0 \right \rangle)$
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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 $
DownLoad: 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 $
DownLoad: 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 $
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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 $
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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
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000 001 010 011 100 101 110 111 $|000000000000\rangle$ $|010000110011\rangle$ $|000000000101\rangle$ $|010000110110\rangle$ $|000000010010\rangle$ $|010000100001\rangle$ $|000000010111\rangle$ $|010000100100\rangle$ $|000000001011\rangle$ $|010000111000\rangle$ $|000000001110\rangle$ $|010000111101\rangle$ $|000000011001\rangle$ $|010000101010\rangle$ $|000000011100\rangle$ $|010000101111\rangle$ $|000010100100\rangle$ $|010010010111\rangle$ $|000010100001\rangle$ $|010010010010\rangle$ $|000010110110\rangle$ $|010010000101\rangle$ $|000010110011\rangle$ $|010010000000\rangle$ $|000010101111\rangle$ $|010010011100\rangle$ $|000010101010\rangle$ $|010010011001\rangle$ $|000010111101\rangle$ $|010010001110\rangle$ $|000010111000\rangle$ $|010010001011\rangle$ $|000101100011\rangle$ $|010101010000\rangle$ $|000101100110\rangle$ $|010101010101\rangle$ $|000101110001\rangle$ $|010101000010\rangle$ $|000101110100\rangle$ $|010101000111\rangle$ $|000101101000\rangle$ $|010101011011\rangle$ $|000101101101\rangle$ $|010101011110\rangle$ $|000101111010\rangle$ $|010101001001\rangle$ $|000101111111\rangle$ $|010101001100\rangle$ $|000111000111\rangle$ $|010111110100\rangle$ $|000111000010\rangle$ $|010111110001\rangle$ $|000111010101\rangle$ $|010111100110\rangle$ $|000111010000\rangle$ $|010111100011\rangle$ $|000111001100\rangle$ $|010111111111\rangle$ $|000111001001\rangle$ $|010111111010\rangle$ $|000111011110\rangle$ $|010111101101\rangle$ $|000111011011\rangle$ $|010111101000\rangle$ $|001001010000\rangle$ $|011001100011\rangle$ $|001001010101\rangle$ $|011001100110\rangle$ $|001001000010\rangle$ $|011001110001\rangle$ $|001001000111\rangle$ $|011001110100\rangle$ $|001001011011\rangle$ $|011001101000\rangle$ $|001001011110\rangle$ $|011001101101\rangle$ $|001001001001\rangle$ $|011001111010\rangle$ $|001001001100\rangle$ $|011001111111\rangle$ $|001011110100\rangle$ $|011011000111\rangle$ $|001011110001\rangle$ $|011011000010\rangle$ $|001011100110\rangle$ $|011011010101\rangle$ $|001011100011\rangle$ $|011011010000\rangle$ $|001011111111\rangle$ $|011011001100\rangle$ $|001011111010\rangle$ $|011011001001\rangle$ $|001011101101\rangle$ $|011011011110\rangle$ $|001011101000\rangle$ $|011011011011\rangle$ $|001100110011\rangle$ $|011100000000\rangle$ $|001100110110\rangle$ $|011100000101\rangle$ $|001100100001\rangle$ $|011100010010\rangle$ $|001100100100\rangle$ $|011100010111\rangle$ $|001100111000\rangle$ $|011100001011\rangle$ $|001100111101\rangle$ $|011100001110\rangle$ $|001100101010\rangle$ $|011100011001\rangle$ $|001100101111\rangle$ $|011100011100\rangle$ $|001110010111\rangle$ $|011110100100\rangle$ $|001110010010\rangle$ $|011110100001\rangle$ $|001110000101\rangle$ $|011110110110\rangle$ $|001110000000\rangle$ $|011110110011\rangle$ $|001110011100\rangle$ $|011110101111\rangle$ $|001110011001\rangle$ $|011110101010\rangle$ $|001110001110\rangle$ $|011110111101\rangle$ $|001110001011\rangle$ $|011110111000\rangle$ $|110001100011\rangle$ $|100001010000\rangle$ $|110001100110\rangle$ $|100001010101\rangle$ $|110001110001\rangle$ $|100001000010\rangle$ $|110001110100\rangle$ $|100001000111\rangle$ $|110001101000\rangle$ $|100001011011\rangle$ $|110001101101\rangle$ $|100001011110\rangle$ $|110001111010\rangle$ $|100001001001\rangle$ $|110001111111\rangle$ $|100001001100\rangle$ $|110011000111\rangle$ $|100011110100\rangle$ $|110011000010\rangle$ $|100011110001\rangle$ $|110011010101\rangle$ $|100011100110\rangle$ $|110011010000\rangle$ $|100011100011\rangle$ $|110011001100\rangle$ $|100011111111\rangle$ $|110011001001\rangle$ $|100011111010\rangle$ $|110011011110\rangle$ $|100011101101\rangle$ $|110011011011\rangle$ $|100011101000\rangle$ $|110100000000\rangle$ $|100100110011\rangle$ $|110100000101\rangle$ $|100100110110\rangle$ $|110100010010\rangle$ $|100100100001\rangle$ $|110100010111\rangle$ $|100100100100\rangle$ $|110100001011\rangle$ $|100100111000\rangle$ $|110100001110\rangle$ $|100100111101\rangle$ $|110100011001\rangle$ $|100100101010\rangle$ $|110100011100\rangle$ $|100100101111\rangle$ $|110110100100\rangle$ $|100110010111\rangle$ $|110110100001\rangle$ $|100110010010\rangle$ $|110110110110\rangle$ $|100110000101\rangle$ $|110110110011\rangle$ $|100110000000\rangle$ $|110110101111\rangle$ $|100110011100\rangle$ $|110110101010\rangle$ $|100110011001\rangle$ $|110110111101\rangle$ $|100110001110\rangle$ $|110110111000\rangle$ $|100110001011\rangle$ $|111000110011\rangle$ $|101000000000\rangle$ $|111000110110\rangle$ $|101000000101\rangle$ $|111000100001\rangle$ $|101000010010\rangle$ $|111000100100\rangle$ $|101000010111\rangle$ $|111000111000\rangle$ $|101000001011\rangle$ $|111000111101\rangle$ $|101000001110\rangle$ $|111000101010\rangle$ $|101000011001\rangle$ $|111000101111\rangle$ $|101000011100\rangle$ $|111010010111\rangle$ $|101010100100\rangle$ $|111010010010\rangle$ $|101010100001\rangle$ $|111010000101\rangle$ $|101010110110\rangle$ $|111010000000\rangle$ $|101010110011\rangle$ $|111010011100\rangle$ $|101010101111\rangle$ $|111010011001\rangle$ $|101010101010\rangle$ $|111010001110\rangle$ $|101010111101\rangle$ $|111010001011\rangle$ $|101010111000\rangle$ $|111101010000\rangle$ $|101101100011\rangle$ $|111101010101\rangle$ $|101101100110\rangle$ $|111101000010\rangle$ $|101101110001\rangle$ $|111101000111\rangle$ $|101101110100\rangle$ $|111101011011\rangle$ $|101101101000\rangle$ $|111101011110\rangle$ $|101101101101\rangle$ $|111101001001\rangle$ $|101101111010\rangle$ $|111101001100\rangle$ $|101101111111\rangle$ $|111111110100\rangle$ $|101111000111\rangle$ $|111111110001\rangle$ $|101111000010\rangle$ $|111111100110\rangle$ $|101111010101\rangle$ $|111111100011\rangle$ $|101111010000\rangle$ $|111111111111\rangle$ $|101111001100\rangle$ $|111111111010\rangle$ $|101111001001\rangle$ $|111111101101\rangle$ $|101111011110\rangle$ $|111111101000\rangle$ $|101111011011\rangle$
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[1] Feynman R P 1982 Int. J. Theor. Phys. 21 467
Google Scholar
[2] Shor P W 1999 SIREV 41 303
Google Scholar
[3] Preskill J 2012 arXiv: 1203.5813v3 [quant-ph
[4] 张诗豪, 张向东, 李绿周 2021 物理学报 70 210301
Google Scholar
Zhang S H, Zhang X D, Li L Z 2021 Acta Phys. Sin. 70 210301
Google Scholar
[5] 周文豪, 王耀, 翁文康, 金贤敏 2022 物理学报 71 240302
Google Scholar
Zhou W H, Wang Y, Weng W K, Jin X M 2022 Acta Phys. Sin. 71 240302
Google Scholar
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Google Scholar
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