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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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$X_{{\mathrm{L}}}$ $Z_{{\mathrm{L}}}$ $X_{{\mathrm{L}}1}=X_{1}X_{3}$ $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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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 $\alpha \left |01 \right \rangle (m\left |0 \right \rangle+n\left |1 \right \rangle )-\beta \left | 11 \right \rangle(m\left | 1 \right \rangle+n\left | 0 \right \rangle)$ DownLoad:
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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:
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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
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 $ DownLoad:
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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 $ DownLoad:
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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 DownLoad:
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000 001 010 011 100 101 110 111 $|000000000000\rangle$ $|010000110011\rangle$ $|000000000101\rangle$ $|010000110110\rangle$ $|000000010010\rangle$ $|010000100001\rangle$ $|000000010111\rangle$ $|010000100100\rangle$ $|000000001011\rangle$ $|010000111000\rangle$ $|000000001110\rangle$ $|010000111101\rangle$ $|000000011001\rangle$ $|010000101010\rangle$ $|000000011100\rangle$ $|010000101111\rangle$ $|000010100100\rangle$ $|010010010111\rangle$ $|000010100001\rangle$ $|010010010010\rangle$ $|000010110110\rangle$ $|010010000101\rangle$ $|000010110011\rangle$ $|010010000000\rangle$ $|000010101111\rangle$ $|010010011100\rangle$ $|000010101010\rangle$ $|010010011001\rangle$ $|000010111101\rangle$ $|010010001110\rangle$ $|000010111000\rangle$ $|010010001011\rangle$ $|000101100011\rangle$ $|010101010000\rangle$ $|000101100110\rangle$ $|010101010101\rangle$ $|000101110001\rangle$ $|010101000010\rangle$ $|000101110100\rangle$ $|010101000111\rangle$ $|000101101000\rangle$ $|010101011011\rangle$ $|000101101101\rangle$ $|010101011110\rangle$ $|000101111010\rangle$ $|010101001001\rangle$ $|000101111111\rangle$ $|010101001100\rangle$ $|000111000111\rangle$ $|010111110100\rangle$ $|000111000010\rangle$ $|010111110001\rangle$ $|000111010101\rangle$ $|010111100110\rangle$ $|000111010000\rangle$ $|010111100011\rangle$ $|000111001100\rangle$ $|010111111111\rangle$ $|000111001001\rangle$ $|010111111010\rangle$ $|000111011110\rangle$ $|010111101101\rangle$ $|000111011011\rangle$ $|010111101000\rangle$ $|001001010000\rangle$ $|011001100011\rangle$ $|001001010101\rangle$ $|011001100110\rangle$ $|001001000010\rangle$ $|011001110001\rangle$ $|001001000111\rangle$ $|011001110100\rangle$ $|001001011011\rangle$ $|011001101000\rangle$ $|001001011110\rangle$ $|011001101101\rangle$ $|001001001001\rangle$ $|011001111010\rangle$ $|001001001100\rangle$ $|011001111111\rangle$ $|001011110100\rangle$ $|011011000111\rangle$ $|001011110001\rangle$ $|011011000010\rangle$ $|001011100110\rangle$ $|011011010101\rangle$ $|001011100011\rangle$ $|011011010000\rangle$ $|001011111111\rangle$ $|011011001100\rangle$ $|001011111010\rangle$ $|011011001001\rangle$ $|001011101101\rangle$ $|011011011110\rangle$ $|001011101000\rangle$ $|011011011011\rangle$ $|001100110011\rangle$ $|011100000000\rangle$ $|001100110110\rangle$ $|011100000101\rangle$ $|001100100001\rangle$ $|011100010010\rangle$ $|001100100100\rangle$ $|011100010111\rangle$ $|001100111000\rangle$ $|011100001011\rangle$ $|001100111101\rangle$ $|011100001110\rangle$ $|001100101010\rangle$ $|011100011001\rangle$ $|001100101111\rangle$ $|011100011100\rangle$ $|001110010111\rangle$ $|011110100100\rangle$ $|001110010010\rangle$ $|011110100001\rangle$ $|001110000101\rangle$ $|011110110110\rangle$ $|001110000000\rangle$ $|011110110011\rangle$ $|001110011100\rangle$ $|011110101111\rangle$ $|001110011001\rangle$ $|011110101010\rangle$ $|001110001110\rangle$ $|011110111101\rangle$ $|001110001011\rangle$ $|011110111000\rangle$ $|110001100011\rangle$ $|100001010000\rangle$ $|110001100110\rangle$ $|100001010101\rangle$ $|110001110001\rangle$ $|100001000010\rangle$ $|110001110100\rangle$ $|100001000111\rangle$ $|110001101000\rangle$ $|100001011011\rangle$ $|110001101101\rangle$ $|100001011110\rangle$ $|110001111010\rangle$ $|100001001001\rangle$ $|110001111111\rangle$ $|100001001100\rangle$ $|110011000111\rangle$ $|100011110100\rangle$ $|110011000010\rangle$ $|100011110001\rangle$ $|110011010101\rangle$ $|100011100110\rangle$ $|110011010000\rangle$ $|100011100011\rangle$ $|110011001100\rangle$ $|100011111111\rangle$ $|110011001001\rangle$ $|100011111010\rangle$ $|110011011110\rangle$ $|100011101101\rangle$ $|110011011011\rangle$ $|100011101000\rangle$ $|110100000000\rangle$ $|100100110011\rangle$ $|110100000101\rangle$ $|100100110110\rangle$ $|110100010010\rangle$ $|100100100001\rangle$ $|110100010111\rangle$ $|100100100100\rangle$ $|110100001011\rangle$ $|100100111000\rangle$ $|110100001110\rangle$ $|100100111101\rangle$ $|110100011001\rangle$ $|100100101010\rangle$ $|110100011100\rangle$ $|100100101111\rangle$ $|110110100100\rangle$ $|100110010111\rangle$ $|110110100001\rangle$ $|100110010010\rangle$ $|110110110110\rangle$ $|100110000101\rangle$ $|110110110011\rangle$ $|100110000000\rangle$ $|110110101111\rangle$ $|100110011100\rangle$ $|110110101010\rangle$ $|100110011001\rangle$ $|110110111101\rangle$ $|100110001110\rangle$ $|110110111000\rangle$ $|100110001011\rangle$ $|111000110011\rangle$ $|101000000000\rangle$ $|111000110110\rangle$ $|101000000101\rangle$ $|111000100001\rangle$ $|101000010010\rangle$ $|111000100100\rangle$ $|101000010111\rangle$ $|111000111000\rangle$ $|101000001011\rangle$ $|111000111101\rangle$ $|101000001110\rangle$ $|111000101010\rangle$ $|101000011001\rangle$ $|111000101111\rangle$ $|101000011100\rangle$ $|111010010111\rangle$ $|101010100100\rangle$ $|111010010010\rangle$ $|101010100001\rangle$ $|111010000101\rangle$ $|101010110110\rangle$ $|111010000000\rangle$ $|101010110011\rangle$ $|111010011100\rangle$ $|101010101111\rangle$ $|111010011001\rangle$ $|101010101010\rangle$ $|111010001110\rangle$ $|101010111101\rangle$ $|111010001011\rangle$ $|101010111000\rangle$ $|111101010000\rangle$ $|101101100011\rangle$ $|111101010101\rangle$ $|101101100110\rangle$ $|111101000010\rangle$ $|101101110001\rangle$ $|111101000111\rangle$ $|101101110100\rangle$ $|111101011011\rangle$ $|101101101000\rangle$ $|111101011110\rangle$ $|101101101101\rangle$ $|111101001001\rangle$ $|101101111010\rangle$ $|111101001100\rangle$ $|101101111111\rangle$ $|111111110100\rangle$ $|101111000111\rangle$ $|111111110001\rangle$ $|101111000010\rangle$ $|111111100110\rangle$ $|101111010101\rangle$ $|111111100011\rangle$ $|101111010000\rangle$ $|111111111111\rangle$ $|101111001100\rangle$ $|111111111010\rangle$ $|101111001001\rangle$ $|111111101101\rangle$ $|101111011110\rangle$ $|111111101000\rangle$ $|101111011011\rangle$ DownLoad:
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