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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.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 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
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
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
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 $
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
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
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
$X_{{\mathrm{L}}}$ $Z_{{\mathrm{L}}}$ $X_{{\mathrm{L}}1}=X_{1}X_{3}$ $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 $
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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 $
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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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