Research advances in continuous-variable quantum computation and quantum error correction

Wang Mei-Hong; Hao Shu-Hong; Qin Zhong-Zhong; Su Xiao-Long

doi:10.7498/aps.71.20220635

Abstract

Quantum computation presents incomparable advantages over classical computer in solving some complex problems. To realize large-scale quantum computation, it is required to establish a hardware platform that is universal, scalable and fault tolerant. Continuous-variable optical system, which has unique advantages, is a feasible way to realize large-scale quantum computation and has attracted much attention in recent years. Measurement-based continuous-variable quantum computation realizes the computation by performing the measurement and feedforward of measurement results in large-scale Gaussian cluster states, and it provides an efficient method to realize quantum computation. Quantum error correction is an important part in quantum computation and quantum communication to protect quantum information. This review briefly introduces the basic principles and research advances in one-way quantum computation based on cluster states, quantum computation based on optical Schrödinger cat states and quantum error correction with continuous variables, and discusses the problems and challenges that the continuous-variable quantum computation is facing.

Keywords:

Authors and contacts

1.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
2.
School of Mathematics and Physics, Anhui University of Technology, Maanshan 243000, China

Corresponding author: Su Xiao-Long, suxl@sxu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11834010, 62005149, 11804001, 11974227), the Fund for Shanxi “1331 Project” Key Subjects Construction, China, and the Fundamental Research Program of Shanxi Province, China (Grant Nos. 20210302121002, 20210302122002, 201901D211164).

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References

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Cited By

图 1 平衡零拍探测系统示意图^[36]

Figure 1. Schematic of balance homodyne detection^[36] .

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图 2 平移算符对量子态的作用效果^[39]

Figure 2. Effect of displacement operation on quantum state^[39]

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图 3 多组份cluster态示意图 (a) 四组份线性cluster态^[45]; (b) 二维cluster态^[45]; (c) 三维cluster态^[45]

Figure 3. Schematic of multipartite cluster entangled states: (a) Linear four-mode cluster state^[45]; (b) two-dimensional cluster state^[45]; (c) three-dimensional cluster state^[45].

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图 4 立方位相门 (a)基于立方位相态实现立方位相门的线路图^[77]; (b)立方位相态的产生方案^[37]

Figure 4. Cubic phase gate: (a) The cubic phase gate by the measurement-induced scheme using the cubic phase state^[77]; (b) the preparation of the cubic phase state^[37].

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图 5 基于光学猫态的Hadamard门方案示意图^[104]

Figure 5. Schematic of Hadamard gate based on optical cat state^[104].

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图 6 基于光学猫态的位相旋转门方案示意图^[95]

Figure 6. Schematic of phase rotation gate based on optical cat state^[95].

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图 7 基于光学猫态的可控位相门方案示意图^[95]

Figure 7. Schematic of controlled phase gate based on optical cat state^[95].

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图 8 基于五波包部分编码方式的连续变量量子纠错方案^[118]

Figure 8. Scheme of CV quantum error correction with five-wave-packet code^[118].

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图 9 GKP量子比特的编码方式^[77]

Figure 9. The codeword for the GKP qubit^[77].

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图 10 基于cluster态的连续变量拓扑误差修正方案 (a) 八组份拓扑结构连续变量cluster 纠缠态的图态表示^[135]; (b) 产生八组份连续变量cluster 纠缠态的分束器网络^[135]

Figure 10. Scheme of topological error correction with CV a Gaussian cluster state: (a) The graph structure of the topological eight-partite CV cluster state; (b) the beam-splitter network for the preparation of the cluster state^[135].

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表 1 离散变量和连续变量量子逻辑门的比较^[37]

Table 1. Comparison between quantum logical gates with describe variables and continuous variables^[37].

	离散变量 (qubits)	连续变量 (qumodes)
计算基矢	$ \{{ \|0 \rangle }_{\mathrm{L}}, { \|1 \rangle }_{\mathrm{L}} \} $	$ \{{{ \|s \rangle }_{x}\}}_{\mathrm{s}\in \mathbb{R}} $
共轭基矢	$ \big\{{{ \|\pm \rangle }_{\mathrm{L}}=( \|0 \rangle }_{\mathrm{L}}\pm { \|1 \rangle }_{\mathrm{L}})/\sqrt{2} \big \} $	${ \bigg\{ { \|t \rangle }_{p}=\dfrac{1}{\sqrt{2\mathrm{\pi } } } \displaystyle\int_{-\infty }^{\infty }\mathrm{d}s{\mathrm{e} }^{\mathrm{i}st}{ \|s \rangle }_{x} \bigg\} }_{t\in \mathbb{R} }$
编码	$ { \|\psi \rangle =\alpha \|0 \rangle }_{\mathrm{L}}+\beta { \|1 \rangle }_{\mathrm{L}} $$ ({ \|\alpha \|}^{2}+{ \|\beta \|}^{2}=1 $)	$\|\psi \rangle = \displaystyle\int_{-\infty }^{\infty }\mathrm{d}s\psi (s ){ \|s \rangle }_{x} \bigg(\displaystyle\int_{-\infty }^{\infty }\mathrm{d}s{ \|\psi (s ) \|}^{2}=1 \bigg)$
探测方式	光子探测	平衡零拍探测
量子逻辑门	Bit-flip: $ {\widehat{X} \|0 \rangle }_{\mathrm{L}}={ \|1 \rangle }_{\mathrm{L}}, {\widehat{X} \|1 \rangle }_{\mathrm{L}}={ \|0 \rangle }_{\mathrm{L}} $	x方向平移: $ \widehat{X} (v ){ \|s \rangle }_{x}={ \|s+v \rangle }_{x} $
	Phase-flip: $ {\widehat{Z} \|0 \rangle }_{\mathrm{L}}={ \|0 \rangle }_{\mathrm{L}}, {\widehat{Z} \|1 \rangle }_{\mathrm{L}}={- \|1 \rangle }_{\mathrm{L}} $	p方向平移: $ \widehat{Z} (u ){ \|t \rangle }_{p}={ \|t+u \rangle }_{p} $
	Hadamard门:$ {\widehat{H} \|0 \rangle }_{\mathrm{L}}={ \|+ \rangle }_{\mathrm{L}}, {\widehat{H} \|1 \rangle }_{\mathrm{L}}={ \|- \rangle }_{\mathrm{L}} $	傅立叶变换: $\widehat{R} ( {\mathrm{\pi } }/{2} ){ \|s \rangle }_{x}={ \|s \rangle }_{p}, \widehat{R} ( {\mathrm{\pi } }/{2} ){ \|t \rangle }_{p}={ \|-t \rangle }_{x}$
	可控非门: $ {\widehat{CX} \|0 \rangle }_{\mathrm{L}}{ \|0 (1 ) \rangle }_{\mathrm{L}}={ \|0 \rangle }_{\mathrm{L}}{ \|0 (1 ) \rangle }_{\mathrm{L}} $	可控X门: $ {\widehat{CX} \|{s}_{1} \rangle }_{{q}_{1}}{ \|{s}_{2} \rangle }_{{q}_{2}}={ \|{s}_{1} \rangle }_{{q}_{1}}{ \|{s}_{2}+{s}_{1} \rangle }_{{q}_{2}} $
	$ {\widehat{CX} \|1 \rangle }_{\mathrm{L}}{ \|0 (1 ) \rangle }_{\mathrm{L}}={ \|1 \rangle }_{\mathrm{L}}{ \|1 (0 ) \rangle }_{\mathrm{L}} $	$ {\widehat{CX} \|{t}_{1} \rangle }_{{p}_{1}}{ \|{t}_{2} \rangle }_{{p}_{2}}={ \|{t}_{1}-{t}_{2} \rangle }_{{p}_{1}}{ \|{t}_{2} \rangle }_{{p}_{2}} $

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	离散变量 (qubits)	连续变量 (qumodes)
计算基矢	$ \{{ \|0 \rangle }_{\mathrm{L}}, { \|1 \rangle }_{\mathrm{L}} \} $	$ \{{{ \|s \rangle }_{x}\}}_{\mathrm{s}\in \mathbb{R}} $
共轭基矢	$ \big\{{{ \|\pm \rangle }_{\mathrm{L}}=( \|0 \rangle }_{\mathrm{L}}\pm { \|1 \rangle }_{\mathrm{L}})/\sqrt{2} \big \} $	${ \bigg\{ { \|t \rangle }_{p}=\dfrac{1}{\sqrt{2\mathrm{\pi } } } \displaystyle\int_{-\infty }^{\infty }\mathrm{d}s{\mathrm{e} }^{\mathrm{i}st}{ \|s \rangle }_{x} \bigg\} }_{t\in \mathbb{R} }$
编码	$ { \|\psi \rangle =\alpha \|0 \rangle }_{\mathrm{L}}+\beta { \|1 \rangle }_{\mathrm{L}} $$ ({ \|\alpha \|}^{2}+{ \|\beta \|}^{2}=1 $)	$\|\psi \rangle = \displaystyle\int_{-\infty }^{\infty }\mathrm{d}s\psi (s ){ \|s \rangle }_{x} \bigg(\displaystyle\int_{-\infty }^{\infty }\mathrm{d}s{ \|\psi (s ) \|}^{2}=1 \bigg)$
探测方式	光子探测	平衡零拍探测
量子逻辑门	Bit-flip: $ {\widehat{X} \|0 \rangle }_{\mathrm{L}}={ \|1 \rangle }_{\mathrm{L}}, {\widehat{X} \|1 \rangle }_{\mathrm{L}}={ \|0 \rangle }_{\mathrm{L}} $	x方向平移: $ \widehat{X} (v ){ \|s \rangle }_{x}={ \|s+v \rangle }_{x} $
	Phase-flip: $ {\widehat{Z} \|0 \rangle }_{\mathrm{L}}={ \|0 \rangle }_{\mathrm{L}}, {\widehat{Z} \|1 \rangle }_{\mathrm{L}}={- \|1 \rangle }_{\mathrm{L}} $	p方向平移: $ \widehat{Z} (u ){ \|t \rangle }_{p}={ \|t+u \rangle }_{p} $
	Hadamard门:$ {\widehat{H} \|0 \rangle }_{\mathrm{L}}={ \|+ \rangle }_{\mathrm{L}}, {\widehat{H} \|1 \rangle }_{\mathrm{L}}={ \|- \rangle }_{\mathrm{L}} $	傅立叶变换: $\widehat{R} ( {\mathrm{\pi } }/{2} ){ \|s \rangle }_{x}={ \|s \rangle }_{p}, \widehat{R} ( {\mathrm{\pi } }/{2} ){ \|t \rangle }_{p}={ \|-t \rangle }_{x}$
	可控非门: $ {\widehat{CX} \|0 \rangle }_{\mathrm{L}}{ \|0 (1 ) \rangle }_{\mathrm{L}}={ \|0 \rangle }_{\mathrm{L}}{ \|0 (1 ) \rangle }_{\mathrm{L}} $	可控X门: $ {\widehat{CX} \|{s}_{1} \rangle }_{{q}_{1}}{ \|{s}_{2} \rangle }_{{q}_{2}}={ \|{s}_{1} \rangle }_{{q}_{1}}{ \|{s}_{2}+{s}_{1} \rangle }_{{q}_{2}} $
	$ {\widehat{CX} \|1 \rangle }_{\mathrm{L}}{ \|0 (1 ) \rangle }_{\mathrm{L}}={ \|1 \rangle }_{\mathrm{L}}{ \|1 (0 ) \rangle }_{\mathrm{L}} $	$ {\widehat{CX} \|{t}_{1} \rangle }_{{p}_{1}}{ \|{t}_{2} \rangle }_{{p}_{2}}={ \|{t}_{1}-{t}_{2} \rangle }_{{p}_{1}}{ \|{t}_{2} \rangle }_{{p}_{2}} $

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