连续变量量子计算和量子纠错研究进展

王美红; 郝树宏; 秦忠忠; 苏晓龙

doi:10.7498/aps.71.20220635

摘要

量子计算机在解决某些复杂问题方面具有经典计算机无法比拟的优势. 实现大规模量子计算需建立具有通用性、可扩展性和容错性的硬件平台. 连续变量光学系统具有独特的优势, 是实现大规模量子计算的一种可行途径, 近年来受到了广泛关注. 基于测量的连续变量量子计算通过对大规模高斯簇态(cluster态)的测量和测量结果的前馈来实现计算, 为实现量子计算提供了一条可行的途径. 量子纠错是量子计算和量子通信中保护量子信息的重要环节. 本文简要介绍了基于cluster态的单向量子计算、基于光学薛定谔猫态的量子计算和连续变量量子纠错的基本原理和研究进展, 并讨论了连续变量量子计算面临的问题和挑战.

关键词:

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:

作者及机构信息

1.
山西大学光电研究所, 量子光学与光量子器件国家重点实验室, 极端光学协同创新中心, 太原　030006

2.
安徽工业大学数理科学与工程学院, 马鞍山　243000

通信作者: 苏晓龙, suxl@sxu.edu.cn

基金项目: 国家自然科学基金(批准号: 11834010, 62005149, 11804001, 11974227)、山西省“1331 工程”重点学科建设经费和山西省基础研究计划(批准号: 20210302121002, 20210302122002, 201901D211164)资助的课题

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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施引文献

图 1 平衡零拍探测系统示意图^[36]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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]

Fig. 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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