Hybrid quantum-classical algorithms: Foundation, design and applications

Chen Ran-Yi-Liu; Zhao Ben-Chi; Song Zhi-Xin; Zhao Xuan-Qiang; Wang Kun; Wang Xin

doi:10.7498/aps.70.20210985

Abstract

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1. 引　言
2. 喷嘴设计
3. 流体模拟
4. 实验结果
5. 结　论

References

Abstract

Quantum computing, as an emerging computing paradigm, is expected to tackle problems such as quantum chemistry, optimization, quantum chemistry, information security, and artificial intelligence, which are intractable with using classical computing. Quantum computing hardware and software continue to develop rapidly, but they are not expected to realize universal quantum computation in the next few years. Therefore, the use of quantum hardware to solve practical problems in the near term has become a hot topic in the field of quantum computing. Exploration of the applications of near-term quantum hardware is of great significance in understanding the capability of quantum hardware and promoting the practical process of quantum computing. Hybrid quantum-classical algorithm (also known as variational quantum algorithm) is an appropriate model for near-term quantum hardware. In the hybrid quantum-classical algorithm, classical computers are used to maximize the power of quantum devices. By combining quantum computing with machine learning, the hybrid quantum-classical algorithm is expected to achieve the first practical application of quantum computation and play an important role in the studying of quantum computing. In this review, we introduce the framework of hybrid quantum-classical algorithm and its applications in quantum chemistry, quantum information, combinatorial optimization, quantum machine learning, and other fields. We further discuss the challenges and future research directions of the hybrid quantum-classical algorithm.

Keywords:

PACS:

52.38.Kd	(Laser-plasma acceleration of electrons and ions)
52.38.-r	(Laser-plasma interactions)

Authors and contacts

Institute for Quantum Computing, Baidu Research, Beijing 100193, China

Corresponding author: Wang Xin, wangxin73@baidu.com

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1. 引　言

激光尾场加速(laser wakefield acceleration, LWFA)是一种新型的加速技术^[1-4], 有望带来加速器领域的革命. 它利用超短脉冲激光在稀薄等离子体中产生的等离子体波加速电子, 可在数毫米的距离内将电子加速到数百兆电子伏特甚至数吉电子伏特. 相比传统射频场加速器, LWFA具有加速梯度高、脉冲短、焦斑小等诸多优势, 吸引了各国研究者的广泛兴趣. 经过40多年的发展, LWFA技术日渐成熟, 在电子束的能量^[5-9]、单能性^[10-13]、稳定性^[14]等指标上都达到了很高的水平. 然而, LWFA的重复频率相比传统加速器还有很大的差距. 目前射频场加速器的重复频率已经可以达到数兆赫兹, 尽管某些少周期激光器能够实现10 Hz甚至1 kHz的重频加速^[15], 但产生的电子能量和电量都比较低, 应用场景受限, 而多数LWFA实验的重复频率甚至不到1 Hz^[14,16-18]. 过低的重复频率已经成为限制LWFA应用的一个重要因素. 可以预见, 高重频的LWFA将是未来重要的发展方向之一.

除了激光器本身, 靶也是限制LWFA重复频率的重要因素. LWFA需要在真空中进行, 真空度一般要求优于10^–2 Pa. 但是LWFA需要用到气体靶, 它被激光预脉冲电离形成稀薄等离子体, 之后与激光主脉冲作用产生高能电子. 气体靶的使用将对真空系统产生影响. 为了保证一致性, 前一发打完后需要尽快把真空抽至10^–2 Pa以下才能进行下一发打靶, 因此喷气量就成了限制重复频率的一个重要因素. 喷气量主要由出口处的气体密度、流速、出口面积等因素决定. 目前尾场加速中常用的靶主要有超音速喷嘴^[19,20]、气室^[21-23]和放电毛细管^[7,9]几种. 采用放电毛细管进行高重频加速有很大的困难, 它口径较小容易被激光打坏, 因此不在这篇文章的讨论范围之内. 剩下的两种靶中, 超音速喷嘴一般长度较短, 比较适合用在中小规模的激光器上进行短距离、较低能量的电子加速, 而气室靶往往尺寸较大, 比较适合在大规模激光器上进行长距离、高能电子加速. 就喷气量而言, 超音速喷嘴的背压较高, 喷气量较大, 而气室靶背压低, 喷气量相对较小. 造成这种差异的主要原因在于两种靶的工作方式. 一般情况下LWFA需要气体密度呈平台状分布, 并且具有较短的上升沿和下降沿. 对于超音速喷嘴, 激光在喷嘴外的开放空间与气流作用, 出口处的长度和密度都由尾场加速的需求决定, 无法随意更改. 另一方面, 气流的速度必须足够高, 才能够抑制横向扩散, 实现所需的平台状密度分布. 上面这些因素决定了超音速喷嘴喷气量可以优化的空间十分有限. 在使用气室时, 激光在气室内部与气体作用, 带来以下好处: 1)气室的出气口只要能保证激光光斑通过即可, 基本不受等离子体长度影响, 所以出气口的面积可大幅减小; 2)出口处并非激光作用的主要区域, 这里的密度和流速可以进行专门优化. 除此之外, 由于气流被气室壁约束, 不需要很高的流速也可以实现平台状的密度分布, 流速降低还能减轻激波、紊流等因素的影响, 提高尾场加速的稳定性.

尽管气室在喷气量上具有一定的优势, 但是以往的气室长度往往在厘米量级, 更适合在中大型激光器上使用, 这类激光装置本身重复频率较低, 对高重频喷嘴的需求并不迫切. 而中小规模的激光器造价较低, 更容易在高校、医院等单位推广, 对重复频率的需求也更为迫切. 目前对气体靶喷气量的研究较为缺乏. 本文针对中小规模激光器设计了一种微气室喷嘴, 并通过流体程序模拟对比了这种微气室喷嘴和同尺寸超音速喷嘴在喷气量上的差异. 结果表明相比超音速喷嘴, 微气室喷嘴的喷气量减小了97%, 同时还能产生更长的平台区. 在随后开展的LWFA实验中, 采用这种微气室喷嘴还能够在多条件下产生高稳定度的电子束. 这将极大地提高LWFA中气体靶重复频率的上限.

2. 喷嘴设计

图1展示了超音速喷嘴(a)与微气室喷嘴(b)的截面示意图. 超音速喷嘴内部为缩放结构, 气流从底部进入, 从上方出口喷出, 激光(图中黑色虚箭头)在出口上方与气流作用. 微气室喷嘴内部为直筒结构, 上方没有开口, 气流进入内部后从左右两侧的开孔流出. 激光从开口处穿过与气体作用, 如图中黑色虚箭头所示. 两侧开孔设计成锥孔, 以便气体快速扩散, 缩短气体密度上升沿和下降沿的长度. 超音速喷嘴的喉部是影响气流分布的关键, 尺寸往往在亚毫米量级, 加工难度较大, 而这种微气室喷嘴结构更加简单, 更容易加工.

$图 1 超音速喷嘴(a)与微气室喷嘴(b)截面示意图\r\nFig. 1. Cross sections of the supersonic nozzle (a) and the micro gas cell (b).$

图 1 超音速喷嘴(a)与微气室喷嘴(b)截面示意图

Fig. 1. Cross sections of the supersonic nozzle (a) and the micro gas cell (b).

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3. 流体模拟

利用流体程序Fluent对微气室喷嘴和超音速喷嘴进行三维模拟, 对两者的喷气量进行分析. 微气室喷嘴外形为长方体, 壁厚1 mm, 内部挖了一个直径4 mm、高18 mm的圆筒. 锥孔为90°, 距喷嘴底部13 mm, 在内部直筒交接处形成了直径约0.5 mm的近似圆形注入孔, 而在外表面形成了直径2.5 mm的圆孔. 超音速喷嘴内部为缩放结构, 入口直径为3 mm, 出口直径为4 mm, 喉部直径为0.5 mm. 收缩段高度为3 mm, 扩张段高度为15 mm. 在真实场景中喷嘴需要连接脉冲电磁阀使用. 模拟中两个喷嘴下方都有一段直径0.7 mm、长2 mm的注入段, 代表电磁阀的出气孔. 利用气体流域的对称性, 只模拟其中1/4的区域. 气体采用N₂, 设为理想气体, 采用k-ε湍流模型. 调整入口的压力, 使N₂分子密度达到5×10¹⁷ cm^–3左右, 这里微气室喷嘴取的是锥孔高度, 超音速喷嘴取的是出口上方2 mm处的数据, 分别如图2(a)和图2(b)中黑色虚线所示. 假设N原子的外壳层电子全部被电离的话, 对应的等离子体密度大约在5×10¹⁸ cm^–3, 是尾场加速中常用的等离子体密度. 为了达到这样的密度, 微气室喷嘴入口压力是5 kPa, 而超音速喷嘴入口压力则达到了200 kPa. 图2(a)和图2(b)分别展示了两个喷嘴内部的密度分布, 请注意两者色标的差异. 图2(c)展示了这两个喷嘴在激光高度的密度分布(对应图2(a)和图2(b)中黑色虚线处). 可以看到, 微气室喷嘴内部气流密度非常均匀, 在锥孔高度上气流呈现了很好的平台型分布, 上升/下降沿的长度约300 μm. 在超音速喷嘴的例子中, 平台区长度不到1 mm, 上升/下降沿的长度超过了1.5 mm (需要说明的是, 这个例子采用了气体N₂, 背压较低, 出口流速低, 造成平台区较短. 在另外一个模拟中采用气体He在1 MPa下可以产生相近的气体密度, 平台区的长度达到了1.5 mm). 超音速喷嘴的喷气量达到了21.4 mg/s, 而微气室喷嘴只有0.73 mg/s, 比超音速喷嘴减小了97%, 这将极大地降低真空系统的负载.

$图 2 超音速喷嘴(a)与微气室(b)气流密度分布的模拟结果; (c)比较了两个喷嘴在激光传播方向上((a)和(b)中黑色虚线位置)的分子密度分布\r\nFig. 2. Density distribution of the supersonic nozzle (a) and the micro gas cell (b); (c) the molecular number density along the laser propagation direction (dash lines) are compared in (c).$

图 2 超音速喷嘴(a)与微气室(b)气流密度分布的模拟结果; (c)比较了两个喷嘴在激光传播方向上((a)和(b)中黑色虚线位置)的分子密度分布

Fig. 2. Density distribution of the supersonic nozzle (a) and the micro gas cell (b); (c) the molecular number density along the laser propagation direction (dash lines) are compared in (c).

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对影响喷气量的因素进行分析, 图3展示了超音速喷嘴出口((a)和(b))和微气室内壁锥孔处((c)和(d))的流速与密度分布(内壁锥孔更有代表性, 它必须远大于激光焦斑, 而外面表锥孔受锥角、壁厚等因素的影响, 不具代表性). 超音速喷嘴出口气体密度最高处为0.025 kg/m³, 流速最高达到721 m/s. 而微气室出口气体密度最高处为0.016 kg/m³, 流速最高达到270 m/s, 相较而言, 气流速度的下降幅度更为明显. 超音速喷嘴的出口面积约为12.5 mm², 微气室喷嘴两个出口面积总共是0.39 mm², 是超音速喷嘴的1/32. 不过从图3可以看出, 超音速喷嘴出口处高流速、高密度区域占出口面积的比例较低, 意味着等效出口面积并没有达到12.5 mm². 从上述结果可以看出, 微气室喷嘴喷气量降低的主要来源是出口面积和出口流速的减小.

$图 3 超音速喷嘴出口(a), (b)和微气室喷嘴内壁锥孔(c), (d)的气体密度(a), (c)和流速(b), (d)分布\r\nFig. 3. Density (a), (c) and velocity (b), (d) distribution at the exit of the supersonic nozzle (a), (b) and the internal interface of the micro gas cell exit (c), (d).$

图 3 超音速喷嘴出口(a), (b)和微气室喷嘴内壁锥孔(c), (d)的气体密度(a), (c)和流速(b), (d)分布

Fig. 3. Density (a), (c) and velocity (b), (d) distribution at the exit of the supersonic nozzle (a), (b) and the internal interface of the micro gas cell exit (c), (d).

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对不同尺寸的微气室喷嘴进行了模拟, 这些喷嘴内部圆筒的直径分别为1, 2, 3和4 mm, 壁厚以及其他参数均与图2(b)中的喷嘴保持一致. 模拟中气体仍然选择N₂, 入口压力都设为5 kPa, 图4展示了不同喷嘴的气流密度分布. 从图4可以看出, 在这些模拟中气体密度保持了较好的平台状分布. 随着内径从1 mm增至4 mm, 气体的中心密度逐渐从8.2×10¹⁷ cm^–3逐渐降低到4.5×10¹⁷ cm^–3. 这是因为电磁阀出口的直径只有0.7 mm, 小于微气室的内径. 气流从电磁阀出口进入气室后, 内径越大, 气体的膨胀越明显, 密度也就越低. 另一方面随着内径增大, 喷气速率从0.56 mg/s提高到0.73 mg/s, 呈缓慢上升趋势. 这些结果表明, 这种构型的喷嘴可以较好地适用于不同的尺寸.

$图 4 不同内径的微气室喷嘴中N2的分子密度分布\r\nFig. 4. Molecular number density in micro gas cells of different inside diameters.$

图 4 不同内径的微气室喷嘴中N₂的分子密度分布

Fig. 4. Molecular number density in micro gas cells of different inside diameters.

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4. 实验结果

测试了这种喷嘴对真空系统的负载. 喷嘴选用的是内径4 mm的微气室, 工作气体为N₂, 背压为10 kPa, 电磁阀开门时间为5 ms. 真空腔体内部尺寸为1.1 m×0.8 m×0.7 m, 用一台Edward STP分子泵接一台Edward XDS机械泵抽真空, 用一个复合真空计测量真空度. 在未开启电磁阀之前, 真空度稳定在4.1×10^–3 Pa. 表1展示了不同重复频率下腔室真空度的最大值, 重复频率在2 Hz以下时, 喷嘴喷气的影响几乎可以忽略. 当重复频率提高到10 Hz时, 真空度最终稳定在了10^–2 Pa. 这表明在当前的真空系统下, 该喷嘴能够实现10 Hz的重频工作.

表 1 使用4 mm微气室不同工作频率下的真空度

Table 1. Vaccum at different repetition rates using 4 mm micro gas cell.

	重复频率/Hz
	1	2	5	10
真空度最大值/(10^–3 Pa)	4.7	4.6	6.7	10

下载: 导出CSV

| 显示表格

开展实验测试了这种构型的喷嘴在激光尾场加速中的效果. 实验在中国工程物理研究院激光聚变研究中心的45 TW飞秒激光器上开展. 激光的波长是800 nm, 脉宽是25 fs. 采用焦距600 mm的离轴抛面镜将激光聚焦在微气室喷嘴的入口处, 激光到靶能量为0.7 J, 激光焦斑为13 μm, 1/e²的能量集中度约50%, 对应激光a₀=1.9. 考虑到电子的失相长度, 选用的是内径1 mm的微气室喷嘴, 气体选择的是N₂. 图5(a)和图5(b)分别展示了在5 kPa和8 kPa下连续10发的电子能谱测量结果. 可以看出, 在5 kPa下, 电子的最高能量达到了250 MeV, 散角较小而能散较大. 当背压提高到8 kPa, 电子的最高能量降低到70 MeV, 散角增大, 并且电子的单能性有了明显的提升, 这可能是由束流负载效应导致的. 在不同的气压点下, 电子束都呈现出了很高的稳定性. 这说明这种微气室喷嘴可以在多工况下产生稳定的气流分布.

$图 5 采用内径1 mm的微气室喷嘴在分别在5 kPa (a)和8 kPa (b)下连续10发获得的电子能谱\r\nFig. 5. Electron sepctra in 10 consecutive shots using a 1 mm micro gas cell backing at 5 kPa (a) and 8 kPa (b).$

图 5 采用内径1 mm的微气室喷嘴在分别在5 kPa (a)和8 kPa (b)下连续10发获得的电子能谱

Fig. 5. Electron sepctra in 10 consecutive shots using a 1 mm micro gas cell backing at 5 kPa (a) and 8 kPa (b).

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5. 结　论

本文针对中小规模激光器对高重复频率气体靶的应用需求, 设计了一种低喷气量的微气室喷嘴, 使之用于短距离尾场加速. 这种喷嘴的喷气量相比同尺寸超音速喷嘴降低了97%, 能够在提供4 mm均匀密度分布的同时实现10 Hz的重复频率. 通过Fluent模拟对比了微气室和超音速喷嘴的喷气量差异, 说明微气室大幅降低喷气量的主要原因在于出气口面积的减小以及出口流速的降低. 在一台中小规模飞秒激光器上开展的实验中, 应用这种喷嘴在多条件下产生了稳定性很好的电子束. 除了喷气量小, 这种喷嘴还具有易加工、气流稳定的优点, 在未来高重频、高稳定性的尾场加速中将起到重要作用.

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图 1 混合量子-经典算法流程

Figure 1. Diagram for hybrid quantum-classical algorithms.

DownLoad: Full-Size Img PowerPoint

图 2 (a) 交换测试电路; (b)破坏性交换测试电路; (c) 哈达玛测试电路

Figure 2. (a) Swap test circuit; (b) destructive swap test circuit; (c) Hadamard test circuit.

DownLoad: Full-Size Img PowerPoint

图 3 (a) 量子感知机; (b)量子卷积神经网络; (c)影子电路

Figure 3. (a) Quantum perceptron; (b) quantum convolutional neural network; (c) shadow circuit.

DownLoad: Full-Size Img PowerPoint

图 4 量子纠错基本框架: 量子态 $\left| \varPsi \right\rangle$ 使用编码信道 ${\cal{U}}$ 编码, 经过噪声信道 ${\cal{N}}$ 后使用纠错信道 ${\cal{W}}$ 纠正错误, 最后使用解码信道 ${\cal{U}}^\dagger$ 解码, 还原输入量子态

Figure 4. Framework of quantum error correction. The quantum state $\left| \varPsi \right\rangle$ first is encoded by the encoding channel ${\cal{U}}$ , then passes the noise channel ${\cal{N}}$ , and then is corrected by the correction channel ${\cal{W}}$ , finally is recovered by the decoding channel ${\cal{U}}^\dagger$ .

DownLoad: Full-Size Img PowerPoint

图 5 无噪(左)与有噪(右)条件下损失函数景观对比

Figure 5. Comparison between noise-free (left) and noisy (right) cost function landscape.

DownLoad: Full-Size Img PowerPoint

表 1 主要符号表

Table 1. Notations

定义	符号
厄米特算符	H
含参数酉算符	${\boldsymbol U}({\boldsymbol{\theta} }), {\boldsymbol V}({\boldsymbol{\theta} })$
不含参数酉算符	W
可调参数	θ
量子态	${\boldsymbol \rho}, {\boldsymbol \sigma}$
量子比特数	n
电路层数	L
损失函数	$C, C({\boldsymbol{\theta}})$
能量	E
泡利算符	P
迹	Tr

DownLoad: CSV

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Vol. 70, Issue 21 - 2021-11-05

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