Review on quantum advantages of sampling problems

Li Ying; Han Ze-Yao; Li Chao-Jian; Lü Jin; Yuan Xiao; Wu Bu-Jiao

doi:10.7498/aps.70.20211428

Abstract

Exploiting the coherence and entanglement of quantum many-qubit states, quantum computing can significantly surpass classical algorithms, making it possible to factor large numbers, solve linear equations, simulate many-body quantum systems, etc., in a reasonable time. With the rapid development of quantum computing hardware, many attention has been drawn to explore how quantum computers could go beyond the limit of classical computation. Owing to the need of a universal fault-tolerant quantum computer for many existing quantum algorithms, such as Shor’s factoring algorithm, and considering the limit of near-term quantum devices with small qubit numbers and short coherence times, many recent works focused on the exploration of demonstrating quantum advantages using noisy intermediate-scaled quantum devices and shallow circuits, and hence some sampling problems have been proposed as the candidates for quantum advantage demonstration. This review summarizes quantum advantage problems that are realizable on current quantum hardware. We focus on two notable problems—random circuit simulation and boson sampling—and consider recent theoretical and experimental progresses. After the respective demonstrations of these two types of quantum advantages on superconducting and optical quantum platforms, we expect current and near-term quantum devices could be employed for demonstrating quantum advantages in general problems.

Keywords:

Authors and contacts

1.
School of Physics, Peking University, Beijing 100871, China
2.
Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
3.
School of Computer Science and Technology, Guangdong University of Technology, Guangzhou 510006, China

Corresponding author: Wu Bu-Jiao, bujiaowu@gmail.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 91964101)

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References

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Cited By

图 1 随机线路的CNOT门的8种不同的摆放方式^[6]. 其中第0层全部摆放H门, 电路中每8层循环一次(重复图中1—8层), 空白节点处随机放置$ {\boldsymbol I}, {\boldsymbol T}, {\boldsymbol X}^{1/2}, {\boldsymbol Y}^{1/2} $门, 两比特门为CZ门

Figure 1. Eight different layouts of the CNOT gate in the random circuit, where all of qubits are performed H gate in the $ 0 $-th layer, and cycle once every 8 layers in the circuit (repeat 1–8 layers of this graph), the blank vertices are laid out $ {\boldsymbol I}, {\boldsymbol T}, {\boldsymbol X}^{1/2}, {\boldsymbol Y}^{1/2} $ randomly, and the two-qubit gates are all CZ gates^[6].

DownLoad: Full-Size Img PowerPoint

图 2 悬铃木处理器随机线路架构^[7]. 其中第0层全部摆放H门, 电路每层迭代重复模式ABCDCDBA, 两个模式中间由一层随机放置的单比特门$ {\boldsymbol X}^{1/2},\; {\boldsymbol Y}^{1/2},\;{\boldsymbol W}^{1/2} $构成, 两比特门为控制相位门和部分$ i $SWAP门的乘积(部分$ i $SWAP门后跟随一个控制相位门构成)

Figure 2. Random circuit architecture for Sycamore processor, where all of qubits are performed H gates in the $ 0 $-th layer, the layer of the circuit iterates and repeats the pattern ABCDCDBA, a layer of random single-qubit gates are performed between two modes, which constructed by $ {\boldsymbol X}^{1/2},\;{\boldsymbol Y}^{1/2},\; {\boldsymbol W}^{1/2} $, the two-qubit gate is the multiplication of the partial-iSWAP gate and control-phase gate (constructed by partial-iSWAP gate followed by a control-phase gate)^[7].

DownLoad: Full-Size Img PowerPoint

图 3 玻色采样模型^[56]. 输入是$ n $个单光子, 经过线性光学网络后, 可在输出的$ m $个模中探测光子

Figure 3. Device of boson sampling^[56]. The input are $ n $ photons, and one can detect photons on the $ m $ output modes through a linear optical network.

DownLoad: Full-Size Img PowerPoint

图 4 SBS的装置简介^[63]. 该模型中输入为$ 2 m $个单模压缩态, 在其进入分束器和相移子装置后产生双模压缩态, 并通过一个额外的测量装置来固定SBS的线性光学装置$ \boldsymbol{U}_m $输入的光子数

Figure 4. Brief introduction of SBS device^[63]. In this model, the input are $ 2 m $ single-mode compressed states, and the two-mode compressed states are generated after entering the beam splitter and the phase-shifting sub-device. An additional measuring device is used to fix the number of input photons for the linear optical device $ \boldsymbol{U}_m $ of SBS.

DownLoad: Full-Size Img PowerPoint

图 5 GBS的装置简介^[63]. 输入端为K个单模压缩态注入线性光学网络, 在输出端的M个模中进行光子数探测

Figure 5. Brief introduction to GBS device^[63]. The input terminal is a K single-mode compressed state injected into the linear optical network, and the photon number is detected in M modes at the output terminal.

DownLoad: Full-Size Img PowerPoint

图 6 量子优越性证明的实验结果^[7]　(a)在经典可验证区, 简单全量子线路的保真度与简单删减量子线路、简单分割量子线路、简单乘积模型的保真度符合得很好, 每个数据点是多个随机量子线路采样的平均值; (b)在量子优越区, 通过更简单的线路和简单乘积模型来估计复杂全量子线路的保真度. 红色时间标志表示经典模拟复杂全量子线路的验证任务需要的时间, 灰色时间标志表示经典模拟相应的采样任务需要的时间

Figure 6. Experimental results of the proof of quantum advantage^[7]: (a) In the classical verifiable region, the fidelity of simple full quantum circuit accords well with that of simple truncated quantum circuit, simple split quantum circuit and simple product model. Each data point is the average of multiple random quantum circuit samples. (b) In the quantum advantage region, the fidelity of complex full quantum circuits is estimated by using simpler circuits and simple product models. The red time label represents the time required for the verification task of the classical simulation complex full quantum circuit, and the grey time label represents the time required for the corresponding sampling task of the classical simulation.

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图 7 九章GBS 实验示意图^[53]

Figure 7. Illustration of the Jiuzhang GBS experiment^[53].

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

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