图 1  实现HHL算法的量子线路图. 其中$ r=2, {t}_{0}=2 $. 单比特门$ {{S}}=\left(\begin{array}{cc}1& 0\\ 0& \mathrm{i}\end{array}\right),\; {{H}}=\dfrac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right),\; {{{R}}}_{y}\left(\theta \right)= $$ \left(\begin{array}{cc}\cos\tfrac{\theta }{2}& -\sin\tfrac{\theta }{2}\\ \sin\frac{\theta }{2}& \cos\frac{\theta }{2}\end{array}\right) $. 与直线相连的$ \times $表示SWAP门^[31]

Fig. 1.  The quantum circuit of the HHL algorithm. Parameter $ r=2, ~~{t}_{0}=2 $. Quantum gate $ {{S}}=\left(\begin{array}{cc}1& 0\\ 0& \mathrm{i}\end{array}\right), \; {{H}}= $$ \dfrac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right), \;{{{R}}}_{y}\left(\theta \right)=\left(\begin{array}{cc}\cos\frac{\theta }{2}& -\sin\tfrac{\theta }{2}\\ \sin\frac{\theta }{2}& \cos\tfrac{\theta }{2}\end{array}\right) $. The symbol $ \times $ connected with the straight line represents the SWAP gate^[31].

图 2  解线性微分方程的量子线路图. 线路中第一个辅助寄存器是单比特, 第二个辅助寄存器为$ T=\mathrm{l}\mathrm{o}{\mathrm{g}}_{2}\left(k+1\right) $比特, 然后是一个工作系统. 所有的辅助寄存器被初始化为$ \left|0\right\rangle {\left|0\right\rangle }^{\mathrm{T}} $, 控制操作$ {U}_{x} $和$ {U}_{b} $分别被用来生成$ \left|{{x}}\left(0\right)\right\rangle $和$ \left|{{b}}\right\rangle $. 在编码和解码期间的演化算子为$\displaystyle \sum\nolimits_{\tau =0}^{k}\left|\tau \right\rangle \left\langle\tau \right|\otimes {U}_{\tau }$. 在线路的结尾, 在所有辅助比特为$ \left| 0 \right\rangle $的子空间中测量工作系统的态矢^[38]

Fig. 2.  Quantum circuit for solving linear differential equations. The first auxiliary register in the circuit is a single bit, and the second auxiliary register is $ T=\mathrm{l}\mathrm{o}{\mathrm{g}}_{2}\left(k+1\right) $ bits, then is a working system $ \left|\phi \right\rangle $. All auxiliary registers are initialized to $ \left|0\right\rangle {\left|0\right\rangle }^{\mathrm{T}} $, and then the operation $ {U}_{x} $ and $ {U}_{b} $ are used to generate $ \left|{{x}}\left(0\right)\right\rangle $ and $ \left|{{b}}\right\rangle $. The evolution operator during encoding and decoding is $\displaystyle \sum\nolimits_{\tau =0}^{k}\left|\tau \right\rangle \left\langle\tau \right|\otimes {U}_{\tau }$. At the end of the circuit, the state vector of the working system is measured in the subspace where all auxiliary bits are $ \left| 0 \right\rangle $^[38].

图 3  手写字符“6”和“9”的识别结果, 第1—4行分别代表手写字符, 实验指示符, 相干项的幅度和识别结果^[42]

Fig. 3.  Recognition results of handwritten characters of “6” and “9”. Lines 1 to 4 represent handwritten characters, experimental indicators, amplitude, and recognition results, respectively^[42].

图 4  通过qPCA实现人脸识别的流程图. 通过混合经典量子控制方法对PQC $ {\cal{U}}\left({{\theta }}\right) $进行迭代优化, 其中在量子处理器上测量目标函数$ L\left({{\theta }}\right) $和梯度$ g\left({{\theta }}\right) $. 参数$ {{\theta }} $的存储和更新在经典计算机上实现. 用优化后的$ {U}_{g} $来计算特征脸矩阵D和协方差矩阵C的特征向量^[45]

Fig. 4.  Workflow for human face recognition via qPCA. The PQC $ {\cal{U}}\left({{\theta }}\right) $ is iteratively optimized via the hybrid classicalquantum control approach, where the objective function $ L\left({{\theta }}\right) $ and the gradient $ g\left({{\theta }}\right) $ are measured on the quantum processor. The storage and update of the parameters $ {{\theta }} $ are implemented on a classical computer. The optimized PQC with the operator $ {U}_{g} $ is applied to compute the eigenvectors of the eigenface matrix $ {{D}} $ and the covariance matrix $ {{C}}={{A}}{{{A}}}^{\mathrm{T}} $^[45].

图 5  (a)金刚石NV色心结构图; (b) NV色心电子能级跃迁过程示意图, $ {}_{ }{}^{3}{\mathrm{A}}_{2} $和$ {}^{3}\mathrm{E} $分别代表基态和激发态, $ {}^{1}{\mathrm{A}}_{1} $和$ {}^{1}\mathrm{E} $为中间亚稳态, 从激发态直接跃迁回基态会发出荧光, 而经中间态回基态不会发出荧光

Fig. 5.  (a) NV color center structure; (b) schematic diagram of the transition process of NV color center electron energy level, $ {}^{3}{\mathrm{A}}_{2} $ and $ {}^{3}\mathrm{E} $ represent the ground state and excited state, respectively, $ {}^{1}{\mathrm{A}}_{1} $ and $ {}^{1}\mathrm{E} $ are the intermediate metastable states, which from the excited state directly transitions back to the ground state and emit fluorescence. But the path througt metastable state returns to the ground state without emitting fluorescence.

图 6  (a)利用共振微波操控NV色心基态能级; (b)可对拓扑相进行分类的3D卷积神经网络的体系结构, 输入是在10 × 10 × 10规则网格上的密度矩阵的实验数据. 每个密度矩阵由八个实数表示. 输出是每个可能相的分类概率^[54]

Fig. 6.  (a) Using resonance microwave to control the ground state energy level of NV color center; (b) architecture of the 3D CNN to classify the topological phases. The input is experimental data of density matrices on a 10 × 10 × 10 regular grid. Each density matrix is represented by eight real numbers^[54].

图 7  迭代次数增加时的训练和验证准确性. 训练和验证准确性在训练过程开始时迅速增加, 然后达到了很高的饱和值(≈ 98%)^[54]

Fig. 7.  The training and verification accuracy when the number of iterations increases. The training and validation accuracy increased rapidly at the beginning of the training process, and then reached a high saturation value (≈ 98%)^[54].

图 8  共振量子主成分分析算法原理图　(a)探针-寄存器耦合系统的能级结构, $ \left|{\lambda }_{i}\right\rangle $是$ {{\rho }} $的第$ i $个本征态, 而$ {\lambda }_{i}\in [0, 1] $是对应的本征值, 如果扫描频率$ \omega \approx {\lambda }_{i} $, 就会引起探针量子位的拉比振荡; (b)使用Suzuki-Trotter分解的RqPCA的量子电路, 对探针量子位进行投影测量得到$ \left|1\right\rangle $表明该算法成功^[57]

Fig. 8.  Algorithm schematic of RqPCA: (a) The energy structure of the coupled probe-register system. $ \left|{\lambda }_{i}\right\rangle $ is the i-th eigenstate of $ {{\rho }} $ and $ {\lambda }_{i}\in [0, 1] $ is the corresponding eigenvalue. Once the scanning frequency $ \omega \approx {\lambda }_{i} $, the Rabi oscillations of the probe qubit is induced; (b) the quantum circuit of RqPCA. The projective measurement of the probe qubit in the state $ \left|1\right\rangle $ indicates success of the algorithm, with principal component being distilled in the register^[57].

图 9  (a)量子自编码器线路图, 通过编码操作$ {{\cal{U}}}_{\cal{E}} $将$ \left|{\varPsi }_{i}\right\rangle $中的信息压缩到$ {\left|\phi \right\rangle }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}} $中, 在需要时通过解码操作$ {{\cal{U}}}_{\cal{D}} $将$ {\left|\phi \right\rangle }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}} $还原为$ \left|{\varPsi }_{f}\right\rangle $; (b)优化编码器的基于梯度算法的HQCA的训练过程, $ {\rho }_{\mathrm{i}\mathrm{n}} $是编码器的输入状态, $ {\rho }_{\mathrm{o}\mathrm{u}\mathrm{t}} $是辅助量子位的输出状态, $ f\left({{\cal{U}}}_{\cal{E}}^{\left(q\right)}\right) $是成本函数, q是迭代次数^[58]

Fig. 9.  (a) Quantum autoencoder circuit. The target information of $ \left|{\varPsi }_{i}\right\rangle $ can be encoded to the code state $ {\left|\phi \right\rangle }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}} $ via the encoder $ {{\cal{U}}}_{\cal{E}} $. $ {\left|\phi \right\rangle }_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}} $ can be reconstructed to $ \left|{\varPsi }_{f}\right\rangle $ when needed by the decoder $ {{\cal{U}}}_{\cal{D}} $. (b) Training process of the gradient-based HQCA to optimize encoder. Here, $ {\rho }_{\mathrm{i}\mathrm{n}} $ is the input state of the encoder, and $ {\rho }_{\mathrm{o}\mathrm{u}\mathrm{t}} $ is the output state on the ancilla qubits. $ f\left({{\cal{U}}}_{\cal{E}}^{\left(q\right)}\right) $ is the cost function, where q is the current iterative number^[58].