初态为 ∣0⋯00⟩|0\cdots00\rangle，电路输出为 ∣ψ⟩=S(x2)†S(x1)∣0⋯00⟩|\psi\rangle=S(x_2)^{\dagger}S(x_1)|0\cdots00\rangle、

示例中构造一个厄米矩阵

M=[10⋯000⋯0⋮⋮⋱⋮00⋯0]=∣0⋯00⟩⟨00⋯0∣M=\begin{bmatrix} 1 &0&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0 \end{bmatrix}=|0\cdots00\rangle\langle 00\cdots 0|

计算投影测量的期望值为

⟨ψ∣M∣ψ⟩=⟨00⋯0∣S(x1)†S(x2)∣0⋯00⟩⟨00⋯0∣S(x2)†S(x1)∣0⋯00⟩\langle\psi|M|\psi\rangle=\langle 00\cdots 0|S(x_1)^{\dagger}S(x_2)|0\cdots00\rangle\langle 00\cdots 0|S(x_2)^{\dagger}S(x_1)|0\cdots00\rangle

以分类任务为例，引入了软间隔和核函数的SVM形式如下：

max⁡α∑i=1mαi−12∑i=1m∑j=1mαiαjyiyjκ(xi,xj) s.t. ∑i=1mαiyi=0,0⩽αi⩽C,i=1,2,…,m\begin{aligned} \max _{\boldsymbol{\alpha}} & \sum_{i=1}^{m} \alpha_{i}-\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m} \alpha_{i} \alpha_{j} y_{i} y_{j} \kappa\left(\boldsymbol{x}_{i},\boldsymbol{x}_{j} \right)\\ \text { s.t. } & \sum_{i=1}^{m} \alpha_{i} y_{i}=0, \\ & 0\leqslant\alpha_{i} \leqslant C, \quad i=1,2, \ldots, m \end{aligned} 

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import numpy as np
from sklearn.svm import SVC
from sklearn import datasets
from sklearn.preprocessing import StandardScaler
import pennylane as qml
from pennylane.kernels import kernel_matrix
from sklearn.model_selection import cross_val_score
from pennylane.templates import AngleEmbedding
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X, y = datasets.load_iris(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
y_scaled = y

使用RBF核

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np.average(cross_val_score(SVC(), X, y))
0.9666666666666666

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n_qubits = 4
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dev_kernel = qml.device('lightning.qubit', wires=n_qubits)
projector = np.zeros((2 ** n_qubits, 2 ** n_qubits))
projector[0, 0] = 1


@qml.qnode(dev_kernel)
def circuit(x1, x2):
    AngleEmbedding(x1, wires=range(n_qubits))
    qml.adjoint(AngleEmbedding)(x2, wires=range(n_qubits))
    return qml.expval(qml.Hermitian(projector, wires=range(n_qubits)))

用量子电路计算核函数

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np.average(cross_val_score(SVC(kernel=lambda X1, X2: kernel_matrix(X1, X2, lambda x1, x2: circuit(x1, x2))), X, y))
0.9800000000000001