Figure 2.

Quantum circuit computes the real part of the inner product $⟨u|v⟩$. The Hadamard gate puts the ancilla qubit ($q_0$) into uniform superposition. A single-controlled unitary gate entangles the exited state of the ancilla qubit with the training data state vector ($\left|u⟩=U,\left|q_1,q_2,q_3,⟩\right)\right)$). The $X$ gate flips the ancilla qubit. Another single unitary controlled gate entangles the state vector of the test data ($\left|v⟩=V,\left|q_1,q_2,q_3,⟩\right)\right)$) with the excited state of the ancilla qubit. A second $X$ gate flips the ancilla qubit. The Hadamard gate on the ancilla qubit interferences train and test data state vectors. The ancilla qubit is measured using a Pauli-$Z$ gate. The real value of $⟨u|v⟩$ is estimated from Eq. (9). The measurement gate is done by a Pauli-$Z$ gate and $Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$.