Mathematical Foundations¶
Core mathematical formulation of statevector spaces, unitary operations, and analytic gradient evaluation.
1. Hilbert Space Representation¶
A register of \(n\) qubits is defined on the complex Hilbert space: $\(\mathcal{H} = \left(\mathbb{C}^2\right)^{\otimes n} \cong \mathbb{C}^{2^n}\)$
A pure state \(|\psi\rangle \in \mathcal{H}\) satisfies: $\(|\psi\rangle = \sum_{b=0}^{2^n - 1} c_b |b\rangle, \quad \sum_{b=0}^{2^n - 1} |c_b|^2 = 1, \quad c_b \in \mathbb{C}\)$
2. Unitary Rotation Operators¶
Evolution operators satisfy \(U^\dagger U = I\). Single-qubit rotation gates generated by the Pauli group \(\mathfrak{su}(2)\): $\(R_x(\theta) = \exp\left(-i \frac{\theta}{2} \sigma^x\right) = \begin{pmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\)$ $\(R_y(\theta) = \exp\left(-i \frac{\theta}{2} \sigma^y\right) = \begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}\)$ $\(R_z(\theta) = \exp\left(-i \frac{\theta}{2} \sigma^z\right) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}\)$
3. Continuous Feature Embedding¶
Continuous features \(x \in \mathbb{R}^n\) are encoded via tensor product rotations: $\(|\psi(x)\rangle = \bigotimes_{j=1}^n R_x(x_j) |0\rangle\)$
The overlap between encoded states defines the kernel: $\(k(x, x') = |\langle \psi(x) | \psi(x') \rangle|^2 = \prod_{j=1}^n \cos^2\left(\frac{x_j - x_j'}{2}\right)\)$
4. Parameter-Shift Differentiation¶
For Pauli generators \(G = \frac{1}{2}\sigma^P\) with eigenvalues \(\pm \frac{1}{2}\), the partial derivative of an expectation value \(\langle H \rangle(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle\) with respect to parameter \(\theta_k\) evaluates analytically: