Solve binary classification with a trainable quantum support vector machineTransparent, Reproducible Results

We use a classical SVM pipeline to solve binary classification problem. The model seeks a set of separating hyperplanes in feature space, \mathbf{w}^\top \phi(\mathbf{x}) + b = 0 , maximizing the geometric margin between classes. When linear separation is difficult, we map data to a higher-dimensional space and apply the kernel trick: instead of building \phi(\cdot) explicitly, we compute inner products k(\mathbf{x},\mathbf{x}')=\langle \phi(\mathbf{x}), \phi(\mathbf{x}')\rangle , which lets the SVM operate implicitly in a Hilbert space \mathcal{H}.

Quantum SVM is implemented using a quantum kernel, where input \mathbf{x} is encoded into a quantum state |\phi(\mathbf{x})\rangle=U(\mathbf{x})|0\rangle^{\otimes n} and using the state overlap (inner product or fidelity) as k(\mathbf{x},\mathbf{x}'). The circuit induces a (potentially very high-dimensional) nonlinear embedding through data-dependent single-qubit rotations and entangling gates; the choice of encoding and entanglement pattern governs which higher-order interactions are represented. These overlaps are estimated by running U(\mathbf{x}')^\dagger U(\mathbf{x}) on quantum hardware; the resulting Gram matrix feeds directly into the standard SVM pipeline.

The circuit can also be generalized to include trainable parameters (e.g., single-qubit rotation angles or entangler weights), enabling data-driven kernel adaptation. In this formulation, the feature map becomes U(\mathbf{x},\boldsymbol{\theta}) , producing states |\phi(\mathbf{x};\boldsymbol{\theta})\rangle=U(\mathbf{x},\boldsymbol{\theta})|0\rangle^{\otimes n} and an induced kernel k_{\boldsymbol{\theta}}(\mathbf{x},\mathbf{x}') \;=\; \bigl|\langle \phi(\mathbf{x};\boldsymbol{\theta}) \mid \phi(\mathbf{x}';\boldsymbol{\theta}) \rangle\bigr|^2. Its geometry is now controlled by the parameter vector \boldsymbol{\theta}. The parameters may be optimized to improve task fit — e.g., by maximizing target alignment, minimizing a margin-based surrogate, or reducing cross-validated risk—using gradient-based or gradient-free routines.