Quantum Binary Classification with LS-QSVM & HHLTransparent, 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.

For this task, the solution used requires linear-system solve, which can be computed via HHL (Harrow–Hassidim–Lloyd), a quantum algorithm for solving A\mathbf{x}=\mathbf{b}. In the LS-SVM/QSVM setting, this corresponds to systems such as (K+\lambda I)\boldsymbol{\alpha}=\mathbf{y}, where K is the kernel Gram matrix; HHL returns a quantum state proportional to the solution vector (from which the needed coefficients are obtained).