QCNN with Rough Path Signature Kernels

Research Approach

This paper addresses the challenge of time series classification, specifically the difficulty of handling time reparameterization invariance—where the essential characteristics of a sequence remain unchanged despite non-linear warping of the time axis. The authors leverage rough path theory, specifically the signature transform, which provides a feature set inherently invariant to these transformations. Because computing signatures directly is computationally expensive, the authors utilize signature kernels, which can be computed by solving a specific Goursat-type partial differential equation (PDE).

Hybrid Quantum Architecture

The proposed architecture consists of two main stages. First, feature layers compute the signature kernel between input paths using either classical methods or a Variational Quantum Linear Solver (VQLS). The VQLS is employed to solve the linear system derived from the finite-difference discretization of the kernel PDE, offering a potential quantum-enhanced alternative to classical solvers. Second, the resulting features are processed by a Quantum Convolutional Neural Network (QCNN). The authors explore two integration strategies: substituting the full kernel matrix as the input to the QCNN, or fusing a scalar representation of the kernel into an auxiliary qubit alongside standard image data.

Findings and Significance

The study evaluates these configurations on a binary classification task using stroke-sequence representations of MNIST digits. By utilizing a hierarchical QCNN, the architecture benefits from logarithmic scaling in circuit depth and parameter count relative to the number of qubits. This work provides a framework for incorporating geometric temporal information into quantum circuits, highlighting both the theoretical potential for quantum-enhanced feature extraction and the practical hurdles—such as noise and circuit depth—associated with running VQLS on near-term quantum devices.