AI Research Portal

This paper proposes a hybrid quantum-classical architecture for time series classification that integrates rough path signatures with Quantum Convolutional Neural Networks (QCNNs). The architecture uses signature kernels, computed either classically or via a Variational Quantum Linear Solver (VQLS), as feature layers followed by a QCNN for downstream learning. Experiments on a binary classification task using handwritten digit stroke sequences demonstrate the potential benefits of signature kernel layers in quantum circuits.

The paper addresses several challenges in time series analysis:

Time Reparameterization Invariance: Standard models struggle with the inherent invariance of sequential data to time warping, making it difficult to extract meaningful temporal features.
Computational Intensity of Signatures: Computing truncated signature features, which are invariant to time reparametrization, is computationally intensive due to an exponential growth in the number of terms with path dimension and truncation order.
Scalability of Signature Kernels: While signature kernels offer a way to compute signature features efficiently, their classical computation via finite-difference schemes for PDEs results in large, sparse linear systems that scale polynomially with grid resolution.
Scalability of Quantum Linear Solvers: Quantum algorithms like HHL offer theoretical speedups for solving linear systems, but their practical implementation on near-term NISQ devices is hindered by requirements for large circuit depth, qubit counts, and noise. VQLS, while more suitable for NISQ devices, faces scalability challenges with increasing system dimensions and condition numbers.

A hybrid quantum-classical architecture for time series classification that combines rough path signatures with QCNNs.
Integration of signature kernels, computed via classical methods or VQLS, as feature layers in a QCNN pipeline.
Evaluation of the proposed architecture on a binary classification task using the MNIST digit stroke sequence dataset.
Analysis of the computational limitations of the VQLS component for signature kernel evaluation.

The proposed method combines rough path signatures with QCNNs for time series classification.

Feature Extraction: Input time series paths are processed to compute signature kernels. This kernel computation can be performed using a classical routine or a Variational Quantum Linear Solver (VQLS). The VQLS approach reformulates the finite-difference discretization of the signature kernel's governing PDE into a linear system of equations (Ax = b) that is then solved variationally.
Quantum Convolutional Neural Network (QCNN): The computed signature kernels serve as input features for a QCNN. The QCNN architecture consists of alternating convolutional and pooling layers, designed to hierarchically extract features and reduce dimensionality. Various QCNN convolutional filter configurations (ansatzes) are explored.
Classification: The QCNN performs downstream learning and classification tasks, ultimately outputting a prediction for the input time series.
Data Encoding: For the QCNN, input data (either raw pixel data or signature kernel matrices) are encoded into an 8-qubit register using amplitude encoding or angle encoding. An auxiliary 9th qubit is used in one variant to encode a scalar representation of the signature kernel.
Training: The model is trained using a Binary Cross-Entropy loss function, with a classical optimizer (Adam or COBYLA) updating the QCNN parameters.

Signature Kernel Computation: The signature kernel is defined as the inner product of two signatures and is related to a second-order hyperbolic linear partial differential equation (PDE). Discretization of this PDE leads to a linear system Ax = b.
VQLS: The Variational Quantum Linear Solver (VQLS) is used to solve the linear system Ax = b. It prepares a quantum state proportional to the solution vector. The VQLS uses a hardware-efficient ansatz with alternating layers of Y rotations and CNOT gates, with LvQLS = 4 layers. A global cost function CG and a local cost function CL are used for optimization.
QCNN Architecture: The QCNN employs a hierarchical, tree-like circuit with alternating convolutional and pooling layers. Various convolutional filter configurations are used, including UTTN, Heuristic Circuits (U5-U15), Uso(4), and Usu(4).
Data Encoding: Amplitude encoding is used for image representations of signature kernels, while angle encoding is used for signature kernels as single numbers. For the QCNN, an 8-qubit register is used.
Loss Function: Binary Cross-Entropy loss (CCE) is used for training.
Optimization: Adam or COBYLA optimizers are used to update gate parameters.
Dataset: The MNIST handwritten digit stroke sequence dataset is used for binary classification (0-vs-1 task). A filtered subset of digits 0 and 1 is used, comprising 12,665 training samples and 2,115 test samples.
Implementation: Qiskit SDK is used for quantum circuits and VQLS configurations. PennyLane framework is used for QCNN implementation. Experiments are conducted on noiseless statevector simulators.

Datasets: MNIST handwritten digit stroke sequence dataset (filtered for digits 0 and 1).
Metrics: Binary classification accuracy.
Baselines: Standard QCNN with raw pixel inputs, pure SigKernel substitution, and SigKernel aux qubit fusion architecture.
Numeric Results: The SigKernel aux qubit architecture consistently outperforms the pure SigKernel architecture across most evaluated ansatzes. Signature kernel features provide a performance enhancement for specific QCNN architectures. Accuracy improvements are observed for some QCNN architectures when including signature kernel layers compared to raw data.
Runtime/Training Cost: Not stated in the paper.

VQLS Scalability: The paper highlights severe scalability bottlenecks for VQLS on near-term quantum devices. The rapid inflation of system dimensions and matrix condition numbers restricts VQLS to heavily compressed sequences.
Classical Evaluation Necessity: Due to VQLS limitations, classical evaluation using libraries like sigkernel is currently the only practical option for multi-sample training loops.
Downsampling Trade-off: Downsampling stroke sequences to a fixed size of 16 points for QCNN input introduces a performance trade-off, as it can discard fine geometric nuances, leading to misclassifications.
Qubit Constraints: Simultaneously utilizing the full kernel matrix and the complete spatial image was not feasible due to qubit limitations. Scaling up the register size could potentially yield uniform performance improvements.

Method evidence: "We propose a hybrid quantum-classical architecture that integrates recent advances in quantum neural networks with the mathematical framework of path signatures, mitigating the impact of time reparametrization invariance." (Abstract)
Result evidence: "Across the majority of the evaluated ansatzes, the Auxiliary Qubit Fusion Architecture (SigKernel aux qubit) consistently outperforms the pure kernel-based SigKernel architecture." (Section 5.2 Performance Evaluation and Comparative Analysis)
Limitation evidence: "The rapid inflation of both system dimensions and matrix condition numbers restricts VQLS to heavily compressed sequences. Consequently, classical evaluation via libraries like sigkernel remains the only practical option for multi-sample training loops at present." (Section 6 Conclusions)

QCNN with Rough Path Signature Kernels

Abstract

Summary

Problem

Key Contributions

Method

Technical Details

Results

Weaknesses / Limitations

Additional Figures