Quantum Path Signatures

TL;DR

This paper introduces a quantum-inspired framework for path signatures using matrix models and quantum circuits, leading to new quantum feature maps and kernels with potential applications in quantum computing.

Contribution

It formulates a novel quantum path signature framework using matrix models, deriving loop equations and defining quantum signature kernels via quantum circuits.

Findings

Defined a quantum path signature feature map.

Developed a quantum algorithm for kernel computation.

Analyzed the Gaussian matrix model with Pauli strings.

Abstract

We elucidate physical aspects of path signatures by formulating randomised path developments within the framework of matrix models in quantum field theory. Using tools from physics, we introduce a new family of randomised path developments and derive corresponding loop equations. We then interpret unitary randomised path developments as time evolution operators on a Hilbert space of qubits. This leads to a definition of a quantum path signature feature map and associated quantum signature kernel through a quantum circuit construction. In the case of the Gaussian matrix model, we study a random ensemble of Pauli strings and formulate a quantum algorithm to compute such kernel.