Quantum Path Signatures

2025/08/07 by Crew, Samuel, Salvi, Cristopher, Turner, William F. +2 · 1 citation
#60L10 #81-08 #81-10 #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2508.05103

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.

Citations

• ParallelFlow: Parallelizing Linear Transformers via Flow Discretization
• Numerical Schemes for Signature Kernels
• Sparse Signature Coefficient Recovery via Kernels
• SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddings
• Exact Gradients for Stochastic Spiking Neural Networks Driven by Rough Signals
• A path-dependent PDE solver based on signature kernels
• Theoretical Foundations of Deep Selective State-Space Models
• Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes
• Free probability, path developments and signature kernels as universal scaling limits
• A Neural RDE approach for continuous-time non-Markovian stochastic control problems
• Non-adversarial training of Neural SDEs with signature kernel scores
• Optimal Stopping via Distribution Regression: a Higher Rank Signature Approach
• Neural signature kernels as infinite-width-depth-limits of controlled ResNets
• New directions in the applications of rough path theory
• A structure theorem for streamed information
• Topologies on unparameterised path space
• Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics
• Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes
• SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data
• SK-Tree: a systematic malware detection algorithm on streaming trees via the signature kernel
• Neural Rough Differential Equations for Long Time Series
• Distribution Regression for Sequential Data
• Sig-SDEs model for quantitative finance
• Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU
• Deep Signature Transforms
• The iisignature library: efficient calculation of iterated-integral signatures and log signatures
• The Signature of a Rough Path: Uniqueness
• Learning from the past, predicting the statistics for the future, learning an evolving system
• Invariants of algebraic curves and topological expansion
• Combinatorial aspects of matrix models
• A planar diagram theory for strong interactions

Cited by

• pySigLib -- Fast Signature-Based Computations on CPU and GPU

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