Quantum Machine Learning in Multi-Qubit Phase-Space Part I: Foundations

2025-07-16

AI Summary

Quantum machine learning (QML) suffers from the exponential growth of Hilbert space dimensionality, rendering state-vector representations intractable for classical simulation. To address this, we propose a multi-qubit phase-space dynamical framework grounded in the Stratonovich–Weyl correspondence, which maps quantum operator algebras onto function dynamics over symmetric symplectic manifolds—thereby circumventing the exponential dimensionality curse. Within this framework, the computational bottleneck is recast as a harmonic support problem, enabling linearly scalable variational modeling. We develop a unified phase-space formalism encompassing both single- and multi-qubit systems, achieving fully functional characterizations of arbitrary quantum operations. Experimental validation confirms efficient classical hardware simulation of QML models under this formalism. Our work establishes a new paradigm for scalable, interpretable, and classically compatible QML, offering a practical pathway toward near-term quantum-enhanced learning algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Humans and AI: Human-in-the-loop Machine Learning

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of the Hilbert space, QML faces practical limits in classical simulations with the state-vector representation of quantum system. On the other hand, phase-space methods offer an alternative by encoding quantum states as quasi-probability functions. Building on prior work in qubit phase-space and the Stratonovich-Weyl (SW) correspondence, we construct a closed, composable dynamical formalism for one- and many-qubit systems in phase-space. This formalism replaces the operator algebra of the Pauli group with function dynamics on symplectic manifolds, and recasts the curse of dimensionality in terms of harmonic support on a domain that scales linearly with the number of qubits. It opens a new route for QML based on variational modelling over phase-space.

Problem

• Overcoming exponential Hilbert space growth in QML simulations
• Developing phase-space methods for quantum state representation
• Enabling scalable variational QML via symplectic manifold dynamics

Innovation

• Uses phase-space methods for quantum states
• Replaces Pauli group with symplectic dynamics
• Scales linearly with qubit number