MAPS: A Novel Multi-Axial Projective Sphere for Geometrically Visualizing Higher d-Valued Quantum State-Space of Qudits

Visualizing the d-valued quantum state-space of quantum systems serves as a foundational pillar for the scientific research and practical applications in quantum computing and information science, where d >= 2. The 2-valued quantum states of a qubit are elegantly visualized on the three-dimensional Bloch sphere. In contrast, expanding this geometrical paradigm to visualize higher d-valued quantum states of a qudit d >= 3, e.g., a qutrit d=3, ququadit d=4, and quintit d=5, leads to severe structural and topological complexities. This paper introduces a new generalized three-dimensional framework to effectively visualize higher d-valued quantum states of a qudit, in the aspects of ease of illustration, structural simplicity, and natural representation for researchers and engineers. We called this new framework the "multi-axial projective sphere (MAPS)", which consists of n projectional intersecting spatial axes, where 0 <= n <= d-1. We also propose a group of novel d-valued phase axial-based gates, to swivel and shift d-valued quantum states of a qudit along these n axes. Our generalized framework could be used for visualizing high-dimensional data for practical applications, e.g., machine learning, quantum machine learning, and quantum chemistry, where every axis of the MAPS represents a single feature of such data with its corresponding distinct values.