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

• URL: http://arxiv.org/abs/2606.15801v2
• Date: Tue, 16 Jun 2026 08:14:17 GMT
• Title: MAPS: A Novel Multi-Axial Projective Sphere for Geometrically Visualizing Higher d-Valued Quantum State-Space of Qudits
• Abstract summary: This paper introduces a new three-dimensional framework to effectively visualize higher d-valued quantum states of a qudit.<n>We call this new framework the "multi-axial projective sphere (MAPS)", which consists of n projectional intersecting spatial axes, where 0 = n = d-1.<n>Our framework could be used for visualizing high-dimensional data for practical applications, e.g., machine learning, quantum machine learning, and quantum chemistry.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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