A Computational Benchmark of Geometric and Algebraic Models for Multi-Qubit State Representation

The Bloch sphere provides a powerful yet fundamentally limited geometric model for quantum states, excelling at single-qubit intuition but failing to represent the multi-qubit entanglement that powers quantum advantage. This work addresses the critical tension between the cognitive simplicity of geometric projections and the informational completeness required to model scalable quantum systems. The analysis herein is confined to pure quantum states, providing a foundational benchmark for future work on more complex mixed states. We introduce a unified computational framework to quantitatively benchmark quantum state representations. Three models—a baseline n-Bloch sphere, a topological model based on the Hopf fibration, and an algebraic model using Geometric Algebra—were evaluated against metrics for state fidelity and entanglement preservation using a series of numerical simulations. Our findings confirm a catastrophic failure of the n-Bloch model to represent entanglement, exhibiting 100% entanglement loss for two-qubit Bell states. We quantify a ‘scalability wall,’ demonstrating that the n-Bloch model’s fidelity remains poor as system size increases, while a principled toy model for advanced representations shows a justifiable decay in fidelity. Furthermore, a redesigned case study on the Quantum Fourier Transform, using superposition inputs, reveals a catastrophic drop in fidelity, demonstrating that the impact of representational error is severe for entanglement-generating algorithms. These results establish a quantifiable basis for the intuition-fidelity trade-off and have significant implications for quantum software development. We conclude that reliance on simple geometric projections is untenable for designing and debugging scalable quantum algorithms, necessitating a shift toward more abstract, representation-aware tools and methodologies.