The Bloch sphere is an elegant tool for representing single-qubit states. However, a widely accepted generalization for multi-qubit systems with entanglement remains absent. We propose a novel geometric model extending the Bloch sphere representation to arbitrary
N-qubit systems using
2N−1 spheres. We demonstrate that any pure 2-qubit state is uniquely described by three spheres: two for individual qubits and a third encapsulating bipartite entanglement. Generalizing this, we establish an
N-qubit parameterization through the hierarchical application of controlled rotation gates along the
Z and
Y axes. We formally prove a strict bijection between the standard state vector representation and our model's angular parameters. This framework provides an intuitive visualization of multiple entanglement, offering potential computational advantages for quantum simulators and new analytical perspectives on quantum gates.
Quantum computing leverages fundamental principles of quantum mechanics to perform computations far exceeding classical architectures. At the core of single-qubit quantum state manipulation is the Bloch sphere, a geometric representation mapping the state space of two-level quantum systems onto a three-dimensional unit sphere. This paper presents a comprehensive theoretical framework of qubits, starting from single-qubit geometric state vectors and expanding to rigorous multi-qubit mathematical formulations. Furthermore, it analyzes single-qubit quantum gates as unitary rotations on the Bloch sphere, evaluates multi-qubit system scaling, and explicitly proves state normalization conditions using Born's rule across multi-dimensional Hilbert spaces.