Visualizing a Multi-Qubit State; an Extension of the Bloch Sphere

The conventional visualization of a single qubit is the Bloch sphere, which has proven to be a successful tool for developing intuition. In addition to showing all the information of an arbitrary state, the Bloch sphere clearly depicts the effect of any 1-qubit gate. Unfortunately, the generalization to multiple qubits is nontrivial. Since nearly all of quantum computing is derived from interactions between qubits, a visualization of multi-qubit states which have similar geometric intuitions to the Bloch sphere would be an effective pedagogical tool. Since any n-qubit gate can be decomposed into combinations of 1- and 2-qubit gates, an intuitive 2-qubit state visualization would be incredibly useful in developing understanding of quantum algorithms.

Here I derive and present a visualization of a pure 2-qubit state consisting of 3 spheres which preserves some of the Bloch sphere's geometric intuitions. Previous visualizations of multi-qubit states have largely focused on directly representing entanglement. This visualization prioritizes geometric intuition with regard to the computational basis states, which gives it higher potential for effectively portraying the effect of an arbitrary 2-qubit gate. The probabilities and phases of each of the four computational basis states are readily apparent in the visual representation, just as they are in the Bloch sphere. Additionally, the depiction can easily be extended to a pure n-qubit state while maintaining these intuitions, using 2ⁿ-1 spheres.