About Qubit States & Bloch Sphere
A qubit is the quantum analog of a classical bit. While a classical bit is either 0 or 1, a qubit can exist in a superposition: |ψ⟩ = α₀|0⟩ + α₁|1⟩, where α₀ and α₁ are complex amplitudes satisfying |α₀|² + |α₁|² = 1. The Bloch sphere is a unit sphere where every point on the surface represents a valid single-qubit pure state. This visualization walks through fundamental qubit states (|0⟩, |1⟩, |+⟩, |−⟩) and shows how quantum gates rotate the state vector on the Bloch sphere.
Complexity Analysis
- Time Complexity
- N/A (state representation)
- Space Complexity
- O(1) per qubit
- Difficulty
- beginner
Key Concepts
Qubit
The fundamental unit of quantum information, capable of existing in a superposition of |0⟩ and |1⟩.
Bloch Sphere
A unit sphere where every point on the surface represents a valid single-qubit pure state. The north pole is |0⟩, the south pole is |1⟩, and the equator contains equal superpositions like |+⟩ and |−⟩.
Superposition
A qubit in state α|0⟩ + β|1⟩ is in a superposition — it's not 'secretly' 0 or 1, but genuinely both until measured.
Measurement Probability
When measured, the probability of getting |0⟩ is |α₀|² and |1⟩ is |α₁|², always summing to 1 (Born rule).
Common Pitfalls
Phase vs Probability
Two states can have the same measurement probabilities but different phases (e.g., |+⟩ and |+i⟩ both give 50/50, but they differ on the Bloch sphere). Phase matters for interference.
Global Phase
Multiplying a state by e^{iγ} doesn't change any observable — the Bloch sphere representation removes global phase.