Curriculum / Quantum Gates Deep Dive / The Bloch Sphere
The Bloch Sphere
Visualize every single-qubit state as a point on a sphere.
The Bloch Sphere
The Bloch sphere is the single most useful picture in quantum computing. It turns the abstract algebra of a single qubit into a concrete geometric object: a point on the surface of a unit sphere. Once you are comfortable reading the sphere, every single-qubit gate becomes a rotation you can visualise, and measurement probabilities stop feeling mysterious.
The Math in One Line
Every pure state of a single qubit can be written as:
|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩
Two real numbers, θ (polar angle, 0 to π) and φ (azimuthal angle, 0 to 2π), are enough to describe the state. That is exactly one point on the unit sphere in 3D. The overall phase is unobservable, so we drop it: the sphere is the physical state space of one qubit.
The Poles and the Equator
- •North pole (|0⟩): θ=0. The qubit is definitely "off", measuring gives 0 with probability 1.
- •South pole (|1⟩): θ=π. The qubit is definitely "on", measuring gives 1 with probability 1.
- •Equator: θ=π/2. Pure superpositions, each with 50/50 measurement probability but different phases φ.
Four special equator points have names you will see constantly:
- •|+⟩ = (|0⟩ + |1⟩)/√2, along the +X axis, φ=0
- •|−⟩ = (|0⟩ − |1⟩)/√2, along the −X axis, φ=π
- •|i⟩ = (|0⟩ + i|1⟩)/√2, along the +Y axis, φ=π/2
- •|−i⟩ = (|0⟩ − i|1⟩)/√2, along the −Y axis, φ=3π/2
All four return 50/50 outcomes in the computational basis. The phase is what distinguishes them, and phase only becomes visible once you interfere the state with another gate.
When we "measure in the computational basis" we are projecting the state onto the Z axis of the Bloch sphere. A state's z-coordinate is exactly P(0) − P(1). The further north, the more likely you see 0; the further south, the more likely you see 1; on the equator both are equally likely.
This is the opening of the lesson. The full walkthrough, the interactive circuit, and the graded challenge continue inside myqubit.
How this lesson works
A guided reading lesson with interactive knowledge checks. Concepts are explained step by step with circuit diagrams and runnable examples, and you confirm understanding before moving on.
Part of: Quantum Gates Deep Dive
Master single-qubit and multi-qubit gates. Understand rotations, phases, and the math behind every gate in Qiskit.
This lesson is part of Pro
Unlock Quantum Gates Deep Dive and all 10 tracks with Pro: $12.99/month, $79/year, or $97 lifetime. Start with the free track first if you are new.