Quantum + Qiskit
Map M0 Repo ↗
Module 02 50 min · load 3/5

Qubit, Bloch Sphere, Basis States

The qubit is not a bit. It's a point on a sphere.

The pop-science lie — 'a qubit is both 0 and 1' — is the biggest source of confusion. Kill it in one image.

Slide 15

The qubit is not a bit

anchor · retrieve · master

You already met a qubit in M0 — you entangled two of them. Now we make the object itself precise.

ARM drillstreak 0/3
0/0

For a valid quantum state |ψ⟩ = α|0⟩ + β|1⟩, what must be true?
Slide 16

State vector = Bloch point

build

Every unit-norm one-qubit state maps to exactly one point on this sphere. Drag to rotate — the gold arrow is |ψ⟩.

xz (|0⟩)y|0⟩|1⟩
θ (polar)0°
φ (azimuth)0°
|ψ⟩ = 1.000 |0⟩ + (0.000 + 0.000i) |1⟩
|α|² + |β|² = 1.000 · drag to rotate view

|0⟩ sits at the north pole, |1⟩ at the south. Antipodal points are orthogonal in state space — so |0⟩ and |1⟩ are 180° apart on the sphere but perpendicular in ℂ².

Pull the sliders: θ (polar, 0→π) tips the vector from pole to pole; φ (azimuth, −π→π) rotates it around the z-axis. The amplitudes |α|, |β| update live.

Slide 17

The six cardinal states

recognize

Six states cover 90% of single-qubit reasoning. Memorize them.

State(α, β)Where it lives
|0⟩(1, 0)north pole
|1⟩(0, 1)south pole
|+⟩(1/√2, 1/√2)+x on the equator
|−⟩(1/√2, −1/√2)−x on the equator
|+i⟩(1/√2, i/√2)+y on the equator
|−i⟩(1/√2, −i/√2)−y on the equator
xz (|0⟩)y|0⟩|1⟩
|ψ⟩ = 0.707 |0⟩ + (0.707 + 0.000i) |1⟩
|α|² + |β|² = 1.000 · drag to rotate view

Which two states are antipodal (orthogonal in ℂ²) pairs?

recognize
Slide 18

Basis states via tensor product

build

Two qubits live in ℂ⁴. Their basis states are tensor products of single-qubit basis states.

basis.py python 
import numpy as np

ket_0 = np.array([[1], [0]])
ket_1 = np.array([[0], [1]])

print(np.kron(ket_0, ket_0).flatten())  # |00⟩ → [1 0 0 0]
print(np.kron(ket_0, ket_1).flatten())  # |01⟩ → [0 1 0 0]
print(np.kron(ket_1, ket_0).flatten())  # |10⟩ → [0 0 1 0]
print(np.kron(ket_1, ket_1).flatten())  # |11⟩ → [0 0 0 1]

Which np.kron call produces |10⟩ (vector [0, 0, 1, 0])?

solve
Slide 19

Bloch sphere breaks after one qubit

reflect

One qubit lives in ℂ² and projects to a sphere. Two qubits live in ℂ⁴. Twenty live in ℂ^(2²⁰) — a million-dimensional complex space.

No sphere. No picture. That’s the whole point of quantum computing.

If you can't visualize a 20-qubit state, how would you convince yourself it's still 'real'?

predict
Slide 20

M2 wrap · metacognition

skillopt · reflect

You can now draw any single-qubit state. Gates come next — they’re just rotations of the sphere.

Bloch geometry40 / 100

60 points to unlock next module.

Explain the Bloch sphere to a smart colleague in 60 seconds. What's your first sentence? Don't lead with 'both 0 and 1' — that's the trap.

metacognition
0/40 words

Save your reflection — the EvoMetaClaw meta-model reads it to route your M3 examples. Colleagues who lead with “a state on a sphere” advance; the “both 0 and 1” trap fires a remediation clip.