Superposition and phase
Recognize why equal probabilities do not imply identical states.
Intuition
|+⟩ and |−⟩ give the same 50/50 computational-basis probabilities. Their relative phase differs, and a later gate can turn that difference into different outcomes.
Prepare |+⟩ with H, then change its relative sign with Z. Try another H: H|+⟩ is |0⟩, while H|−⟩ is |1⟩. Interference makes phase observable through a suitable experiment.
The mathematics
Look a little deeper
On the Bloch sphere these two states lie at opposite points on the x axis. Global phase rotates every amplitude equally and has no observable effect; relative phase changes their relationship.
See a single qubit
The interactive state explorer requires the quantum simulator, which is not hosted in this preview.
Use it
Prepare |−⟩ in at most two gates using H, X, or Z. Inspect phase in Statevector and Bloch.
This exercise uses the same server-side state and circuit checks as its linked practice problem.
Build your circuit
Click a gate to append it, or drag it onto a wire. CX uses the selected control; SWAP uses it as the second operand. Basis order: |qₙ … q₀⟩.
Returned logical circuit · before transpilation · q0 on top
Connect to code
qc.h(0) qc.z(0)
These operations go inside a circuit-building program. Practice problems provide a complete solve() template.
Check your understanding
First complete the circuit exercise successfully.