Harrow-Hassidim-Lloyd algorithm: solve Ax=b for a 2×2 system A=diag(1,2), b=|0⟩ using 4 qubits.

Determine in one query whether a 3-bit Boolean function is constant or balanced (oracle: f = x₀ ⊕ x₁ ⊕ x₂).

Find the hidden period s='11' of a 2-to-1 function f(x) = f(x⊕s) with exponential quantum speedup.

Core period-finding subroutine of Shor's algorithm: find the period r of f(x) = 7^x mod 15.