1. Complexity measures

Quantum algorithms may reduce oracle queries while still requiring expensive state preparation, arithmetic, or measurement. Distinguish asymptotic gate count, depth, qubits, T-count, shots, and wall-clock time.

A speedup claim must specify the resource being improved.

2. Data loading and readout

Amplitude encoding can compress classical data into a quantum state, but preparing that state may erase the theoretical advantage. Likewise, quantum output is sampled, so extracting a full classical vector may be expensive.

Input and output assumptions often determine whether an algorithm is practical.

3. Dequantization and classical competition

Some proposed quantum advantages inspire improved classical algorithms, tensor-network methods, randomized numerical linear algebra, or specialized hardware. Benchmarks must evolve as classical methods improve.

Quantum advantage is a moving comparison, not a fixed label.

4. Applied activity

Advanced exercise: derive or simulate one representative result from this module, document assumptions, and produce a one-page technical interpretation for a non-specialist decision maker.