3. Complexity: BQP vs P vs NP

Quantum Complexity Theory

Where does quantum computing fit in the landscape of solvable problems?

• P (Polynomial): Problems solvable quickly by a classical computer (Multiplication).
• NP (Nondeterministic Polynomial): Problems where a solution can be verified quickly (Sudoku, Traveling Salesman).
• BQP (Bounded-error Quantum Polynomial): Problems solvable quickly by a quantum computer.

The Reality: $P \subseteq BQP$. Quantum computers can solve everything classical computers can. But can they solve NP-Complete problems? Probably not.

They excel at specific "hidden structure" problems (like Factoring) that are likely in NP-Intermediate.