Lecture 022 - Complexity Classes

Complexity Classes:

• P: polynomial with no random classical computer

• NP: polynomial verification, or, equivalently, there exists a randomized algorithm can try to find the answer with non-zero probability

• NP-complete: hardest in NP

• BPP: polynomial with random classical computer (BPP = P)

• BQP: polynomial with random quantum computer

  • Quantum-Eval in BQP-hard
  • BQP not in NP
  • BQP in EXPTIME
  • BQP in PSPACE (polynomial space: by running only one branch once)

• PP: quantum computer can get $0.5 + \gamma$ probability, but classical can get $0.5 + \gamma / 2^n$ probability.

Reduced QC: H, NOT, CNOT, 1QubitRot(pi/8) is quantum-complete

• If no H: classic

• If only NOT, CNOT, H: classic

Classical Simulation: if there is a quantum decision problem

• if QC: $Pr\{1\} = \frac{1}{2} + q$ with $-\frac{1}{2} \leq q \leq \frac{1}{2}$

• then classic: $Pr\{1\} = \frac{1}{2} + \frac{q}{2^h}$ with $h = \text{ number of H gates}$