Understanding the relationship between BQP, P, and NP is a key part of quantum complexity theory. Each of these classes represents a set of problems based on how efficiently they can be solved using different types of computers: classical vs. quantum.
Let’s break this down clearly and step by step.
1. What Are These Complexity Classes?
P (Polynomial Time)
- Stands for problems solvable by a classical computer in “reasonable” time (polynomial time).
- These are problems where we can find solutions efficiently on today’s computers.
- Examples: Sorting numbers, checking if a number is even, etc.
NP (Nondeterministic Polynomial Time)
- These are problems where if you are given a solution, you can verify it quickly using a classical computer.
- Includes problems that might take a long time to solve, but if someone shows you a correct answer, you can check it fast.
- Example: Solving a Sudoku puzzle — hard to solve, easy to verify.
BQP (Bounded-Error Quantum Polynomial Time)
- Represents problems a quantum computer can solve efficiently (in polynomial time), with a small chance of error (that can be reduced).
- Includes many problems that are not known to be solvable efficiently by classical computers.
- Example: Factoring large numbers (important for breaking encryption) can be done using Shor’s algorithm, which is in BQP.
2. How Do These Classes Relate?
The exact relationships are not fully known yet, but here’s what researchers believe and what we do know:
P ⊆ BQP
- Everything you can do efficiently on a classical computer (P) can also be done on a quantum computer.
- So, BQP includes P.
Is P = BQP?
- Not likely. Quantum computers are believed to be more powerful than classical ones for some problems.
- So most experts believe P ≠ BQP.
Is NP inside BQP?
- Unknown.
- Some NP problems may be in BQP, but it’s believed that most NP-complete problems are not solvable efficiently by quantum computers.
- Example: Solving general Sudoku or Traveling Salesman problems may still be hard even for quantum computers.
Does BQP contain NP?
- Probably not the whole of NP.
- Quantum computers give speedups for some structured problems, but NP-complete problems are believed to be hard even for quantum systems.
3. Visual Representation (Venn Diagram Style)
NP
┌─────┐
│ │
│ P │
│ ┌─────┐
│ │ BQP │
│ └─────┘
└─────┘
Explanation:
- P is inside both NP and BQP.
- BQP may overlap with part of NP, but not necessarily contain it.
- Some problems in BQP might not be in NP because they’re not easy to verify classically.
4. Example Problem Comparisons
| Problem | In P? | In NP? | In BQP? | Notes |
|---|---|---|---|---|
| Sorting numbers | Yes | Yes | Yes | Easy problem |
| Sudoku (general case) | No | Yes | Unknown | NP-complete |
| Integer factoring | No | Yes | Yes | In BQP (Shor’s algorithm) |
| Grover’s Search (unsorted search) | No | Yes | Yes | Quantum gives square-root speedup |
5. Key Insights
- Quantum speedup is real, but not magic — it doesn’t solve everything fast.
- BQP helps with certain structured problems, not necessarily with all hard problems.
- Factoring, once considered hard, is now easy for quantum — which threatens current encryption systems.
- NP-complete problems seem to remain difficult even with quantum power — but this isn’t formally proven.
6. Summary
| Complexity Class | Description |
|---|---|
| P | Problems solvable efficiently by classical computers. |
| NP | Problems where solutions can be verified efficiently by classical computers. |
| BQP | Problems solvable efficiently by quantum computers with a small chance of error. |
What We Know
- P is a subset of BQP.
- Some NP problems are in BQP (like factoring).
- BQP probably does not include all of NP.
- The relationships P vs. NP and NP vs. BQP remain unsolved mysteries in computer science.