What is BQP (Bounded-Error Quantum Polynomial time)?

September 02, 2025

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🔑 Definition of BQP

BQP (Bounded-Error Quantum Polynomial time) is the class of decision problems (yes/no problems) that a quantum computer can solve in polynomial time with an error probability of at most 1/3 for all inputs.

“Bounded-error” → The quantum algorithm may be wrong sometimes, but the probability of error is bounded (≤ 1/3).
“Polynomial time” → The number of computation steps grows polynomially with input size (efficient).
“Decision problem” → A problem with a YES or NO answer.

Formally:
A language $L$ is in BQP if there exists a quantum algorithm such that:

If $x \in L$ , the algorithm accepts with probability ≥ 2/3.
If $x \notin L$ , the algorithm accepts with probability ≤ 1/3.

(These constants 2/3 and 1/3 can be replaced with any values > 1/2 and < 1/2, because probabilities can be amplified by repeating the computation.)

⚖️ Analogy with Classical Complexity Classes

P → Problems solvable by a classical computer in polynomial time (deterministic).
BPP → Problems solvable by a classical computer in polynomial time with bounded error (probabilistic).
BQP → Quantum analogue of BPP.

So:

$P \subseteq BPP \subseteq BQP$

🚀 Examples of Problems in BQP

Integer Factorization → Shor’s algorithm solves it efficiently on a quantum computer (classically believed hard).
Discrete Logarithms.
Simulating Quantum Systems → Naturally efficient for quantum machines.
Search Problems → Grover’s algorithm provides quadratic speedup.

🧩 Why BQP Matters

BQP defines what quantum computers can realistically achieve. It helps compare the power of quantum vs classical computation and sets boundaries for quantum advantage.

👉 In short: BQP is the set of problems a quantum computer can solve efficiently, with a small chance of error, making it the quantum counterpart of probabilistic polynomial-time (BPP).

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