Containment of NP in BQP

Ascertain whether the nondeterministic polynomial-time class NP is contained within the bounded-error quantum polynomial-time class BQP; specifically, determine if every NP decision problem can be solved by a quantum algorithm in polynomial time with bounded error.

Background

While introducing BQP, the author highlights key unresolved relationships between quantum and classical complexity classes. Among them is whether NP lies within BQP, a question central to understanding the ultimate scope of efficient quantum computation.

Establishing NP ⊆ BQP (or ruling it out) would clarify the limits of quantum algorithms relative to classical verification and has direct implications for which hard problems could become tractable on quantum hardware.

References

Briefly, there are huge, century-old open questions in the theory of computer science that would net you fame and fortune if you solved them. One such example is, does P = NP? Another: is NP contained in BQP? We suspect the answer is "no" to both of these questions, but there is currently no mathematical proof.

— What You Shouldn't Know About Quantum Computers (2405.15838 - Ferrie, 24 May 2024) in Myth 5: Quantum Computers Will Replace Digital Computers — BQP subsection