In computational complexity theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. [1] . It is the quantum analogue to the complexity class BPP.

计算机科学家们已经证明，bqp包含在pspace中，且bqp包含p。 但是他们不知道BQP是否包含在NP中，但是他们相信这两类是不可比的：存在NP中的问题但不是BQP，反之亦然。

BQP is a class of languages L ⊆ (0, 1)∗, decidable with bounded error probability ( say 1/3 ) by a uniform family of polynomial-size quantum circuit over some universal family of gate. In today’s lecture, we will see where this BQP sits in inclusion diagram of complexity classes.

We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH.

bqp 和 np 之间的关系是量子复杂性理论研究的一个活跃领域。 开放性问题包括 BQP 是否包含在 NP 中、BQP 完全问题的存在性以及 BQP 是否等于 P 或 NP。 解决这些问题将加深我们对量子计算的能力和局限性的理解，并对包括密码学在内的各个领域产生影响。

2024年1月8日 · What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting.

2023年3月4日 · There is no known relationship of BQP and NP. The wikipedia page is up to date on the relationship of BQP to other classes. The pdf you linked is not a peer-reviewed publication, and should not be believed.

在计算复杂度理论内，有限错误量子多项式时间（英语： bounded error quantum polynomial time ， BQP ）是一个决定性问题的复杂度类，并且其内的问题可以在多项式时间内以量子电脑解决，错误的机率小于1/3。

2020年7月28日 · The class of problems efficiently solvable by a quantum computer is called $\mathsf{BQP}$, and so the question would ask whether $\mathsf{NP}\subseteq\mathsf{BQP}$. As is indicted in the comments, quantum computers are not known to efficiently solve arbitrary $\mathsf{NP}$ problems, and are suspected not to be able to.

以目前有的结果来看，NP 和 BQP 似乎更像是互不包含。 Scott Aaronson 有一篇survey（有点老）可以读一读 [quant-ph/0502072] NP-complete Problems and Physical Reality. 说一下我的见解，我认为就算量子计算机成功在多项式时间解决了某个NP完全问题那也不能说P=NP。 一切从定义出发，P表示确定性图灵机多项式可解问题集合，NP表示非确定性图灵机多项式可解问题集合。 P=?NP是问：在非确定性图灵机上多项式可解的问题是不是在确定性图灵机上也多项式可解？ …