We define rewinding operators that invert quantum measurements. Then, we define complexity classes ${\sf RwBQP}$, ${\sf CBQP}$, and ${\sf AdPostBQP}$ as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that ${\sf BPP}^{\sf PP}\subseteq{\sf RwBQP}={\sf CBQP}={\sf AdPostBQP}\subseteq{\sf PSPACE}$. As a byproduct of this result, we show that any problem in ${\sf PostBQP}$ can be solved with only postselections of outputs whose probabilities are polynomially close to one. Under the strongly believed assumption that ${\sf BQP}\nsupseteq{\sf SZK}$, or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. In addition, we consider rewindable Clifford and instantaneous quantum polynomial time circuits.

翻译：我们定义可倒带算子以逆转量子测量。接着，我们定义复杂度类${\sf RwBQP}$、${\sf CBQP}$和${\sf AdPostBQP}$，分别为由具有多项式数量可倒带算子、克隆算子和自适应后选择的多项式规模量子电路可解的判定问题集合。我们的主要结果是${\sf BPP}^{\sf PP}\subseteq{\sf RwBQP}={\sf CBQP}={\sf AdPostBQP}\subseteq{\sf PSPACE}$。作为该结果的副产品，我们证明${\sf PostBQP}$中的任何问题仅需对概率多项式接近1的输出进行后选择即可求解。在强烈相信的假设${\sf BQP}\nsupseteq{\sf SZK}$（即最短独立向量问题难以用量子计算机高效求解）下，我们还表明单个倒带算子足以实现量子计算难以处理的任务。此外，我们考虑了可倒带克利福德门和瞬时量子多项式时间电路。