We propose a set of protocols for verifying quantum computing at any time after the computation itself has been performed. We provide two constructions: one requires five entangled provers and a completely classical verifier; the other requires a single prover, a verifier, who is restricted to measuring qubits in the $X$ or $Z$ basis, and one-way quantum communication from the prover to the verifier. These results demonstrate that the verification can be achieved independently from the blindness. We also show that a constant round protocol with a single prover and a completely classical verifier is not possible, unless bounded error quantum polynomial time (BQP) is contained in the third level of the polynomial hierarchy.

中文翻译：

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量子计算事后验证

我们提出了一套协议，用于在计算本身执行后的任何时间验证量子计算。我们提供了两种构造：一种需要五个纠缠的证明者和一个完全经典的验证者；另一个则需要一个证明者，即一个验证者，该证明者仅限于测量$X$ 或者 $ž$基础，以及从证明者到验证者的单向量子通信。这些结果表明，可以独立于盲目进行验证。我们还表明，除非有限制的误差量子多项式时间（BQP）包含在多项式层次结构的第三级中，否则不可能使用具有单个证明者和完全经典的验证者的恒定舍入协议。