论文标题
在量子电路中对少量大错误的轻量级检测
Lightweight Detection of a Small Number of Large Errors in a Quantum Circuit
论文作者
论文摘要
假设我们要实现一个统一的$ u $，例如用于某些量子算法的电路。假设我们的实际实现是统一的$ \ tilde {u} $，我们只能将其应用于黑框。通常，确定$ \ tilde {u} $等于预期的$ u $，还是在最坏情况下有很大差异。在本文中，我们考虑了两种特殊情况，这些任务存在相对高效且轻巧的程序。 首先，我们在假设$ u $和$ \ tilde {u} $（现在可以用作黑色框）的假设下给出了一个有效的程序，或者仅在一个$ k $ qubit Gate中相同，或者在一个$ k = o（1）$（$ k $ qubits）中只有显着差异。其次，在假设$ u $和$ \ tilde {u} $的假设是Clifford电路相等或以任意方式不同的假设（现在是经典给出$ u $的规范，而$ \ tilde {u} $仍然只能用作黑色单盒），我们给出了更轻巧的程序。这两个过程只需要运行$ \ tilde {u} $恒定数量即可在最坏情况下检测恒定误差。我们注意到，Clifford的结果也来自Flammia和Liu的早期工作以及Da Silva，Landon-Cardinal和Poulin。 在Clifford情况下，我们的错误检测过程还使我们有效地学习（正确）$ \ tilde {u} $，如果我们有一小部分可能发生在$ u $的可能发生的错误列表；例如，如果我们知道$ \ tilde {u} $的门的$ o（1）$是错误的，那么此列表将很小，我们可以用$ \ tilde {u} $测试每个可能的错误版本的$ u $。
Suppose we want to implement a unitary $U$, for instance a circuit for some quantum algorithm. Suppose our actual implementation is a unitary $\tilde{U}$, which we can only apply as a black-box. In general it is an exponentially-hard task to decide whether $\tilde{U}$ equals the intended $U$, or is significantly different in a worst-case norm. In this paper we consider two special cases where relatively efficient and lightweight procedures exist for this task. First, we give an efficient procedure under the assumption that $U$ and $\tilde{U}$ (both of which we can now apply as a black-box) are either equal, or differ significantly in only one $k$-qubit gate, where $k=O(1)$ (the $k$ qubits need not be contiguous). Second, we give an even more lightweight procedure under the assumption that $U$ and $\tilde{U}$ are Clifford circuits which are either equal, or different in arbitrary ways (the specification of $U$ is now classically given while $\tilde{U}$ can still only be applied as a black-box). Both procedures only need to run $\tilde{U}$ a constant number of times to detect a constant error in a worst-case norm. We note that the Clifford result also follows from earlier work of Flammia and Liu, and da Silva, Landon-Cardinal, and Poulin. In the Clifford case, our error-detection procedure also allows us efficiently to learn (and hence correct) $\tilde{U}$ if we have a small list of possible errors that could have happened to $U$; for example if we know that only $O(1)$ of the gates of $\tilde{U}$ are wrong, this list will be polynomially small and we can test each possible erroneous version of $U$ for equality with $\tilde{U}$.
