The hope of the quantum computing field is that quantum architectures are able to scale up and realize fault-tolerant quantum computing. Due to engineering challenges, such ''cheap'' error correction may be decades away. In the meantime, we anticipate an era of ''costly'' error correction, or $\textit{early fault-tolerant quantum computing}$. Costly error correction might warrant settling for error-prone quantum computations. This motivates the development of quantum algorithms which are robust to some degree of error as well as methods to analyze their performance in the presence of error. Several such algorithms have recently been developed; what is missing is a methodology to analyze their robustness. To this end, we introduce a randomized algorithm for the task of phase estimation and give an analysis of its performance under two simple noise models. In both cases the analysis leads to a noise threshold, below which arbitrarily high accuracy can be achieved by increasing the number of samples used in the algorithm. As an application of this general analysis, we compute the maximum ratio of the largest circuit depth and the dephasing scale such that performance guarantees hold. We calculate that the randomized algorithm can succeed with arbitrarily high probability as long as the required circuit depth is less than 0.916 times the dephasing scale.

中文翻译：

---

论证明早期容错量子计算机算法的鲁棒性


量子计算领域的希望是量子架构能够扩展并实现容错量子计算。由于工程挑战，这种“廉价”的纠错可能需要几十年的时间。与此同时，我们预计会出现一个“成本高昂”的纠错或 $\textit{早期容错量子计算}$ 的时代。成本高昂的纠错可能需要满足于容易出错的量子计算。这推动了量子算法的发展，这些算法在一定程度的误差下具有鲁棒性，以及在存在误差的情况下分析其性能的方法。最近开发了几种这样的算法;缺少的是分析其稳健性的方法。为此，我们引入了一种用于相位估计任务的随机算法，并分析了它在两个简单噪声模型下的性能。在这两种情况下，分析都会得到一个噪声阈值，低于该阈值可以通过增加算法中使用的样本数量来实现任意高的准确性。作为这种一般分析的一个应用，我们计算了最大电路深度和去相尺度的最大比率，以便保证性能。我们计算出，只要所需的电路深度小于去相尺度的 0.916 倍，随机算法就可以以任意高的概率成功。