Quantum protocols commonly require a certain number of quantum resource states to be available simultaneously. An important class of examples is quantum network protocols that require a certain number of entangled pairs. Here, we consider a setting in which a process generates a quantum resource state with some probability $p$ in each time step, and stores it in a quantum memory that is subject to time-dependent noise. To maintain sufficient quality for an application, each resource state is discarded from the memory after $w$ time steps. Let $s$ be the number of desired resource states required by a protocol. We characterise the probability distribution $X_{(w,s)}$ of the ages of the quantum resource states, once $s$ states have been generated in a window $w$. Combined with a time-dependent noise model, the knowledge of this distribution allows for the calculation of fidelity statistics of the $s$ quantum resources. We also give exact solutions for the first and second moments of the waiting time $\tau_{(w,s)}$ until $s$ resources are produced within a window $w$, which provides information about the rate of the protocol. Since it is difficult to obtain general closed-form expressions for statistical quantities describing the expected waiting time $\mathbb{E}(\tau_{(w,s)})$ and the distribution $X_{(w,s)}$, we present two novel results that aid their computation in certain parameter regimes. The methods presented in this work can be used to analyse and optimise the execution of quantum protocols. Specifically, with an example of a Blind Quantum Computing (BQC) protocol, we illustrate how they may be used to infer $w$ and $p$ to optimise the rate of successful protocol execution.

翻译：量子协议通常要求同时准备好一定数量的量子资源态。一个重要的实例是量子网络协议，其需要特定数量的纠缠对。本文考虑以下场景：在每个时间步，一个过程以概率 $p$ 生成量子资源态，并将其存储在受时间相关噪声影响的量子存储器中。为维持应用所需的足够质量，每个资源态在存储 $w$ 个时间步后被丢弃。设 $s$ 为协议所需资源态的数量。我们刻画了在时间窗口 $w$ 内生成 $s$ 个资源态后，这些量子资源态年龄的概率分布 $X_{(w,s)}$。结合时间相关噪声模型，该分布的知识可用于计算 $s$ 个量子资源的保真度统计量。我们还给出了等待时间 $\tau_{(w,s)}$（即在窗口 $w$ 内生成 $s$ 个资源所需时间）的一阶矩和二阶矩的精确解，这提供了协议速率的信息。由于难以获得描述期望等待时间 $\mathbb{E}(\tau_{(w,s)})$ 和分布 $X_{(w,s)}$ 的统计量的通用闭合形式表达式，我们提出了两个新结果，以辅助在特定参数范围内的计算。本文提出的方法可用于分析和优化量子协议的执行。具体而言，通过一个盲量子计算（BQC）协议的示例，我们展示了如何利用这些方法推断 $w$ 和 $p$，以优化协议成功执行的速率。