Tools for the analysis of quantum protocols requiring state generation within a time window

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 <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$w$</tex-math></inline-formula> time steps. Let <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> be the number of desired resource states required by a protocol. We characterise the probability distribution <inline-formula><tex-math notation="LaTeX">$X_{(w,s)}$</tex-math></inline-formula> of the ages of the quantum resource states, once <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> states have been generated in a window <inline-formula><tex-math notation="LaTeX">$w$</tex-math></inline-formula>. Combined with a time-dependent noise model, knowledge of this distribution allows for the calculation of fidelity statistics of the <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> quantum resources. We also give exact solutions for the first and second moments of the waiting time <inline-formula><tex-math notation="LaTeX">$\tau _{(w,s)}$</tex-math></inline-formula> until <inline-formula><tex-math notation="LaTeX">$s$</tex-math></inline-formula> resources are produced within a window <inline-formula><tex-math notation="LaTeX">$w$</tex-math></inline-formula>, 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 <inline-formula><tex-math notation="LaTeX">$\mathbb {E}(\tau _{(w,s)})$</tex-math></inline-formula> and the distribution <inline-formula><tex-math notation="LaTeX">$X_{(w,s)}$</tex-math></inline-formula>, 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 <inline-formula><tex-math notation="LaTeX">$w$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> to optimise the rate of successful protocol execution.