# Optimal quantum spatial search on random temporal networks

## TL;DR

This paper demonstrates that quantum spatial search on a dynamic Erdős-Rényi network can be optimized by tuning the temporal interval, achieving quadratic speedup even when static counterparts are sub-optimal.

## Contribution

It analytically shows how the interplay of network temporality and connectivity enables optimal quantum search on evolving random networks, a novel insight in quantum network theory.

## Key findings

- Optimal search time $\\mathcal{O}(\sqrt{n})$ for certain temporal regimes
- Sub-optimal search performance in some regimes despite high connectivity
- Temporality can be exploited to enhance quantum information tasks

## Abstract

To investigate the performance of quantum information tasks on networks whose topology changes in time, we study the spatial search algorithm by continuous time quantum walk to find a marked node on a random temporal network. We consider a network of $n$ nodes constituted by a time-ordered sequence of Erd\"os-R\'enyi random graphs $G(n,p)$, where $p$ is the probability that any two given nodes are connected: after every time interval $\tau$, a new graph $G(n,p)$ replaces the previous one. We prove analytically that for any given $p$, there is always a range of values of $\tau$ for which the running time of the algorithm is optimal, i.e.\ $\mathcal{O}(\sqrt{n})$, even when search on the individual static graphs constituting the temporal network is sub-optimal. On the other hand, there are regimes of $\tau$ where the algorithm is sub-optimal even when each of the underlying static graphs are sufficiently connected to perform optimal search on them. From this first study of quantum spatial search on a time-dependent network, it emerges that the non-trivial interplay between temporality and connectivity is key to the algorithmic performance. Moreover, our work can be extended to establish high-fidelity qubit transfer between any two nodes of the network. Overall, our findings show that one can exploit temporality to achieve optimal quantum information tasks on dynamical random networks.