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# System prepended metadata

title: Quantum Routing

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## Problem Statement: Routing and Optimal Ordering for Quantum Networks

Consider an underlying **grid graph** $G = (V, E)$, where the set of nodes is $V = \{1, 2, \dots, n\}$ and edges $E$ represent possible quantum links between nodes.

Each edge $(u, v) \in E$ is characterized by:

- an elementary entanglement generation probability $p_{u,v} \in [0,1]$, and 
- a link capacity $C_{u,v} > 0$, denoting the maximum number of concurrent entanglement attempts per time slot.

Each node $w \in V$ has an associated entanglement swapping (Bell-state measurement) success probability $q_w \in [0,1]$.

### Demand Matrix

A **demand matrix** $D = [D_{i,j}]$ specifies the required number of end-to-end entangled pairs (or *entanglement requests*) between each node pair $(i, j)$ per time slot.

### Objective

The objective is to determine an **optimal routing and ordering matrix** $O = [O_{i,j}]$ that specifies, for each demand pair $(i,j)$:

1. The optimal **path or set of paths** through $G$ to fulfill the entanglement demand $D_{i,j}$. 
2. The optimal **ordering or scheduling** of entanglement generation and swapping operations across the network.
