Optimal Fidelity-Aware Entanglement Distribution in Linear Quantum Networks

TL;DR

This paper addresses end-to-end entanglement distribution in a two-hop quantum network with memory decoherence by framing purification and swapping as combinatorial optimization problems. It shows that optimal swapping corresponds to a max-weight matching on a bipartite graph, while optimal purification corresponds to a max-weight matching on a non-bipartite graph, and introduces two polynomial algorithms, PtS and StP, that differ in the order of operations. Through numerical experiments, it demonstrates that PtS typically outperforms StP and that omitting purification in StP can yield substantial gains, with swap-only policies performing best under additive utilities. The framework provides a foundation for scalable memory-management and purification strategies in larger networks and can be extended to joint routing and dynamic multi-commodity scenarios.

Abstract

We study the problem of entanglement distribution in terms of maximizing a utility function that captures the total fidelity of end-to-end entanglements in a two-link linear quantum network with a source, a repeater, and a destination. The nodes have several quantum memories, and the problem is how to coordinate entanglement purification in each of the links, and entanglement swapping across links, so as to achieve the goal above. We show that entanglement swapping (i.e, deciding on the pair of qubits from each link to perform swapping on) is equivalent to finding a max-weight matching on a bipartite graph. Further, entanglement purification (i.e, deciding which pairs of qubits in a link will undergo purification) is equivalent to finding a max-weight matching on a non-bipartite graph. We propose two polynomial algorithms, the Purify-then-Swap (PtS) and the Swap-then-Purify (StP) ones, where the decisions about purification and swapping are taken with different order. Numerical results show that PtS performs better than StP, and also that the omission of purification in StP gives substantial benefits.

Figures (3)

• Figure 1: (a) Quantum Teleportation, with which one bit of information is transmitted from A to B, if A,B share an entangled pair of qubits; (b) Entanglement swapping that leads to end-to-end entanglement between A and B, through BSM measurement at the qubits in the memories of repeater R. After end-to-end entanglement is established, A can send an information bit to B through teleportation.
• Figure 2: Upper part (a): Equivalence of the optimal purification problem to max-weight matching in a non-bipartite graph. We show a hypothetical best solution: LLEs b and c are purified, while LLE a is not. This corresponds to a max-weight matching in the non-bipartite graph on the right. Bottom part (b): Equivalence of the optimal swapping problem to max-weight matching in a bipartite graph. We show a hypothetical best solution: LLEs a,b are swapped, and so are LLEs b,h, and LLEs c,e. This corresponds to the max-weight matching in the bipartite graph on the right.
• Figure 3: Example of the Purify-then-Swap algorithm. Purification: In link $(s,r)$, LLEs a and b are purified, while c is not. The new purified LLE is stored in memory 1 of s and 1 of r. In link $(r,d)$, LLEs e and h are purified, while q is not. The new purified LLE is stored in memory 4 or r and 2 of d. Swapping:LLEs a and q are swapped, and also LLEs c and e are swapped. The outcome is two end-to-end entanglements.