# Transforming graph states to Bell-pairs is NP-Complete

## TL;DR

This paper proves that determining whether a complex quantum entangled state can be converted into Bell pairs between specific nodes using limited operations is an NP-Complete problem, highlighting computational challenges in quantum network management.

## Contribution

It establishes the NP-Completeness of transforming graph and stabilizer states into Bell pairs with restricted local operations and classical communication.

## Key findings

- The problem is NP-Complete.
- Transformation feasibility is computationally hard.
- Implications for quantum network protocols.

## Abstract

Critical to the construction of large scale quantum networks, i.e. a quantum internet, is the development of fast algorithms for managing entanglement present in the network. One fundamental building block for a quantum internet is the distribution of Bell pairs between distant nodes in the network. Here we focus on the problem of transforming multipartite entangled states into the tensor product of bipartite Bell pairs between specific nodes using only a certain class of local operations and classical communication. In particular we study the problem of deciding whether a given graph state, and in general a stabilizer state, can be transformed into a set of Bell pairs on specific vertices using only single-qubit Clifford operations, single-qubit Pauli measurements and classical communication. We prove that this problem is NP-Complete.