How to transform graph states using single-qubit operations: computational complexity and algorithms

TL;DR

This paper investigates the complexity of transforming quantum graph states using specific local operations, proving NP-Completeness in general but providing efficient algorithms for certain practical cases involving low entanglement or small target states.

Contribution

It establishes the NP-Completeness of graph state transformations under LC+LPM+CC and offers efficient algorithms for special cases with practical relevance.

Findings

Deciding graph state transformability is NP-Complete.

Efficient algorithms exist for states with Schmidt-rank width one transforming into GHZ states.

Algorithms also work efficiently when transforming states with unbounded Schmidt-rank width into small GHZ states.

Abstract

Graph states are ubiquitous in quantum information with diverse applications ranging from quantum network protocols to measurement based quantum computing. Here we consider the question whether one graph (source) state can be transformed into another graph (target) state, using a specific set of quantum operations (LC+LPM+CC): single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. We first show that deciding whether a graph state |G> can be transformed into another graph state |G'> using LC+LPM+CC is NP-Complete, even if |G'> is restricted to be the GHZ-state. However, we also provide efficient algorithms for two situations of practical interest: 1. |G> has Schmidt-rank width one and |G'> is a GHZ-state. The Schmidt-rank width is an entanglement measure of quantum states, meaning this algorithm is efficient if the original state has little entanglement. Our algorithm has runtime O(|V(G')||V(G)|^3), and is also efficient in practice even on small instances as further showcased by a freely available software implementation. 2. |G> is in a certain class of states with unbounded Schmidt-rank width, and |G'> is a GHZ-state of a constant size. Here the runtime is O(poly(|V(G)|)), showing that more efficient algorithms can in principle be found even for states holding a large amount of entanglement, as long as the output state has constant size. Our results make use of the insight that deciding whether a graph state |G> can be transformed to another graph state |G'> is equivalent to a known decision problem in graph theory, namely the problem of deciding whether a graph G' is a vertex-minor of a graph G. Many of the technical tools developed to obtain our results may be of independent interest.