Transforming graph states using single-qubit operations

TL;DR

This paper presents an efficient algorithm to determine and implement transformations between stabilizer states using local Clifford operations, measurements, and classical communication, with applications in quantum networks and error correction.

Contribution

It introduces a graph-theoretic algorithm for stabilizer state transformations based on vertex-minor problems, with a software implementation and constant-time application method.

Findings

Decides stabilizer state transformations efficiently

Provides a polynomial-time algorithm for vertex-minor problem with bounded rank-width

Offers a practical tool for quantum information processing

Abstract

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM), and classical communication (CC) between sites holding the individual qubits. What's more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool - for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G' is a vertex-minor of another graph G. Here we show that the vertex-minor problem can be solved in time O(|G|^3) where |G| is the size of the graph G, whenever the rank-width of G and the size of G' are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information.