Print Email Facebook Twitter Transforming graph states using single-qubit operations Title Transforming graph states using single-qubit operations Author Dahlberg, E.A. (TU Delft QID/Wehner Group; TU Delft QuTech Advanced Research Centre) Wehner, S.D.C. (TU Delft Quantum Internet Division; TU Delft Quantum Information and Software; TU Delft QuTech Advanced Research Centre) Department Quantum Internet Division Date 2018-07-13 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 is 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. The vertex-minor problem is, in general, NP-Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. 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. This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'. Subject Computational complexityGraph statesRank-widthSingle-qubit Clifford operationsSingle-qubit Pauli measurements To reference this document use: http://resolver.tudelft.nl/uuid:2ee9a0ce-21cc-4e01-af04-fd1f9a87223e DOI https://doi.org/10.1098/rsta.2017.0325 ISSN 1364-503X Source Royal Society of London. Philosophical Transactions A. Mathematical, Physical and Engineering Sciences, 376 (2123) Part of collection Institutional Repository Document type journal article Rights © 2018 E.A. Dahlberg, S.D.C. Wehner Files PDF courcelle1.pdf 552.5 KB Close viewer /islandora/object/uuid:2ee9a0ce-21cc-4e01-af04-fd1f9a87223e/datastream/OBJ/view