SAMOP 2023 – scientific programme
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Q: Fachverband Quantenoptik und Photonik
Q 34: Quantum Communication (joint session QI/Q)
Q 34.4: Talk
Wednesday, March 8, 2023, 15:30–15:45, B305
Easy-to-compute local Clifford invariants for graph states — •Frederik Hahn1 and Adam Burchardt2,3,4 — 1Technische Universität Berlin, Berlin, Deutschland — 2Universität Amsterdam, Amsterdam, Niederlande — 3QuSoft, Amsterdam, Niederlande — 4CWI, Amsterdam, Niederlande
In this work, we study easy-to-compute LC-invariants of graph states. Although previous studies have already led to finite sets of invariants that fully characterize the LC-equivalence classes of graph states, these invariants are computationally inefficient. Their computation requires knowledge of the given state's full stabilizer set, which is exponential in the number of its qubits n.
In this paper, without the need to calculate this entire stabilizer set, we instead present an easy-to-calculate LC-invariant of order O(n^3). It is closely related to the so-called foliage of a graph and has a simple graphical interpretation in terms of leaves, axils, and twins: For any graph, we define a partition of the set of its vertices based on a simple equivalence relation and call it the foliage partition of this graph. We further show that foliage partitions remain invariant under any local complementation of the corresponding graph. Foliage partitions then represent simple LC-invariants for graph states, since there is a one-to-one correspondence between LC-operations on a graph state and local complementations of its graph. Finally, we generalize foliage partitions from qubits to qudits and prove their invariance under the generalized local complementation operations.