Abstract
Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the 587 orbits up to 9 qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.
Highlights
Graph states provide a language of entanglement between qubits and are at the core of modern quantum computing and communication architectures across all qubit platforms [1,2,3,4,5,6,7]
For example we display the Schmidt measure, ES, which is known to be a useful entanglement monotone for graph states [11,28], encoding the strength of error correcting codes built from the state 29
It is known that to be a universal resource for quantum computation, lattice graph states must have unbounded rank-width as they increase in size 21
Summary
Graph state entanglement is well studied [11,12,13,14,15], with each of the ∼1.6×1012 non-isomorphic graph states up to 12 qubits classified into ∼1.3 × 106 LC-inequivalent classes [16,17]. We refer to the object that links graphs via local comeplemtation as their ‘orbit’, and we refer to those component graphs as ‘graph states’ These orbits, which are wildly complex, give a fresh perspective for the study of stabiliser entanglement and graph states, while providing new tools for optimising quantum protocols. We identify promising applications of local complementation in both quantum secret sharing and compilation of measurement-based protocols By mapping these orbits we expose the exquisite structure of graph state orbits and present them as promising avenues for further study. Repeated application of local complementation is guaranteed to hit every member of a entanglement class of LC-equivalent graph states, given any member of that class as a starting point [11,12] This defines graph (and stabiliser) entanglement classes, each with their own orbit under local complementation.
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