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.
However, many graph states are locally equivalent to one another, that is, they possess the same type of entanglement. Graph states which are locally equivalent can be transformed into one another by successive applications of the graph operation local complementation (example shown above). Using this operation, we can analyse only graph structure of the state, which is much simpler than analysing the exponentially large quantum state vector. This equivalence of graph states has been studied previously, with all graph states up to 12 qubits classified.
However, local complementation gives us more than sets of locally equivalent graphs: it also gives us an orbit (example shown above) which tells us how different graphs are related via local complementation. In this work we study these orbits, and relate their properties to properties of the entangled quantum states they contain. We find that orbit properties, such as colourability, correlate with entanglement properties, such as schmidt measure, and discuss applications of local complementation in quantum technology.
 Alexander, R. N. et al. One-way quantum computing with arbitrarily large time-frequency continuous-variable cluster states from a single optical parametric oscillator. Physical Review A 94, 032327 (2016).
 Cabello, A., López-Tarrida, A. J., Moreno, P. & Portillo, J. R. Entanglement in eight-qubit graph states. Physics Letters A 373, 2219–2225 (2009).
 Van den Nest, M., Dehaene, J. & De Moor, B. Graphical description of the action of local Clifford transformations on graph states. Physical Review A 69, 022316 (2004).
 Danielsen, L. E. & Parker, M. G. On the classification of all self-dual additive codes over GF(4) of length up to 12. Journal of Combinatorial Theory, Series A 113, 1351–1367 (2006).
 Van den Nest, M., Dehaene, J. & De Moor, B. Efficient algorithm to recognize the local clifford equivalence of graph states. Physical Review A 70, 034302 (2004).
 Dahlberg, A., Helsen, J. & Wehner, S. How to transform graph states using single-qubit operations: computational complexity and algorithms. arXiv preprint arXiv:1805.05306 (2018).
 Van den Nest, M., Dür, W., Vidal, G. & Briegel, H. Classical simulation versus universality in measurement-based quantum computation. Physical Review A 75, 012337 (2007).
 Dahlberg, A., Helsen, J. & Wehner, S. Counting single-qubit clifford equivalent graph states is #p-complete. arXiv preprint arXiv:1907.08024 (2019).
 Adcock, J. C., Morley-Short, S., Silverstone, J. W. & Thompson, M. G. Hard limits on the postselectability of optical graph states. Quantum Science and Technology 4, 015010 (2019).
 Joo, J. & Feder, D. L. Edge local complementation for logical cluster states. New Journal of Physics 13, 063025 (2011).
 Gimeno-Segovia, M., Shadbolt, P., Browne, D. E. & Rudolph, T. From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation. Physical Review Letters 115, 020502 (2015).
 Morley-Short, S. et al. Physical-depth architectural requirements for generating universal photonic cluster states. Quantum Science and Technology 3, 015005 (2017).
 Morley-Short, S., Gimeno-Segovia, M., Rudolph, T. & Cable, H. Physical-depth architectural requirements for generating universal photonic cluster states. Quantum Science and Technology (2018).
 Alexander Pickston, Joseph Ho, Andrés Ulibarrena, Federico Grasselli, Massimiliano Proietti, Christopher L. Morrison, Peter Barrow, Francesco Graffitti, and Alessandro Fedrizzi, "Conference key agreement in a quantum network", npj Quantum Information 9 1, 82 (2023).
 Thomas J. Bell, Love A. Pettersson, and Stefano Paesani, "Optimizing Graph Codes for Measurement-Based Loss Tolerance", PRX Quantum 4 2, 020328 (2023).
 Zheng-Hao Liu, Jie Zhou, Hui-Xian Meng, Mu Yang, Qiang Li, Yu Meng, Hong-Yi Su, Jing-Ling Chen, Kai Sun, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo, "Experimental test of the Greenberger–Horne–Zeilinger-type paradoxes in and beyond graph states", npj Quantum Information 7 1, 66 (2021).
 Carlo Cafaro, "Geometric graph-theoretic aspects of quantum stabilizer codes", Physica Scripta 97 7, 075105 (2022).
 Sebastiano Corli and Enrico Prati, 2022 IEEE International Conference on Rebooting Computing (ICRC) 1 (2022) ISBN:979-8-3503-4709-8.
 Sitong Liu, Naphan Benchasattabuse, Darcy QC Morgan, Michal Hajdušek, Simon J. Devitt, and Rodney Van Meter, "A Substrate Scheduler for Compiling Arbitrary Fault-tolerant Graph States", arXiv:2306.03758, (2023).
The above citations are from Crossref's cited-by service (last updated successfully 2023-09-28 02:06:55) and SAO/NASA ADS (last updated successfully 2023-09-28 02:06:55). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.