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QI: Fachverband Quanteninformation

QI 9: Quantum Entanglement II

QI 9.1: Vortrag

Dienstag, 11. März 2025, 11:00–11:15, HS IX

Full classification of Pauli Lie algebras — •Gerard Aguilar, Simon Cichy, Jens Eisert, and Lennart Bittel — Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, their classification and characterization often becomes impractical for large systems. This work provides a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. Mapping the problem to a graph setting, we identify a reduced set of equivalence classes for connected graphs: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis, the Lie algebra of symplectic Paulis, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Out of these, we distinguish 6 Clifford inequivalent cases, for which we give a physical interpretation of their dynamics. We then extend this result to general graphs with arbitrarily many connected components. Our findings reveal a no-go result for the existence of small Lie algebras beyond the free-fermionic case in the Pauli setting and offer efficiently computable criteria for universality and extendibility of gate sets. These results bear significant impact in ideas in a number of fields like quantum control, quantum machine learning, or classical simulation of quantum circuits

Keywords: Lie algebra characterization; Pauli operators