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This paper establishes an invariant-based framework for analyzing Pauli Lie algebras and their reachability within quantum dynamics, offering efficient algorithms for identifying orbits and symmetries. By linking Pauli strings to Clifford group subgroups via symplectic properties, the work provides a simplified approach to studying structured generating sets relevant to variational quantum algorithms and many-body systems. The results also demonstrate that specific Clifford subgroups generated by transvections function as 3-designs, bridging theoretical group structures with practical quantum circuit applications.
From Pauli Strings to Quantum Dynamics: A Unified Characterization
arXiv:2606.09773v1 Announce Type: new Abstract: Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
Abstract:Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
arXiv:2606.09773 [quant-ph]
(or arXiv:2606.09773v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.09773 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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