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Full classification of Pauli Lie algebras
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Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, classification and characterization methods for these structures are often impractical for larger systems. In this work, we provide a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. By mapping the problem to a graph setting, we identify a reduced set of equivalence classes: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n qubits, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Moreover, out of these, we distinguish 6 Clifford inequivalent cases and find a simple set of canonical operators for each, which allow us to give a physical interpretation of the dynamics of each class. 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.
Forward citations
Cited by 7 Pith papers
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Enabling Lie-Algebraic Classical Simulation beyond Free Fermions
Symmetry-adapted Pauli-orbit and modified Gell-Mann bases make polynomial-dimensional dynamical Lie algebras practically simulable beyond free fermions.
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Modifying DLA generators by tensoring with a Hermitian operator of K distinct eigenvalues yields K copies of the algebra (or its commutator part) using about log K extra qubits.
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Optimal Haar random fermionic linear optics circuits
The paper constructs optimal-depth, optimal-gate-count quantum circuits that sample Haar-random active and passive fermionic linear optics unitaries directly from angle distributions, plus an optimal Clifford FLO sampler.
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An efficient algorithm for approximate shadow Hamiltonian simulation
Pruned shadow Hamiltonian simulation approximates observable dynamics under interacting Hamiltonians by keeping only the most relevant operators, cutting the qubit register needed for shadow evolution.
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A Dynamical Lie-Algebraic Framework for Hamiltonian Engineering and Quantum Control
A Lie-algebra toolkit that composes, preserves, and reduces Hamiltonian generator sets, including a nearest-neighbor su(2^N) generating set and a filtering-operator reduction, though the reduction proof and one error-...
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Fast numerical generation of Lie closure
Replacing rank-based linear independence checks with orthonormal projection or Gram matrix inversion accelerates numerical construction of dynamical Lie algebras.
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An Optimized Construction of Lie Algebra Generator Pools for Variational Quantum Eigensolvers in Chemistry
The proposed rank test does not determine the Lie algebra generated by a Pauli pool, so the claimed polynomial MCP construction is not established.
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