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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 15 Pith papers
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Necessary and Sufficient Conditions for Universal Gates with Pauli Strings and Beyond
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New Pauli orbit and modified Gell-Mann bases enable polynomial-cost Lie-algebraic simulation for permutation-equivariant and bounded-excitation quantum dynamics.
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Exponentially many initializations to avoid barren plateaus
A first-moment operator diagnostic reveals exponentially many inequivalent initialization distributions avoid barren plateaus in variational quantum algorithms, with numerics indicating distinct attained minima.
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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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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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Closest Accessible Symmetry reduction: a tool for Hamiltonian interpolation analysis
Introduces Closest Accessible Symmetry reduction to analyze spectra of Hamiltonian interpolations by projecting onto closest accessible symmetries, yielding weakly coupled sectors that capture quantum phase transition...
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From Pauli Strings to Quantum Dynamics: A Unified Characterization
Develops an invariant-based framework connecting Pauli Lie algebras to transvection-generated Clifford subgroups for quantum reachability and dynamics analysis.
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Stabilizers for Compiling Logical Circuits under Hardware Constraints
Stabilizer redundancy from error-correcting codes reduces the choice of physical operators for a logical target to a least-squares problem with closed-form solution, allowing native hardware Hamiltonians to replace co...
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On the Complexity of Quantum States and Circuits from the Orthogonal and Symplectic Groups
Random states from symplectic and orthogonal unitaries show exponentially large strong state complexity and near-orthogonality, with average-case hardness for learning circuits from these groups.
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Frustration graph formalism for qudit observables
A frustration graph formalism for prime-d qudit observables yields a unitary map to generalized Pauli form plus bounds on observable sums used for qudit entanglement quantification.
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Symmetries and overparametrization properties of Hamiltonian variational ansatzes for the $(1+1)$d $\mathbb{Z}_2$ lattice gauge theory
Numerical study of five symmetry-preserving HVAs for Z2 gauge theory finds overparametrization eliminates local minima and loss decay rate scales linearly with number of parameters.
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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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Quantum simulation of massive Thirring and Gross--Neveu models for arbitrary number of flavors
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