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.
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.
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.
We study the set of n-qubit pure quantum states. Specifically, we consider their incarnation as density matrices in Bloch representation. Building on prior work, we define a family of quadratic functions in Pauli expectation values, and fully characterise all instances that result in state-independent purity constraints. The resulting family of pure-state invariants is surprisingly rich in structure, with strong ties to the geometry of symplectic polar spaces. After charting the general landscape, we define a sub-family of pure-state invariants that satisfies the dynamical constraint of being stable under state update. We prove that it corresponds with (different embeddings of) symplectic Lie algebras sp(2n−1)⊂su(2n), which admit a generating set of Pauli operators. We carefully distinguish the symplectic case from its orthogonal counterpart, using the Arf invariant of quadratic refinements of the symplectic form on the projective Pauli group, which we further identify with quaternionic structures on C2n that are both generated by and restricted to Pauli operators. As a consequence of this analysis, we reveal previously largely overlooked structure in the n-qubit Pauli group. We highlight this in a number of examples. We characterise all state-independent uncertainty relations in variance for Pauli observables, as well as all Pauli channels for which the output fidelity and output purity are state-independent. We further show that the orbits of the projective stabiliser groups of pure-state invariants, which for n≥2 are proper subgroups of the Clifford group, form complex projective state 2-designs, while their unitary lifts form symplectic 2-designs.
The dynamical Lie algebra (DLA) governs the controllability, expressibility, and simulation complexity of a quantum system. Explicitly computing it has been a major computational bottleneck: brute-force Lie closure scales exponentially in the number of qubits n. Many applications, however, consult only the isomorphism type of the DLA. We introduce PauLie, an open-source framework that decides this isomorphism type for DLAs generated by arbitrary Pauli strings, building on the anticommutation-graph reduction of Aguilar et al. PauLie runs in O(n∣G∣max(n,∣G∣)) time, where ∣G∣ is the number of generators, turning DLA classification into a routine preprocessing step. We demonstrate its use as a structural oracle for routing Lie-algebraic simulation and Cartan decomposition, diagnosing barren plateaus in variational quantum algorithms, and engineering universal Pauli string generator sets with optimal generation rate.
We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.
Determining the unitary dynamics accessible from finite Hamiltonian resources is a central problem in Hamiltonian engineering and quantum control. Dynamical Lie algebras (DLAs) connect available control Hamiltonians with the reachable dynamics, but their use as a design tool for modifying Hamiltonian generator sets remains less developed. In this work, we develop a finite-dimensional DLA framework for three generator-set operations: composition, invariance, and reduction. For composition, we construct direct sums of component DLAs using spectral projectors on an auxiliary register. For invariance, we analyze when modifications of Pauli-string generating sets preserve the generated Lie algebra, and introduce algebraic diagnostics for added generators. For reduction, we consider compact reductive DLAs and show how projection onto selected simple ideals gives reduced generating sets whose Lie closures are the corresponding ideal sums. We illustrate these results with finite-dimensional examples and numerical checks, including direct-sum dimension addition, central-spin invariance diagnostics, and DLA-based ansatz reduction for block-local Hamiltonians. The results show how DLA structure can be used to diagnose controllability and guide Hamiltonian generator design under explicit algebraic assumptions.
Dynamical Lie algebras generated by Pauli strings and quadratic spaces over F2
Dynamical Lie algebras, i.e. Lie subalgebras of su(2n), generated by Pauli strings have recently been studied intensively. They are also called Pauli Lie algebras or Hamiltonian Lie algebras.
In this paper we provide a uniform mathematical approach to various recent results on Pauli Lie algebras. Moreover, we present an algorithm that on input of a set of Pauli strings determines the isomorphism type of the dynamical Lie algebra generated by these Pauli's in time O(max(n,m)3) where m is the size of the generating set.