Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension

Abstract

Much is understood about 1-dimensional spin chains in terms of entanglement properties, physical phases, and integrability. However, the Lie algebraic properties of the Hamiltonians describing these systems remain largely unexplored. In this work, we provide a classification of all Lie algebras generated by translation-invariant 2-local spin chain Hamiltonians, or so-called dynamical Lie algebras. We consider chains with open and periodic boundary conditions and find 17 unique dynamical Lie algebras. Our classification covers some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that cannot be identified easily. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches. We discuss the practical implications of our work in the context of quantum control, variational quantum computing, and the spin chain literature.

References in corpus (27)

• Classification of topological insulators and superconductors in three spatial dimensions
• A Quantum Approximate Optimization Algorithm
• The Variational Quantum Eigensolver: a review of methods and best practices
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
• Theory of overparametrization in quantum neural networks
• A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits
• Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks
• Classically optimized Hamiltonian simulation
• Avoiding barren plateaus via transferability of smooth solutions in Hamiltonian Variational Ansatz
• The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze
• Non-Abelian eigenstate thermalization hypothesis
• Noncommuting conserved charges in quantum thermodynamics and beyond
• Non-Abelian symmetry can increase entanglement entropy
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution
• Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach
• From Symmetries to Commutant Algebras in Standard Hamiltonians
• Optimizing quantum circuits with Riemannian gradient flow
• Learning Unitaries by Gradient Descent
• An Algebraic Quantum Circuit Compression Algorithm for Hamiltonian Simulation
• Quantum simulations under translational symmetry
• A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers
• Riemannian optimization of photonic quantum circuits in phase and Fock space
• Measurement-induced entanglement phase transitions in variational quantum circuits
• Integrable spin chains and the Clifford group
• Characterization of variational quantum algorithms using free fermions
• Algebraic Compression of Free Fermionic Quantum Circuits: Particle Creation, Arbitrary Lattices and Controlled Evolution
• Scaling adaptive quantum simulation algorithms via operator pool tiling

Cited by in corpus (19)

• Barren Plateaus in Variational Quantum Computing
• Does provable absence of barren plateaus imply classical simulability?
• Lie-algebraic classical simulations for quantum computing
• Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras
• Geometric Quantum Machine Learning with Horizontal Quantum Gates
• Scalability Challenges in Variational Quantum Optimization under Stochastic Noise
• Backpropagation scaling in parameterised quantum circuits
• Architectures and random properties of symplectic quantum circuits
• Optimally generating $\mathfrak{su}(2^N)$ using Pauli strings
• Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor
• The Lie Algebra of XY-mixer Topologies and Warm Starting QAOA for Constrained Optimization
• Non-Universality from Conserved Superoperators in Unitary Circuits
• Expanding the Class of Free Fermions via Twin-Collapse Methods
• Scaling of symmetry-restricted quantum circuits
• Quantum simulation of massive Thirring and Gross--Neveu models for arbitrary number of flavors
• A graph-theoretic approach to chaos and complexity in quantum systems
• Fast-forwardability of Qubit-mapped Fermion models based on Cartan decomposition
• Interpolation-based coordinate descent method for parameterized quantum circuits
• Direct Gradient Computation for Barren Plateaus in Parameterized Quantum Circuits