[Paper Review] Classification of dynamical Lie algebras generated by spin interactions on undirected graphs

Efekan Kökcü, Roeland Wiersema+2 more|arXiv (Cornell University)|Sep 29, 2024

Molecular spectroscopy and chiralityChemistry3 citations

TL;DR

This paper classifies dynamical Lie algebras (DLAs) generated by 1- and 2-local Pauli spin interactions on arbitrary undirected graphs, showing that for non-one-dimensional graphs, the DLA structure depends solely on whether the graph is bipartite or not. The key result is that polynomial-sized DLAs are restricted to one-dimensional topologies, with higher connectivity in complex graphs leading to larger, less symmetric algebras.

ABSTRACT

We provide a classification of all dynamical Lie algebras generated by 2-local spin interactions on undirected graphs. Building on our previous work where we provided such a classification for spin chains, here we consider the more general case of undirected graphs. As it turns out, the one-dimensional case is special; for any other graph, the dynamical Lie algebra solely depends on whether the graph is bipartite or not. An important consequence of this result is that the cases where the dynamical Lie algebra is polynomial in size are special and restricted to one dimension.

Motivation & Objective

• To extend the classification of dynamical Lie algebras (DLAs) from spin chains to general undirected graphs with 1- and 2-local Pauli interactions.
• To determine how graph topology—particularly bipartiteness—affects the structure and dimensionality of DLAs.
• To identify which DLAs remain polynomial-sized, as these are critical for efficient quantum simulation and variational quantum algorithms.
• To establish a framework using interaction graphs that enables local equivalence relations and simplifies subgraph analysis compared to frustration graphs.

Proposed method

• Use of interaction graphs where vertices represent qubits and edges represent 2-local spin interactions, with symmetric generators (e.g., XY and YX) required.
• Classification of DLAs into a-type (generated solely by 2-local terms) and b-type (requiring 1-local terms), based on the structure of the interaction graph.
• Employment of frustration graphs as a complementary tool, with colored vertex operations (add/remove) to verify membership of Pauli strings in the Lie algebra.
• Application of the Baker–Campbell–Hausdorff formula to generate nested commutators and explore the closure of the algebra under Lie bracket operations.
• Use of equivalence relations on subgraphs to simplify analysis, enabling local reasoning about algebraic structure.
• Explicit verification of Pauli string membership in DLAs via frustration graph coloring sequences, such as moving colored vertices to derive new terms (e.g., X1X3 ∈ aΩ_14).

Experimental results

Research questions

• RQ1 How does the topology of an undirected interaction graph influence the structure of the dynamical Lie algebra generated by 1- and 2-local Pauli operators?
• RQ2 What determines whether a DLA generated by such interactions is polynomial-sized or exponentially large?
• RQ3 To what extent do bipartite versus non-bipartite graphs lead to distinct algebraic structures in the DLA?
• RQ4 Can the classification be extended beyond one-dimensional chains using interaction graphs, and how does this compare to frustration graph-based approaches?
• RQ5 What role do local operations on subgraphs play in determining the closure and dimension of the DLA?

Key findings

• For any undirected graph that is not one-dimensional, the structure of the dynamical Lie algebra depends exclusively on whether the graph is bipartite or non-bipartite.
• The one-dimensional case (e.g., spin chains) is exceptional: it supports a rich variety of DLAs, including polynomial-sized ones, which are absent in higher-dimensional or more connected graphs.
• Polynomial-sized DLAs are restricted to one-dimensional topologies; all other graphs—especially those with vertices of degree >2—generate larger, non-polynomial algebras.
• The DLA aG_k for k ∈ {0,2,4,6,7,14,16,20,22} is generated by 2-local terms and depends only on the graph’s bipartite status; for example, aG_14 arises from {XX, XY, YX} and is closed under commutators.
• Explicit derivations via frustration graph coloring confirm that specific Pauli strings like X1X3 and X1X4 belong to aΩ_14 and aΣ_14, respectively, by constructing valid sequences of vertex additions and removals.
• The use of interaction graphs enables local equivalence reasoning (e.g., Lemma III.3), which is not generally possible with frustration graphs, offering a more intuitive and practical framework for subgraph analysis.