A graph-theoretic approach to chaos and complexity in quantum systems

TL;DR

The paper introduces the commutator graph as a fine-grained refinement of the dynamical Lie algebra for ensembles generated by Pauli strings. It connects graph structure to scrambling metrics (OTOC), design properties (frame potential), and operator growth (Krylov complexity), providing exact relations such as $F_G^{(2)}=(\#\text{isolated vertices})(\#\text{components})$ and a coherence phenomenon between symmetry-related components. It defines graph-based complexity $\mathsf{G}(p_t)$ and proves it is a tight bound for Krylov complexity, with universal short-time scaling $\mathsf{G}(p_t)=\Theta(t^2)$ for non-symmetric operators. The results are illustrated across universal, matchgate, Ising-like, orthogonal, and symplectic DLAs, showing how component structure governs average OTOCs and long-time dynamics, and highlighting the graph perspective as a powerful tool for diagnosing chaos, complexity, and classical simulability in quantum many-body dynamics.

Abstract

There has recently been considerable interest in studying quantum systems via dynamical Lie algebras (DLAs) -- Lie algebras generated by the terms which appear in the Hamiltonian of the system. However, there are some important properties that are revealed only at a finer level of granularity than the DLA. In this work we explore, via the commutator graph, average notions of scrambling, chaos and complexity over ensembles of systems with DLAs that possess a basis consisting of Pauli strings. Unlike DLAs, commutator graphs are sensitive to short-time dynamics, and therefore constitute a finer probe to various characteristics of the corresponding ensemble. We link graph-theoretic properties of the commutator graph to the out-of-time-order correlator (OTOC), the frame potential, the frustration graph of the Hamiltonian of the system, and the Krylov complexity of operators evolving under the dynamics. For example, we reduce the calculation of average OTOCs to a counting problem on the graph; separately, we connect the Krylov complexity of an operator to the module structure of the adjoint action of the DLA on the space of operators in which it resides, and prove that its average over the ensemble is lower bounded by the average shortest path length between the initial operator and the other operators in the commutator graph.

Figures (10)

• Figure 1: (a) The commutator graph of a family of generators $\mathcal{G}={\rm span}_{\space\mathbb{R}}\{ H_\ell \}_\ell$ (with each $H_\ell$ a Pauli string) possesses a node for each $n$-qubit Pauli string, with edges connecting vertices $p$ and $q$ that are linked by the adjoint action of a generator $H_\ell$ of the dynamics, i.e. $[H_\ell,p]\propto q$. The graph breaks up into connected components corresponding to representations of the adjoint action of the dynamical Lie algebra (DLA) $\mathfrak{g}=\expval{\mathcal{G}}_{\rm Lie}$ on the space of linear operators $\mathcal{L}={\rm End\ }\mathcal{H}$ on the Hilbert space $\mathcal{H}$ of the system. (b) Importantly, the DLA does not fix the graph structure, which depends explicitly on $\mathcal{G}$. While different sets of generators can produce (under the Lie closure) identical DLAs, they do not lead to identical commutator graphs; indeed the graphs are not even necessarily isomorphic. All that the DLA does specify is the set of connected components. (c) The out-of-time-order correlator (OTOC) $F(W,V_t)=d^{-1}{\rm tr} [WV_tW^\dagger V_t^\dagger]$ gives a common diagnostic of scrambling in quantum systems: the exponential decay of the OTOC between two initially commuting operators. Properties of the commutator graph allow us to evaluate the average OTOC between Pauli strings for a wide class of dynamics. (d) An operator which is initially a Pauli string will spread across the connected component of the graph to which it belongs via Heisenberg time evolution. The connected components therefore contain the Krylov spaces of their constituent nodes. At long times, the details of the internal edge structure of a connected component on average "wash out", and all of the remaining information is captured at the level of the DLA.
• Figure 2: The OTOC between $e^{i\sum_j c_jH_j} (ZXYZIY) e^{-i\sum_j c_jH_j}$ and a representative Pauli string from each connected component of the commutator graph (indexed by $\kappa$, ordered arbitrarily), for 100 (uniformly) random choices of the coefficients $c_j$ appearing in the Hamiltonian. The empirical average value is plotted in blue, with the standard deviation indicated, and compared to the analytical calculation of the Haar average value (in red) given by Prop. \ref{['crllr:counting']}. Interestingly, we find that this induced distribution well-approximates (at least the second moment of) the uniform distribution, suggesting that our results will with high probability be accurate for Hamiltonians with uniformly randomly sampled coefficients. In small (but non-trivial) components one can find larger variances; for example the "8th" component of the middle dynamics in the (arbitrary) ordering of this figure corresponds to the two-element component consisting of $YXIIII$ and $ZXIIII$.
• Figure 3: The average path length between vertices in the commutator graph for $Z,XX$ dynamics, as a function of component index $\kappa$ and number of sites $n$. The $\kappa$th component, of dimension $2n \choose \kappa$, contains the Pauli strings which are a product of $\kappa$ distinct Majorana fermions diaz2023showcasing. For $|\kappa\mathrm{\ mod\ }2n|\leq 2$ the linear growth with the system size $n$ agrees with our analytic calculations; for $|\kappa\mathrm{\ mod\ }2n|> 2$ the growth also appears to be linear. The isomorphic components with $\kappa>n$ are not shown.
• Figure 4: Spanning sets for the first and second order commutants of the orthogonal and (unitary) symplectic groups; for both groups the elements are indexed respectively by the set of pairs of the sets of two and four elements collins2006integrationcollins2009somegarcia2024architectures. Here we highlight the connection between this abstract presentation and the concrete realisation afforded by the tensor-network notation. The operator $\Omega$ that appears in the symplectic case is the canonical symplectic form.
• Figure 5: The (non-trivial components) of the commutator graphs in the case of $\mathfrak{a}_{16}(n)\cong \mathfrak{so}(2^n)$, where for the graphs of the top, middle and bottom rows we take a generating set given respectively by $\{XY,YX,YZ,ZY\}$, the proceeding set augmented with its first order commutators, and the Lie closure of the set.

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