Local-operator entanglement (LOE) dictates the complexity of simulating Heisenberg evolution using tensor network methods, and it bears witness to many-body chaos for local dynamics. We show that LOE is also sensitive to how non-Clifford a unitary is-its magic resources. In particular, we prove that LOE is always upper-bound by three distinct magic monotones: T-count, unitary nullity, and operator-stabilizer Rényi entropy. Moreover, in the average case for large, random circuits, LOE and magic monotones approximately coincide. Our results imply that an operator evolution that is expensive to simulate using tensor network methods must also be inefficient using both stabilizer and Pauli truncation methods. In terms of a previous conjecture on the characteristic scaling of LOE, our results imply that nonintegrable spin chains cannot be simulated classically. Entanglement in operator space therefore measures a unified picture of nonclassical resources, in stark contrast to the Schrödinger picture.
Local-operator entanglement (LOE) dictates the complexity of simulating Heisenberg evolution using tensor network methods, and it bears witness to many-body chaos for local dynamics. We show that LOE is also sensitive to how non-Clifford a unitary is-its magic resources. In particular, we prove that LOE is always upper-bound by three distinct magic monotones: T-count, unitary nullity, and operator-stabilizer Rényi entropy. Moreover, in the average case for large, random circuits, LOE and magic monotones approximately coincide. Our results imply that an operator evolution that is expensive to simulate using tensor network methods must also be inefficient using both stabilizer and Pauli truncation methods. In terms of a previous conjecture on the characteristic scaling of LOE, our results imply that nonintegrable spin chains cannot be simulated classically. Entanglement in operator space therefore measures a unified picture of nonclassical resources, in stark contrast to the Schrödinger picture.
Local-operator entanglement (LOE) quantifies the nonlocal structure of Heisenberg operators and serves as a diagnostic of many-body chaos. We provide rigorous bounds showing when an operator can be well-approximated by a matrix-product operator (MPO), given asymptotic scaling of its LOE α-Rényi entropies. Specifically, we prove that a volume law scaling for α≥1 implies that the operator cannot be approximated efficiently as an MPO while faithfully reproducing all expectation values. On the other hand, if we restrict to correlations over a relevant sub-class of (ensembles of) states, then logarithmic scaling of the α<1 entropies implies MPO simulability. This result covers a range of relevant quantities, including infinite temperature autocorrelation functions, out-of-time-ordered correlators, and average-case expectation values over ensembles of computational basis states. Beyond this regime, we provide numerical evidence together with a random matrix model to argue that this simulability result also typically holds for arbitrary states. Our results put on firm footing the heuristic expectation that a low operator entanglement implies efficient tensor network representability, extending celebrated foundational results from the theory of matrix-product states and providing a formal link between quantum chaos and classical simulability.
We study a nonstabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer Rényi entropy analog in operator space is a good magic monotone satisfying the usual conditions while inheriting efficient computability properties and providing a tight lower bound to the minimum number of non-Clifford gates in a circuit. Operationally, this measure quantifies how well an operator can be approximated by one with only a few Pauli strings-analogous to how entanglement entropy relates to tensor-network truncation. A notable advantage of operator stabilizer entropies is their inherent locality, as captured by a Lieb-Robinson bound. This feature makes them particularly suited for studying local dynamical magic resource generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that under random evolution, operator magic typically reaches near-maximal value for all Rényi indices, and we evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we solve the operator stabilizer entropy for interacting integrable XXZ circuit, finding that it quickly saturates to a constant value. Overall, this measure sheds light on the structural properties of many-body nonstabilizerness generation and can inspire Clifford-assisted tensor network methods.
The local-operator entanglement (LOE) measures the classical simulability of a Heisenberg operator and is conjectured to witness many-body chaos in locally interacting systems. Using tools from free probability, we analytically compute its asymptotic value for Haar random dynamics for all Rényi indices. We find that to the first two orders in large dimension, for traceless operators it exactly reproduces the Page curve for random states. In contrast to higher-order out-of-time-ordered correlators (OTOCs), which depend on more complex operator correlations via free cumulants, the leading-order LOE is independent of the initial operator. \add{We detail exactly how these higher-order correlations are encoded in the coefficients of the finite-size corrections. Therefore, guided by our Haar result, we argue that the long-time value of the LOE entropies in chaotic systems is robust: they depend only on a simple sub-sector of the hierarchy of correlations encoded by the full Eigenstate Thermalization Hypothesis. Our results therefore provide a conceptual boundary between operator simulability and higher-order scrambling diagnostics.
Entanglement and non-stabilizerness are complementary resources underlying the complexity of quantum states, yet extracting either quantity from large quantum circuits generally requires exponentially scaling resources. In diagrammatic approaches to quantum computation, such as the ZX-calculus, both quantities likewise typically require expensive tensor contractions that are not native to graphical rewriting. Here we show that both entanglement and magic can be efficiently estimated directly from ZX-diagrams. Flow enables efficient extraction of a normal form separating a potentially extensively entangled graph-state backbone from non-Clifford Pauli gadgets. This structure yields additive upper and lower bounds on bipartite entanglement entropy, which we further tighten through a preprocessing procedure that removes or merges redundant non-Clifford contributions. The same normal form gives an upper bound on logarithmic stabilizer extent. We benchmark these methods in random unitary and monitored circuits, and a circuit combining Trotterized Hamiltonian evolution with Clifford layers, finding that the resulting bounds remain informative even at large qubit numbers, circuit depths, and internal spider counts. Our results provide scalable probes of entanglement and non-stabilizerness of ZX-diagrams and establish a bridge between ZX-calculus and quantum many-body resource characterization, with applications across quantum computing, quantum information, and condensed-matter physics.
Matchgate circuits describe classically tractable dynamics in a physically relevant setting while nonetheless exhibiting features characteristic of complex quantum systems, such as high amounts of magic. This invites the broader question: to what extent are typical complexity measures saturated under trivial dynamics? Here, we address this by studying the behaviour of different Heisenberg-picture complexity measures in random matchgate ensembles. Our main technical contribution is a tractable and generalisible free-fermionic unitary Weingarten calculus formulated entirely in the Majorana basis. This method is tailored to computing moments of operators, and simplifies to a combinatorics problem which depends on the boundary conditions prescribed by a given operator functional. Using this framework, we derive closed-form expressions for the typical local-operator entanglement, operator stabiliser entropy, and higher-order out-of-time-ordered correlators. For extensive initial Majorana strings, both operator entanglement and stabiliser entropy obey volume laws, even though the underlying circuits remain classically simulable. Out-of-time-ordered correlators can likewise become exponentially small, showing that their late-time saturation value alone does not witness computational hardness and constituting a classically tractable example of emergent free independence. Finally, we relate these separations to operator resource theories, showing how operator entanglement beyond the Gaussian baseline lower-bounds the number of non-Gaussian gates in doped matchgate circuits. Our results present a practical toolbox for computing exact averages over random matchgates, while clarifying the distinction between quantum resources versus simulation complexity, and their dependence on the representation being exploited.
Nonstabilizerness is a quantum property of states associated with the non-Clifford resources required for their preparation. As a resource, nonstabilizerness complements entanglement, and the interplay between these two concepts has garnered significant attention in recent years. In this work, we establish an exact correspondence between the generation of nonlocal nonstabilizerness and operator entanglement under unitary evolutions. Nonlocal nonstabilizerness refers to nonstabilizerness that cannot be erased via local operations, while operator entanglement generalizes entanglement to operator space, characterizing the complexity of operators across a bipartition. Specifically, we prove that a unitary map generates nonlocal nonstabilizerness if and only if it generates operator entanglement on Pauli strings. Guided by this result, we introduce an average measure of a unitary's Pauli-entangling power, serving as a proxy for nonlocal nonstabilizerness generation. We derive analytical formulas for this measure and examine its properties, including its typical value and upper bounds in terms of the nonstabilizerness properties of the evolution.