Classical Simulability from Operator Entanglement Scaling

TL;DR

This paper establishes rigorous bounds linking operator entanglement scaling to the classical simulability of quantum many-body operators, showing that low operator entanglement implies efficient matrix-product operator approximations.

Contribution

It provides the first rigorous bounds connecting operator entanglement scaling with the efficient classical simulability of quantum operators, extending foundational tensor network results.

Findings

Volume law LOE scaling for α ≥ 1 prevents efficient MPO approximation.

Logarithmic LOE scaling for α < 1 implies MPO simulability.

Numerical and model evidence suggest logarithmic LOE scaling ensures simulability even out of equilibrium.

Abstract

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 $\alpha$-R\'enyi entropies. Specifically, we prove that a volume law scaling for $\alpha\geq 1$ 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 $\alpha < 1$ R\'enyi LOE 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, also for out-of-equilibrium expectation values, logarithmic scaling for $\alpha < 1$ R\'enyi LOE typically guarantees simulability. 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.