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arXiv:2609.35114

Accurate and bounded approximation of quantum correlation functions

Alexander van Lomwel, Florian Mintert

quant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

The exact classical simulation of non-integrable quantum systems rapidly becomes intractable beyond small system sizes, yet its role is central to predicting many-body dynamics. Approximate methods alleviate this cost but typically lack rigorous guarantees on their deviation from the exact result. For the infinite-temperature correlation function, we derive a rigorous error bound for approximating the full dynamics within a computable finite-sized region. Applied to autocorrelations, this finite-region approach is especially accurate across one-dimensional systems and beyond, and can further provide rigorous error bounds for other approximate methods, demonstrated here for matrix-product-operator simulations.

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The exact classical simulation of non-integrable quantum systems rapidly becomes intractable beyond small system sizes, yet its role is central to predicting many-body dynamics. Approximate methods alleviate this cost but typically lack rigorous guarantees on their deviation from the exact result. For the infinite-temperature correlation function, we derive a rigorous error bound for approximating the full dynamics within a computable finite-sized region. Applied to autocorrelations, this finite-region approach is especially accurate across one-dimensional systems and beyond, and can further provide rigorous error bounds for other approximate methods, demonstrated here for matrix-product-operator simulations.
We describe a numerical method which allows to go beyond the classical approximation for the real-time dynamics of many-body systems by approximating the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state approximation is accurate for significantly smaller field strengths and longer times than the classical one. Applying this approximation to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λL < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.
We describe a numerical method which allows to go beyond the classical approximation for the real-time dynamics of many-body systems by approximating the many-body Wigner function by the most general Gaussian function with time-dependent mean and dispersion. On a simple example of a classically chaotic system with two degrees of freedom we demonstrate that this Gaussian state approximation is accurate for significantly smaller field strengths and longer times than the classical one. Applying this approximation to matrix quantum mechanics, we demonstrate that the quantum Lyapunov exponents are in general smaller than their classical counterparts, and even seem to vanish below some temperature. This behavior resembles the finite-temperature phase transition which was found for this system in Monte-Carlo simulations, and ensures that the system does not violate the Maldacena-Shenker-Stanford bound λL < 2πT, which inevitably happens for classical dynamics at sufficiently small temperatures.
Finite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are ubiquitous in numerical simulations of quantum many-body systems.Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing.Here we derive rigorous upper bounds on the FSE of local observables in real-time quantum dynamics simulations initialized from a product state.In d-dimensional locally interacting systems with a finite local Hilbert space, our bound implies | Ŝ(t ) L -Ŝ(t ) ∞ | C(2vt/L) cL-μ , with v, C, c, μ constants independent of L and t, which we compute explicitly.For periodic boundary conditions (PBCs), the constant c is twice as large as that for open boundary conditions (OBCs), suggesting that PBCs have smaller FSEs than OBCs at early times.The bound can be generalized to a large class of correlated initial states as well.As a byproduct, we prove that the FSE of local observables in ground-state simulations decays exponentially with L under a suitable spectral gap condition.Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.
We present a method to compute dynamical correlation functions in quantum many-body systems at zero temperature. The ground state of a finite system is evaluated exactly by a modified Lanczos method and from it the real-frequency correlation functions are obtained by a projective technique for the memory-function formalism. We apply the method to the density-density correlation function of a one-dimensional spinless fermion system. Exact results are recovered in the noninteracting limit; in the strong-coupling limit our results are compared with Monte Carlo simulation and analytical approximations.
Recovering properties of correlation functions is typically challenging. On the one hand, experimentally, it requires measurements with a temporal resolution finer than the system's dynamics. On the other hand, analytical or numerical analysis requires solving the system evolution. Here, we use recent results of quantum metrology with continuous measurements to derive general lower bounds on the relaxation and second-order correlation times that are both easy to calculate and measure. These bounds are based solely on steady-state expectation values and their derivatives with respect to a system parameter, and can be readily extended to the autocorrelation of arbitrary observables. We validate our method on two examples of critical quantum systems: a critical driven-dissipative resonator, where the bound matches analytical results for the dynamics, and the infinite-range Ising model, where only the steady state is solvable, and thus the bound provides information beyond the reach of existing analytical approaches. Our results can be applied to the experimental characterization of ultrafast systems and to the theoretical analysis of many-body models whose dynamics are hard to compute.
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 ααα-Rényi entropies. Specifically, we prove that a volume law scaling for α≥1α\geq 1α≥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<1α< 1α<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.