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arXiv:2605.03829

A Berry-Esseen Bound for Quantum Lattice Systems

Marcus Cramer, Fernando G. S. L. Brandão, Mădălin Guţă, Álvaro M. Alhambra, Matteo Scandi

quant-phcond-mat.stat-mechSubmitted May 5, 2026• Announced May 6, 2026• 0 citations
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Abstract

It is expected that the statistical fluctuations of local observables in large quantum systems obey the central limit theorem, and approximate a normal distribution as their size grows. Here, we prove a version of the Berry-Esseen theorem for quantum lattice systems, which strengthens that central limit theorem by providing a rigorous convergence estimate towards the normal distribution for large but finite system size. Given a local quantum Hamiltonian on $N$ particles and a quantum state with a finite correlation length, the result states that the measurement of local observables such as the energy follows a normal distribution, up to an error scaling as $\mathcal{O}\left(N^{-\frac{1}{2}} \text{polylog}(N)\right)$, which is optimal up to logarithmic factors.

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31 pages, 1 figure. Comments welcome

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We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of NNN spins arranged on a ddd-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to O(N1/(d+1))O(N^{1/(d+1)})O(N1/(d+1)) spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.
The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.
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