Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Abstract

We present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least Ω(log n) decays exponentially. We can improve the factor of log n to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.

Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature

“…Note added.-Regarding the classical simulations of quantum Gibbs states, we identified a related result obtained using a similar approach [100] at the same time of our submission.…”

Exponential decay of mutual information for Gibbs states of local Hamiltonians

“…Let Φ be a finite-range interaction on Z. If Φ satisfies uniform clustering with decay ε( ) in (24), then for every finite interval I ⊂ Z split into three subintervals I = ABC with |B| ≥ 2 ≥ 0 and every pair of observables Proof. We illustrate the proof of the first inequality, the other one is completely analogous.…”

“…Remarkably, Araki's result was extended for different correlation functions to higher dimensional systems above a threshold temperature in a series of papers [22,31,39,44], both for classical and quantum systems. In a recent work [24], the exponential decay of correlations property was related to the absence of complex zeroes of the partition function close to the real axis. Moreover, it is known that systems with a positive spectral gap exhibit exponential decay of correlations [29].…”

The complexity of approximating the complex-valued Potts model

“…The framework of viewing partition functions as polynomials in the complex plane of the underlying parameters has been wellexplored in statistical physics and has recently gained traction in computer science as well in the context of approximate counting. On the positive side, zero-free regions in the complex plane translate into efficient algorithms for approximating the partition function (Barvinok 2017;Patel & Regts 2017) and this scheme has led to a broad range of new algorithms even for positive real values of the underlying parameters (Guo et al 2021(Guo et al , 2020Harrow et al 2020;Liu et al 2019;Liu et al 2019a,b;Peters & Regts 2018. On the negative side, the presence of zeros poses a barrier to this approach and, in fact, it has been demonstrated that zeros mark the onset of computational hardness for the approximability of the partition function (Bezáková et al 2020;Bezáková et al 2021;Goldberg & Guo 2017;Goldberg & Jerrum 2014).…”