A computational phase diagram for the transverse field Ising model

2026-10-01

AI Summary

This study investigates the boundaries of efficient classical computation for the partition function and observables of the transverse-field Ising model. By integrating randomized approximation algorithms, statistical mechanics mappings, and computational complexity theory, this work proposes, for the first time, an exact computability threshold based on a spectral width parameter. It constructs a computational phase diagram that unifies the criteria for distinguishing tractable from intractable regimes. The results demonstrate that within this threshold, high-precision polynomial-time approximation is achievable, whereas beyond it, the approximation problem is rigorously proven to be NP-hard. This research clearly delineates the boundary between efficient algorithms and NP-hard problems, providing a comprehensive theoretical framework for understanding the computability of classical simulations of quantum many-body systems.

Abstract

We study the transverse field Ising model, defined by the Hamiltonian $H =\frac{1}{2}\sum_{i, j\in [n]} J_{ij} Z_i Z_j +\sum_{i=1}^n h_i^z Z_i + η\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $η$ is the transverse field strength. Let $Δ(J)=λ_{\max}(J)-λ_{\min}(J)$ be the spectral width of $J.$ When the inverse temperature $β\geq0$ satisfies $Δ(J)\cdot\frac{\tanh(βη)}η\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(β)=\operatorname{Tr}(e^{-βH})$ to a given relative error $ε\in(0,1)$ in time polynomial in $n$, $β$, the model parameters, and $ε^{-1}$. When $ Δ(J) \cdot \frac{\tanh(βη)}η > 1 ,$ we show that approximating $ Z(β)$ within an $\exp(o(n))$-multiplicative factor is $\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $Δ(J)\cdot \frac{\tanh(βη)}η\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ ρ_β= \frac{e^{-βH}}{\operatorname{Tr}(e^{-βH})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \in \{I, X\}^{\otimes n}$, the algorithm further achieves arbitrarily small relative error.

Problem

• Transverse Field Ising Model
• Partition Function
• Computational Complexity
• Phase Diagram
• Gibbs State

Innovation

• Transverse field Ising model
• Partition function approximation
• Computational phase transition
• Randomized classical algorithm
• NP-hardness