A computational phase diagram for the transverse field Ising model
Abstract
We study the transverse field Ising model, defined by the Hamiltonian H =1/2∑i, j∈ [n] Jij Zi Zj +∑i=1n hiz Zi + η∑i Xi where $J $ is the symmetric interaction matrix, and η is the transverse field strength. Let Δ(J)=λmax(J)−λmin(J) be the spectral width of J. When the inverse temperature β≥0 satisfies Δ(J)·tanh(βη)/η≤1, we give a randomized classical algorithm that approximates the partition function Z(β)=Tr(e−βH) to a given relative error ε∈(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 NP-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime Δ(J)· tanh(βη)/η≤ 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 ∈ {I, X}⊗ n, the algorithm further achieves arbitrarily small relative error.
Related
- Phase diagram and fluctuations in the cubic mean field Ising model
- Analytical results for the Shannon entropy of the critical transverse-field Ising chain
- Phase transition in the antiferromagnetic Ising model with external field
- Detailed balance and phase transition in Nash equilibria of mean-field Ising game
- Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram
BibTeX
@misc{vuong2026computational,
title = {{A computational phase diagram for the transverse field Ising model}},
author = {Thuy-Duong Vuong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02079},
eprint = {2610.02079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02079},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A computational phase diagram for the transverse field Ising model
AU - Thuy-Duong Vuong
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02079
UR - https://arxiv.org/abs/2610.02079
AB - We study the transverse field Ising model, defined by the Hamiltonian H =1/2∑_i, j∈ [n] J_ij Z_i Z_j +∑_i=1^n h_i^z Z_i + η∑_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≥0 satisfies Δ(J)·tanh(βη)/η≤1, we give a randomized classical algorithm that approximates the partition function Z(β)=Tr(e^−βH) to a given relative error ε∈(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 NP-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime Δ(J)· tanh(βη)/η≤ 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 ∈ {I, X}^⊗ n, the algorithm further achieves arbitrarily small relative error.
ER -