High-temperature partition functions and classical simulatability of long-range quantum systems

TL;DR

The paper investigates high-temperature thermodynamics of long-range quantum spin systems with interactions decaying as $1/r^{α}$. In the weak long-range regime $α> D$, it proves that the partition function $Z_ρ$ is analytic for small inverse temperature $β$ and provides a convergent quantum cluster expansion, enabling a classical subexponential-time algorithm to approximate $\log Z_ρ$. It further shows ensemble equivalence and Gaussian density of states, with numerical support, and extends the KP convergence criterion to quantum long-range models. The results constrain the occurrence of various phase transitions in the strong long-range regime and offer a bridge between classical simulability and quantum hardness, suggesting a potential quantum advantage for certain regimes and outlining directions to extend the framework to broader settings.

Abstract

Long-range quantum systems, in which the interactions decay as $1/r^α$, are of increasing interest due to the variety of experimental set-ups in which they naturally appear. Motivated by this, we study fundamental properties of long-range spin systems in thermal equilibrium, focusing on the weak regime of $ α>D$. Our main result is a proof of analiticity of their partition functions at high temperatures, which allows us to construct a classical algorithm with sub-exponential runtime $\exp(\mathcal{O}(\log^2(N/ε)))$ that approximates the log-partition function to small additive error $ε$. As by-products, we establish the equivalence of ensembles and the Gaussianity of the density of states, which we verify numerically in both the weak and strong long-range regimes. This also yields constraints on the appearance of various classes of phase transitions, including thermal, dynamical and excited-state ones. Our main technical contribution is the extension to the quantum long-range regime of the convergence criterion for cluster expansions of Kotecký and Preiss.

Figures (9)

• Figure 1: Illustration of a set of polymers $\Gamma=\{\gamma_1,\gamma_2,\gamma_3\}$. In (a) the set is admissible, as they are disconnected, while in (b) the set is a cluster as they are connected.
• Figure 2: The density of states of the long-range transverse-field Ising model Eq. \ref{['eq:def-TFI']} varying $N$ for $\alpha=0.5, \, 1.5, \, 2.5$ and the short-ranged transverse-field Ising model ($\alpha \rightarrow \infty$). We show both the numerical estimation (markers) and the fit to a Gaussian curve (solid lines). The variance decreases with system size due to the rescaling of the Hamiltonian $H \rightarrow H/\norm{H}$.
• Figure 3: Representation of the residuals $\epsilon = \frac{1}{R \max_i{y_i}} \sum_i^R |y_i -g(x_i)|^2$, where $g(x)$ is the Gaussian fit, as a function of system sizes and for different values of $\alpha$. For more details on the simulations see Appendix \ref{['app: sim']}.
• Figure 4: Scheme of a polymer containing a hyperedge $Z^*$. It has multiplicity $m=\norm{\gamma}=5$, and it is formed by 5 hyperedges. It can also be seen as a set of three subpolymers $\gamma_1, \gamma_2, \gamma_3$ connected to sites $i_1, i_2, i_3$.
• Figure 5: (a) Example of a polymer $\gamma$, a tuple of hyperedges, being associated with two different connectivity polymers $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$, tuples of edges. (b) Example of a given connectivity polymer $\tilde{\gamma}$ with different polymers $\gamma_1$ and $\gamma_2$ associated.

Theorems & Definitions (16)

• Lemma 1
• Theorem 2
• proof
• Theorem 3
• Lemma 4
• proof
• Theorem 5
• proof
• Corollary 1
• Corollary 2