Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature

TL;DR

This work presents a quasi-polynomial-time classical algorithm to construct matrix product operators that exactly approximate the quantum Gibbs state of one-dimensional long-range interacting systems at any temperature. By introducing a DMRG-inspired truncation strategy and a controlled high-to-low temperature merging process, the authors establish rigorous Schatten-norm error bounds and derive explicit bond-dimension and time-complexity scalings: in general, the construction scales as $e^{C_0 eta \ln^3(n/\\epsilon)}$, and for 2-local Hamiltonians the bound improves to $e^{\\tilde{O}(\beta \ln^2(n/\\epsilon) \ln\ln(n/\\epsilon))}$. The method yields efficient imaginary-time evolution and extends to real-time dynamics, with explicit MPO-expansion schemes for long-range, power-law decaying interactions. By providing detailed proofs and explicit bond-dimension estimates, the work delivers rigorous efficiency guarantees for simulating 1D long-range quantum systems across temperatures, with practical implications for quantum many-body physics and quantum information processing.

Abstract

We introduce a method that ensures efficient computation of one-dimensional quantum systems with long-range interactions across all temperatures. Our algorithm operates within a quasi-polynomial runtime for inverse temperatures up to $β={\rm poly}(\ln(n))$. At the core of our approach is the Density Matrix Renormalization Group algorithm, which typically does not guarantee efficiency. We have created a new truncation scheme for the matrix product operator of the quantum Gibbs states, which allows us to control the error analytically. Additionally, our method can be applied to simulate the time evolution of systems with long-range interactions, achieving significantly better precision than that offered by the Lieb-Robinson bound.

Figures (1)

• Figure 1: Illustration of the Matrix Product Operator (MPO) construction for approximating the quantum Gibbs state. (a) High-Temperature Quantum Gibbs States: The process begins with the product state of quantum Gibbs states across eight blocks. In the initial merging process $\{\Psi_1^{(1)},\Psi_2^{(1)},\Psi_3^{(1)},\Psi_4^{(1)}\}$, the block size is doubled, yielding quantum Gibbs states $\{e^{-\beta_0 H_1^{(2)}},e^{-\beta_0 H_2^{(2)}},e^{-\beta_0 H_3^{(2)}},e^{-\beta_0 H_4^{(2)}}\}$. After $q$ merging processes, with each process doubling the block size, the final block size becomes $2^q$ times the original. To cover the total system size $n$, $\log_{2}(n)$ merging processes are required. Each merging operator $\{\Psi_{s}^{(q)}\}_{s,q}$ is approximated by MPOs $\{\tilde{\Psi}_{s}^{(q)}\}_{s,q}$ with truncated bond dimensions [see Eq. \ref{['tilde_Psi_M_0']}], facilitating the construction of the MPO $M_{\beta_0}$ for the high-temperature quantum Gibbs state $e^{-\beta_0 H}$. Approximation errors are iteratively estimated as detailed in Proposition 2, noting that all approximation errors stem from these merging operators. (b) Low-Temperature Quantum Gibbs States: By iteratively combining high-temperature quantum Gibbs states, the Gibbs state at any desired temperature $\beta$ is achieved by exponentiating $e^{-\beta_0 H}$ to $\beta/\beta_0$, where $\beta_0$ is chosen so that $\beta/\beta_0$ is an integer. The approximation error between $e^{-\beta H}$ and $(M_{\beta_0})^{\beta/\beta_0}$ is managed by controlling the error between $e^{-\beta_0 H}$ and $M_{\beta_0}$ in terms of an arbitrary Schatten $p$-norm, as achieved in \ref{['Schatten_p/high/temp']}.

Theorems & Definitions (5)

• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2
• Lemma 3