Advanced measurement techniques in quantum Monte Carlo: The permutation matrix representation approach

TL;DR

This work establishes a rigorous, general framework for estimating arbitrary static and dynamic observables within permutation matrix representation quantum Monte Carlo (PMR-QMC). By decomposing the Hamiltonian and observables in a common PMR basis and employing the off-diagonal series expansion with divided differences, it provides unbiased, closed-form estimators, including for imaginary-time correlators and integrated susceptibilities. The authors introduce a canonical PMR form to avoid division-by-zero pathologies, prove existence and uniqueness aspects, and demonstrate practical efficacy on transverse-field Ising models and large non-local toy models, with open-source code to enable broad adoption. The approach supports automated construction of estimators for complex, non-local operators, offering substantial speedups and expanding PMR-QMC’s applicability to a wide class of quantum many-body problems. This work thus bridges formal group-theoretic PMR structure with practical, scalable estimators, enabling a black-box workflow for a broad range of Hamiltonians and observables.

Abstract

In a typical finite temperature quantum Monte Carlo (QMC) simulation, estimators for simple static observables such as specific heat and magnetization are known. With a great deal of system-specific manual labor, one can sometimes also derive more complicated non-local or even dynamic observable estimators. Within the permutation matrix representation (PMR) flavor of QMC, however, we show that one can derive formal estimators for arbitrary static observables. We also derive exact, explicit estimators for general imaginary-time correlation functions and non-trivial integrated susceptibilities thereof. We demonstrate the practical versatility of our method by estimating various non-local, random observables for the transverse-field Ising model on a square lattice.

Figures (5)

• Figure 1: We demonstrate clear agreement between PMR-QMC estimates and exact calculation (just direct numerical linear algebra as described in \ref{['app:code-details']}) for a wide variety of observables. Calculations are performed for the $3\times 3$ square TFIM with open boundary conditions in Eq. (\ref{['eq:tfim']}) for $\beta = 1.0, \lambda = 0.5$. QMC points and error bars represent the average and thrice the standard deviation, $3\sigma$, over 100 independent runs with different random seeds.
• Figure 2: We estimate static, dynamic, and derived Hamiltonian-based observables for the $8\times 8$ square TFIM in \ref{['eq:tfim']} with open boundary conditions as a function of transverse field strength. Points and error bars represent the average and twice the standard deviation, $2\sigma$, over 100 independent runs with different random seeds.
• Figure 3: We estimate custom static and dynamic observables defined in terms of $A$ and $B$ given in \ref{['eq:larger-A', 'eq:larger-B']}, respectively for the $8\times 8$ square TFIM in \ref{['eq:tfim']} with open boundary conditions as a function of transverse field strength. Points and error bars represent the average and twice the standard deviation, $2\sigma$, over 100 independent runs with different random seeds.
• Figure 4: We verify that our method outputs the same result up to $3 \sigma$ error bars in the estimation of direct or rotated observables on $H$ and $H_U$ defined in \ref{['eq:prl-model', 'eq:random-H-and-O']}, respectively. (a) A comparison of $\langle O \rangle_H$ (direct) and $\langle O_U \rangle_{H_U}$ (rotated) as a function of $\beta$. (b) A comparison of $f(\beta)$ and $h(\beta)$ defined in \ref{['eq:f-and-h']} as a function of $\beta$.
• Figure 5: A brief summary of wall-clock times used to generate data in this work. (Left) Wall-clock time and $\langle q \rangle$ as a function of $\lambda$ for the $8 \times 8$ TFIM data to collect all measurement estimates plotted in \ref{['fig:standard-obs', 'fig:custom-obs']}. (Right) Wall-clock time and $\langle q \rangle$ as a function of $\beta$ in the study of the direct and rotated models in \ref{['fig:comparison-of-O-and-OU']}. Error bars are 95% intervals, i.e., the band in which 95% of all empirical wall-clock or $\langle q \rangle$ values lie.

Theorems & Definitions (11)

• Definition 1: Permutation matrix representation (PMR) form
• Theorem 1: Existence of PMR form
• proof
• Corollary 1: Products of PMR permutations have no fixed points
• proof
• Theorem 2: Computing $D_\sigma's$ generally
• proof
• Theorem 3: Hermiticity in the PMR
• proof
• Corollary 2: Alternate way to compute $D_\sigma's$