Advanced measurement in quantum Monte Carlo

Finite temperature quantum Monte Carlo (QMC) is a class of numerical techniques used to estimate the properties of quantum systems in thermal equilibrium. In most QMC schemes, simple observables such as average energy, specific heat, and local expectation values can be measured. However, deriving estimators for arbitrary observables is generally not possible. In contrast, we demonstrate that within the permutation matrix representation (PMR) framework of QMC, formal estimators for arbitrary observables can always be derived.

For models like the transverse-field Ising model (TFIM), we prove that these formal estimators should work in principle and show they work well in practice. Specifically, we successfully estimate non-local Pauli operators, two-point imaginary-time correlation functions of non-local Pauli operators, and integrated susceptibilities. Furthermore, for other models, we provide a sufficient condition under which these formal estimators may fail and provide an example of an observable that cannot be accurately measured for a simple model within PMR-QMC.

In summary, our work develops a rigorous and systematic framework for observable estimation in PMR-QMC, offering both theoretical insights and practical tools for a wide range of quantum systems.