Addressing general measurements in quantum Monte Carlo.

Abstract

Quantum Monte Carlo is one of the most promising approaches for dealing with large-scale quantum many-body systems. It has played an extremely important role in understanding strongly correlated physics. However, two fundamental problems, namely the sign problem and general measurement issues, have seriously hampered its scope of application. We propose a universal scheme to tackle the problems of general measurement. The target observables are expressed as the ratio of two types of partition functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⟨</mo> <mi>O</mi> <mo>⟩</mo> <mo>=</mo> <mover><mrow><mi>Z</mi></mrow> <mo>¯</mo></mover> <mo>/</mo> <mi>Z</mi></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"> <mover><mrow><mi>Z</mi></mrow> <mo>¯</mo></mover> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mrow><mo>(</mo> <mrow> <msup><mrow><mi>O</mi> <mi>e</mi></mrow> <mrow><mo>-</mo> <mi>β</mi> <mi>H</mi></mrow> </msup> </mrow> <mo>)</mo></mrow> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mrow><mo>(</mo> <mrow> <msup><mrow><mi>e</mi></mrow> <mrow><mo>-</mo> <mi>β</mi> <mi>H</mi></mrow> </msup> </mrow> <mo>)</mo></mrow> </math> . These two partition functions can be estimated separately within the reweight-annealing frame, and then be connected by an easily solvable reference point. We have successfully applied this scheme to XXZ model and transverse field Ising model, from 1D to 2D systems, from two-body to multi-body correlations and even non-local disorder operators, and from equal-time to imaginary-time correlations. The reweighting path is not limited to physical parameters, but also works for space and time. Essentially, this scheme solves the long-standing problem of calculating the overlap between different distribution functions in mathematical statistics, which can be widely used in statistical problems, such as quantum many-body computation, big data and machine learning.