Random phase vector for calculating the trace of a large matrix

TL;DR

This paper demonstrates that using random phase vectors minimizes statistical error in trace calculations of large matrices, enhancing the accuracy of quantum system analyses.

Contribution

It introduces the use of random phase vectors as optimal for trace estimation, reducing statistical error compared to other random vectors.

Findings

Random phase vectors yield the smallest statistical error in trace calculations.

Supports the application of random phase vectors in quantum density of states computations.

Improves accuracy in linear response function calculations for large quantum systems.

Abstract

We derive an estimate of statistical error in calculating the trace of a large matrix by using random vector, and show that {\em random phase vector} gives the results with the smallest statistical error for a given basis set. This result supports use of random phase vectors in the calculation of density of states and linear response functions of large quantum systems.