Specifically, the diagonal elements(can possibly both positive and negative)of the matrix can be computed efficiently but the total number is impossible to enumerate. My first thought about this was running a MCMC with weight set to be the absolute value of each element, however, this just exactly leads to the notorious sign problem. Thus I'm wondering is there any smarter way to tackle that.
2026-03-29 14:58:54.1774796334
How to stocahstically estimate the trace of matrix
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There are randomized algorithms for estimating elements of the diagonal of a matrix or the trace of a matrix from matrix-vector products with random vectors. See for example:
Bekas, Costas, Effrosyni Kokiopoulou, and Yousef Saad. "An estimator for the diagonal of a matrix." Applied numerical mathematics 57.11 (2007): 1214-1229.