Extending Monte Carlo Methods to Factor Graphs with Negative and Complex Factors

The partition function of a factor graph can sometimes be accurately estimated by Monte Carlo methods. In this paper, such methods are extended to factor graphs with negative and complex factors.

💡 Research Summary

The paper addresses a fundamental limitation of Monte Carlo (MC) techniques for estimating the partition function of factor graphs: traditional MC methods assume that all factor values are non‑negative real numbers, which allows them to be interpreted directly as probabilities. In many modern applications—quantum circuit simulation, tensor‑network contractions with complex phases, statistical models that involve sine or cosine terms, and complex‑valued error‑correcting codes—factors can be negative or even complex. When such factors appear, the naïve MC estimator loses its probabilistic meaning, leading to severe variance inflation and, in the worst case, complete failure of the estimator.

To overcome this, the authors propose a two‑stage generalization. First, they construct an auxiliary probability distribution (q(x)=|f(x)|/\widehat Z), where (f(x)) is the original (possibly signed or complex) factor product and (\widehat Z) is an estimate of its absolute‑value normalizer. Sampling is performed from (q) using standard Gibbs or Metropolis–Hastings kernels, which remain valid because (q) is a proper probability distribution. Second, each sampled configuration (x) is re‑weighted by the “phase” (\phi(x)=\arg f(x)) (for real‑negative factors (\phi(x)=\pi) or (0)). The estimator for the true partition function becomes
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