# A Structured View on Weighted Counting with Relations to Counting, Quantum Computation and Applications

## TL;DR

This paper introduces a structured framework for weighted counting problems, exploring their complexity, relationships, and applications across computational complexity, quantum computing, and probabilistic models.

## Contribution

It defines classes of weighted counting problems based on weight ranges, analyzes their relationships, and applies this framework to simplify and resolve open questions in various fields.

## Key findings

- Weighted counting classes generalize traditional counting problems.
- Exponential small additive approximation remains as hard as standard counting.
- Framework simplifies understanding of complexity in probabilistic and quantum models.

## Abstract

Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path of polynomial-time non-deterministic Turing machines and the goal is to compute the sum of the weights of all paths (instead of just computing the number of accepting paths). Useful closure properties and plenty of applications make weighted counting problems interesting. The general definition of these problems captures even undecidable problems, but it turns out that obtaining an exponentially small additive approximation is just as hard as solving conventional counting problems. In many cases such an approximation is sufficient and working with weighted counting problems tends to be very convenient. We present a structured view on weighted counting by defining classes that depend on the range of the function that assigns weights to paths and by showing the relationships between these different classes. These classes constitute generalizations of the usual counting problems. Weighted counting allows us to easily cast a number of famous results of computational complexity in its terms, especially regarding counting and quantum computation. Moreover, these classes are flexible enough and capture the complexity of various problems in fields such as probabilistic graphical models and stochastic combinatorial optimization. Using the weighted counting terminology and our results, we are able to simplify and answer some open questions in those fields.