# Complexity of full counting statistics of free quantum particles in product states

## TL;DR

This paper investigates the computational complexity of calculating expectation values in multi-particle quantum states, revealing conditions under which these calculations are computationally feasible or hard, with implications for quantum computing resources.

## Contribution

It establishes that full counting statistics are efficiently computable in finite modes for certain quantum states, regardless of initial complexity, advancing understanding of quantum computational resources.

## Key findings

- Expectation values can be either easy or hard to compute depending on initial states.
- Full counting statistics in finite modes are efficiently computable for fermionic and single-boson states.
- Results have implications for using multi-particle states as quantum computing resources.

## Abstract

We study the computational complexity of quantum-mechanical expectation values of single-particle operators in bosonic and fermionic multi-particle product states. Such expectation values appear, in particular, in full-counting-statistics problems. Depending on the initial multi-particle product state, the expectation values may be either easy to compute (the required number of operations scales polynomially with the particle number) or hard to compute (at least as hard as a permanent of a matrix). However, if we only consider full counting statistics in a finite number of final single-particle states, then the full-counting-statistics generating function becomes easy to compute in all the analyzed cases. We prove the latter statement for the general case of the fermionic product state and for the single-boson product state (the same as used in the boson-sampling proposal). This result may be relevant for using multi-particle product states as a resource for quantum computing.