Permanents in linear optical networks

TL;DR

This paper explores the connection between linear optical networks and matrix permanents, revealing computational hardness and providing new insights into the properties of unitary matrices in quantum optics.

Contribution

It establishes a fundamental link between matrix elements in optical networks and permanents, highlighting computational complexity and deriving properties of unitary matrices.

Findings

Calculating matrix elements in optical networks is computationally hard due to permanents.

The permanent of any unitary matrix can take values across the entire unit disk.

Quantum mechanics offers simpler derivations for certain matrix analysis results.

Abstract

We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to the computation of permanents. An implication of this remarkable fact is that all calculations that are based on evaluating matrix elements are generically computationally hard. Moreover, quantum mechanics provides simpler derivations of certain matrix analysis results which we exemplify by showing that the permanent of any unitary matrix takes its values across the unit disk in the complex plane.