Quantum-inspired permanent identities

TL;DR

This paper introduces quantum-inspired proofs for various permanent identities, including the MacMahon master theorem, highlighting their significance in quantum computing and classical sampling complexity.

Contribution

It provides novel quantum-inspired proofs for existing and new permanent identities, including generalizations of the MacMahon master theorem, linking combinatorics and quantum computing.

Findings

Quantum-inspired proofs for permanent identities

New generalizations of the MacMahon master theorem

Demonstration of classical hardness in quantum sampling

Abstract

The permanent is pivotal to both complexity theory and combinatorics. In quantum computing, the permanent appears in the expression of output amplitudes of linear optical computations, such as in the Boson Sampling model. Taking advantage of this connection, we give quantum-inspired proofs of many existing as well as new remarkable permanent identities. Most notably, we give a quantum-inspired proof of the MacMahon master theorem as well as proofs for new generalizations of this theorem. Previous proofs of this theorem used completely different ideas. Beyond their purely combinatorial applications, our results demonstrate the classical hardness of exact and approximate sampling of linear optical quantum computations with input cat states.