A monomial matrix formalism to describe quantum many-body states

TL;DR

This paper introduces a monomial matrix framework to describe and simulate a broad class of quantum many-body states, unifying various important state families and analyzing their properties and computational complexity.

Contribution

The paper develops the concept of M-spaces based on monomial unitary matrices, unifies multiple quantum state families, and provides methods for their eigenbasis construction and classical simulation.

Findings

M-spaces include stabilizer states, AKLT, and anyon models.

A procedure for eigenbasis construction of M-spaces is derived.

Certain M-spaces can be efficiently simulated classically.

Abstract

We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called here "M-spaces"; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M-spaces encompass various important state families, such as all Pauli stabilizer states and codes, the AKLT model, Kitaev's (abelian and non-abelian) anyon models, group coset states, W states and the locally maximally entanglable states. We furthermore show how basic properties of M-spaces can transparently be understood by manipulating their monomial stabilizer groups. In particular we derive a unified procedure to construct an eigenbasis of any M-space, yielding an explicit formula for each of the eigenstates. We also discuss the computational complexity of M-spaces and show that basic problems, such as estimating local expectation values, are NP-hard. Finally we prove that a large subclass of M-spaces---containing in particular most of the aforementioned examples---can be simulated efficiently classically with a unified method.