A monomial matrix formalism to describe quantum many-body states
Maarten Van den Nest · New Journal of Physics · 2011
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 Affleck–Kennedy–Lieb–Tasaki (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 be understood transparently 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.