Preparation and verification of tensor network states

TL;DR

This paper introduces a method to efficiently prepare and verify tensor network states on regular lattices, using semi-definite programming to assess adiabatic gaps and a set of observables for state characterization.

Contribution

It provides a scalable approach to verify the adiabatic preparation of tensor network states and introduces a complete set of efficiently computable observables for state verification.

Findings

Efficient computation of lower bounds to the gap of parent Hamiltonians.

Development of a complete set of observables for state characterization.

Identification of a subset of observables suitable for practical verification.

Abstract

We consider a family of tensor network states defined on regular lattices that come with a natural definition of an adiabatic path to prepare them. This family comprises relevant classes of states, such as injective Matrix Product and Projected Entangled-Pair States, and some corresponding to classical spin models. We show how uniform lower bounds to the gap of the parent Hamiltonian along the adiabatic trajectory can be efficiently computed using semi-definite programming. This allows one to check whether the adiabatic preparation can be performed efficiently with a scalable effort. We also derive a set of observables whose expectation values can be easily determined and that form a complete set, in the sense that they uniquely characterize the state. We identify a subset of those observables which can be efficiently computed if one has access to the quantum state and local measurements, and analyze how they can be used in verification procedures.