Efficient Direct Tomography for Matrix Product States

TL;DR

This paper introduces an efficient method for reconstructing matrix product states (MPS) using local measurements, which is significantly faster than traditional methods and can certify the state as an MPS.

Contribution

The authors present a direct tomography technique for MPS that is exponentially faster and more versatile than existing variational approaches, capable of handling a broader class of states.

Findings

Method is exponentially faster than standard tomography.

Can certify the unknown state as an MPS.

Applicable to a wide range of states including GHZ and W states.

Abstract

In this note, we describe a method for reconstructing matrix product states from a small number of efficiently-implementable measurements. Our method is exponentially faster than standard tomography, and it can also be used to certify that the unknown state is an MPS. The basic idea is to use local unitary operations to measure in the Schmidt basis, giving direct access to the MPS representation. This compares favorably with recently and independently proposed methods that recover the MPS tensors by performing a variational minimization, which is computationally intractable in certain cases. Our method also has the advantage of recovering any MPS, while other approaches were limited to special classes of states that exclude important examples such as GHZ and W states.