Permutationally invariant state reconstruction

TL;DR

This paper introduces an efficient method for reconstructing permutationally invariant quantum states, significantly reducing computational complexity and enabling rapid analysis of large quantum systems like 20 qubits.

Contribution

The authors develop a scalable state reconstruction scheme using a special state representation and convex optimization, compatible with common reconstruction methods like maximum likelihood.

Findings

Reconstruction of 20-qubit states in a few minutes on standard computers.

Significant reduction in computational complexity for permutationally invariant states.

Compatibility with existing reconstruction principles such as maximum likelihood.

Abstract

Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, also an efficient way to reconstruct the density operator from the observed finite data set. Since state reconstruction typically requires the solution of a non-linear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efficient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today's experiments. This high efficiency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer.