Nielsen complexity of coherent spin state operators

TL;DR

This paper computes Nielsen's circuit complexity for coherent spin state operators, relating it to squeezing and entanglement, and extends the analysis to various Hamiltonians including the LMG model.

Contribution

It introduces a method to calculate Nielsen complexity for spin states using Euler angles and extends it to complex Hamiltonians, addressing challenges with the LMG model.

Findings

Complexity relates to squeezing parameters in one-axis twisted Hamiltonians.

Extended complexity calculations to systems with external fields and non-linear interactions.

Resolved computational issues for the LMG model by using Tait-Bryan parametrisation.

Abstract

We calculate Nielsen's circuit complexity of coherent spin state operators. An expression for the complexity is obtained by using the small angle approximation of the Euler angle parametrisation of a general $SO(3)$ rotation. This is then extended to arbitrary times for systems whose time evolutions are generated by couplings to an external field, as well as non-linearly interacting Hamiltonians. In particular, we show how the Nielsen complexity relates to squeezing parameters of the one-axis twisted Hamiltonians in a transverse field, thus indicating its relation with pairwise entanglement. We further point out the difficulty with this approach for the Lipkin-Meshkov-Glick model, and resolve the problem by computing the complexity in the Tait-Bryan parametrisation.