Quantum brachistochrone problem for two spins-1/2 with anisotropic Heisenberg interaction

TL;DR

This paper investigates the quantum brachistochrone problem for a two-spin-1/2 system with anisotropic Heisenberg interaction, deriving optimal evolution times and generating quantum gates within specific subspaces.

Contribution

It provides a detailed analysis of the brachistochrone evolution for anisotropic Heisenberg models and constructs quantum gates in separated subspaces.

Findings

Derived explicit solutions for brachistochrone evolution times.

Constructed quantum gates such as entangler, SWAP, and iSWAP.

Analyzed the evolution in two distinct subspaces separately.

Abstract

We study the quantum brachistochrone evolution for a system of two spins-$\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\|\uparrow\uparrow\rangle$, $\|\downarrow\downarrow\rangle$ and $\|\uparrow\downarrow\rangle$, $\|\downarrow\uparrow\rangle$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.