Constructions of Optimal-Speed Quantum Evolutions: A Comparative Study

TL;DR

This paper compares two methods for constructing optimal-speed quantum evolutions on the Bloch sphere, showing their equivalence under certain conditions and deriving explicit optimal Hamiltonians and evolutions.

Contribution

It provides a detailed comparison of two approaches for optimal quantum evolutions, extending Mostafazadeh's method to nonzero trace Hamiltonians and establishing their equivalence.

Findings

Explicit optimal Hamiltonians derived for both approaches

Optimal unitary operators are rotations about axes orthogonal to initial and final states

The two approaches are equivalent when extended and initial state is on the north pole

Abstract

We present a comparative analysis of two different constructions of optimal-speed quantum Hamiltonian evolutions on the Bloch sphere. In the first approach (Mostafazadeh's approach), the evolution is specified by a traceless stationary Hermitian Hamiltonian and occurs between two arbitrary qubit states by maximizing the energy uncertainty. In the second approach (Bender's approach), instead, the evolution is characterized by a stationary Hermitian Hamiltonian which is not traceless and occurs between an initial qubit state on the north pole and an arbitrary final qubit state. In this second approach, the evolution occurs by minimizing the evolution time subject to the constraint that the difference between the largest and the smallest eigenvalues of the Hamiltonian is kept fixed. For both approaches we calculate explicitly the optimal Hamiltonian, the optimal unitary evolution operator and, finally, the optimal magnetic field configuration. Furthermore, we show in a clear way that Mostafazadeh's and Bender's approaches are equivalent when we extend Mostafazadeh's approach to Hamiltonians with nonzero trace and, at the same time, focus on an initial quantum state placed on the north pole of the Bloch sphere. Finally, we demonstrate in both scenarios that the optimal unitary evolution operator is a rotation about an axis that is orthogonal to the unit Bloch vectors that correspond to the initial and final qubit states.