Time-dependent Hamiltonians with 100% evolution speed efficiency

TL;DR

This paper explores the limits of evolution speed in quantum systems, introducing a method to construct time-dependent Hamiltonians that achieve 100% efficiency, including non-Hermitian cases, with minimal resource use.

Contribution

It presents a novel approach to design time-dependent Hamiltonians that guarantee 100% speed efficiency in quantum evolution, applicable to both Hermitian and non-Hermitian systems.

Findings

Derived resource-based upper bounds on evolution speed.

Introduced quantum speed efficiency as a resource measure.

Provided a recipe for constructing maximally efficient Hamiltonians.

Abstract

The evolution speed in projective Hilbert space is considered for Hermitian Hamiltonians and for non-Hermitian (NH) ones. Based on the Hilbert-Schmidt norm and the spectral norm of a Hamiltonian, resource-related upper bounds on the evolution speed are constructed. These bounds are valid also for NH Hamiltonians and they are illustrated for an optical NH Hamiltonian and for a non-Hermitian $\mathcal{PT}-$symmetric matrix Hamiltonian. Furthermore, the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective Hilbert space. A recipe for the construction of time-dependent Hamiltonians which ensure 100% speed efficiency is given. Generally these efficient Hamiltonians are NH but there is a Hermitian efficient Hamiltonian as well. Finally, the extremal case of a non-Hermitian non-diagonalizable Hamiltonian with vanishing energy difference is shown to produce a 100% efficient evolution with minimal resources consumption.