Optimal parametrizations of adiabatic paths

TL;DR

This paper investigates optimal parametrizations of adiabatic quantum paths, revealing that dephasing Lindblad evolutions have unique optimal solutions characterized by constant tunneling rates, with implications for quantum search algorithms.

Contribution

It characterizes optimal adiabatic paths under dephasing Lindblad evolutions using Euler-Lagrange equations, highlighting their unique properties and applications to quantum algorithms.

Findings

Dephasing Lindblad evolutions have unique optimal parametrizations.

Optimal paths maintain constant tunneling rates regardless of the spectral gap.

Application to quantum search recovers Grover's algorithm results.

Abstract

The parametrization of adiabatic paths is optimal when tunneling is minimized. Hamiltonian evolutions do not have unique optimizers. However, dephasing Lindblad evolutions do. The optimizers are simply characterized by an Euler-Lagrange equation and have a constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing. Dephasing rates that beat Grover imply hidden resources in Lindblad operators.