Hidden local adiabatic ramp in the modulated time evolution and the quantum approximate optimization algorithm

TL;DR

This paper introduces a modulated time evolution approach with a local adiabatic ramp to improve adiabatic state preparation and draws parallels to the QAOA, providing insights into its physical interpretation.

Contribution

It presents a novel modulated evolution method with a local adiabatic ramp and links QAOA's shape to this approach for better understanding.

Findings

Optimized modulated evolution closely resembles local adiabatic ramps.

QAOA's shape is similar to the local adiabatic ramp, aiding interpretation.

Enhanced robustness against small energy gaps in state preparation.

Abstract

Adiabatic state preparation provides an analytical solution for generating the ground state of a target Hamiltonian, starting from an easily prepared ground state of the initial Hamiltonian. While effective for time-dependent Hamiltonians with an energy gap to the first coupled excited state, the process becomes exceedingly slow as the gap narrows. Rather than strictly following the adiabatic theorem, a more robust approach allows controlled diabatic excitations during the evolution and numerically optimizes the path to eliminate these excitations by the end. In this work, this is achieved via modulated time evolution, using a time dependent oscillating field lambda(t) to modulate the Hamiltonian, in conjunction with a transverse field B(t) whose optimized shape closely resembles a local adiabatic ramp. Beyond modulated time evolution, the quantum approximate optimization algorithm (QAOA), which also employs a transverse field defined as beta(t) over gamma(t), exhibits a shape similar to the local adiabatic ramp. This resemblance offers a more intuitive and physically motivated way to understand the QAOA algorithm through the lens of time evolution.