The Discretized Adiabatic Theorem

TL;DR

This paper introduces a discretized version of the adiabatic theorem, using a stepwise measurement process to evolve a quantum state towards the ground state of a final operator, with implications for quantum state preparation.

Contribution

It presents a novel discretized approach to the adiabatic theorem utilizing incremental measurements and a new rule relating operators to expectation values and variances.

Findings

The method successfully guides the state to the ground state of the final operator.

The process maintains the state in the ground state with high probability as step size approaches zero.

The approach connects measurement theory with adiabatic evolution in quantum systems.

Abstract

A discretized version of the adiabatic theorem is described with the help of a rule relating a Hermitian operator to its expectation value and variance. The simple initial operator X with known ground state is transformed in a series of N small steps into a more complicated final operator Z with unknown ground state. Each operator along the discretised path in the space of Hermitian matrices is used to measure the state, initially the ground state of X. Measurements similar to the Zeno effect or Renninger's negative measurements modify the state incrementally. This process eventually leads to an eigenstate combination of Z. In the limit of vanishing step size the state stays with overwhelming probability in the ground state of each of the N observables.