Relating computational complexity and quantum spectral complexity

TL;DR

This paper investigates the transition in spectral complexity of the AQC 3-SAT problem, linking it to classical phase transitions and exploring implications for quantum computation and experiments.

Contribution

It establishes a connection between spectral fluctuations in quantum algorithms and classical computational phase transitions, providing new insights into quantum complexity.

Findings

Spectral fluctuations transition from Poisson to Random Matrix Theory form.

Transition correlates with classical 3-SAT phase transition.

Discusses implications for Gaussian Processes and experimental applications.

Abstract

It is found that the statistical level fluctuations of the AQC 3-SAT problem undergo a transition from a poisson (regular) fluctuation form to a form consistent with the predictions of Random Matrix Theory. We present data which suggests this transition correlates with the computational phase transition in the classical 3-SAT problem. Application to Gaussian Processes and implication for experiment is discussed.