Bounding first-order quantum phase transitions in adiabatic quantum computing

TL;DR

This paper develops a graph-theoretic framework to analyze and bound the occurrence of first-order quantum phase transitions in adiabatic quantum computing, aiming to improve understanding and design of more efficient algorithms.

Contribution

It introduces a novel approach linking graph properties of Hamiltonians to quantum phase transitions, providing bounds on spectral gaps to enhance AQC performance analysis.

Findings

Derived bounds on the minimal spectral gap location.

Linked graph properties to phase transition occurrence.

Enhanced analysis tools for AQC algorithm optimization.

Abstract

In the context of adiabatic quantum computation (AQC), it has been argued that first-order quantum phase transitions (QPTs) due to localisation phenomena cause AQC to fail by exponentially decreasing the minimal spectral gap of the Hamiltonian along the annealing path as a function of the qubit number. The vanishing of the spectral gap is often linked to the localisation of the ground state in a local minimum, requiring the system to tunnel into the global minimum at a later stage of the annealing. Recent methods have been proposed to avoid this phenomenon by carefully designing the involved Hamiltonians. However, it remains a challenge to formulate a comprehensive theory of the effect of the various parameters and the conditions under which QPTs make the AQC algorithm fail. Equipped with concepts from graph theory, in this work we link graph quantities associated to the Hamiltonians along the annealing path with the occurrence of QPTs. These links allow us to derive bounds on the location of the minimal spectral gap along the annealing path, augmenting the toolbox for the analysis of strategies to improve the runtime of AQC algorithms.