Lower bounds for adiabatic quantum algorithms by quantum speed limits

TL;DR

This paper presents a straightforward method to estimate lower bounds on the runtime of adiabatic quantum algorithms, using variance of the final Hamiltonian, and applies it to problems like finding k-cliques in graphs.

Contribution

Introduces a simple variance-based framework for lower bounds in adiabatic quantum algorithms, connecting it with spectral gap analysis in certain cases.

Findings

Derived lower bounds for adiabatic algorithms solving k-clique problems

Established equivalence between variance-based bounds and spectral gap analysis for specific Hamiltonians

Applied the framework to analyze adiabatic versions of key quantum algorithms

Abstract

We introduce a simple framework for estimating lower bounds on the runtime of a broad class of adiabatic quantum algorithms. The central formula consists of calculating the variance of the final Hamiltonian with respect to the initial state. After examining adiabatic versions of certain keystone circuit-based quantum algorithms, this technique is applied to adiabatic quantum algorithms with undetermined speedup. In particular, we analytically obtain lower bounds on adiabatic algorithms for finding k-clique in random graphs. Additionally, for a particular class of Hamiltonian, it is straightforward to prove the equivalence between our framework and the conventional approach based on spectral gap analysis.