On the efficiency of Hamiltonian-based quantum computation for low-rank matrices

TL;DR

This paper explores the efficiency and limitations of Hamiltonian-based quantum algorithms, especially adiabatic quantum computing, for low-rank matrix problems and unstructured search, highlighting conditions for quantum speedup and control precision constraints.

Contribution

It extends AQC algorithms to unknown marked items, provides lower bounds on search times, and analyzes the impact of control errors on quantum speedup.

Findings

Quantum speedup requires very precise control conditions.

Lower bounds on search and evolution times are established.

Quantum advantage may be limited by control error constraints.

Abstract

We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.