Quantum adiabatic evolutions that can't be used to design efficient algorithms

TL;DR

This paper demonstrates that certain simple initial or final Hamiltonians in quantum adiabatic evolution lead to exponentially small gaps, making them unsuitable for efficient quantum algorithms.

Contribution

It establishes limitations on the effectiveness of quantum adiabatic algorithms with simple Hamiltonians, highlighting cases where they cannot be efficient.

Findings

Simple initial Hamiltonians lead to exponentially small gaps.

Simple final Hamiltonians also result in long running times.

Some functions cannot be minimized efficiently by any adiabatic evolution.

Abstract

Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.