On the Universality of the Quantum Approximate Optimization Algorithm

TL;DR

This paper proves the universality of the quantum approximate optimization algorithm (QAOA) for certain Hamiltonians, showing its capability to implement any quantum unitary under specific conditions, thus broadening its theoretical foundation.

Contribution

The paper provides a complete proof of QAOA's universality for 1D local Hamiltonians and extends this to more complex graph-structured Hamiltonians, clarifying its theoretical scope.

Findings

QAOA can be universal for 1D local Hamiltonians with ZZ terms.

Universality extends to Hamiltonians with ZZ and ZZZ terms on certain graphs.

The paper establishes precise conditions for QAOA's universality.

Abstract

The quantum approximate optimization algorithm (QAOA) is considered to be one of the most promising approaches towards using near-term quantum computers for practical application. In its original form, the algorithm applies two different Hamiltonians, called the mixer and the cost Hamiltonian, in alternation with the goal being to approach the ground state of the cost Hamiltonian. Recently, it has been suggested that one might use such a set-up as a parametric quantum circuit with possibly some other goal than reaching ground states. From this perspective, a recent work [S. Lloyd, arXiv:1812.11075] argued that for one-dimensional local cost Hamiltonians, composed of nearest neighbor ZZ terms, this set-up is quantum computationally universal, i.e., all unitaries can be reached up to arbitrary precision. In the present paper, we give the complete proof of this statement and the precise conditions under which such a one-dimensional QAOA might be considered universal. We further generalize this type of universality for certain cost Hamiltonians with ZZ and ZZZ terms arranged according to the adjacency structure of certain graphs and hypergraphs.