The Quantum Approximate Optimization Algorithm Can Require Exponential Time to Optimize Linear Functions

TL;DR

This paper demonstrates that the Quantum Approximate Optimization Algorithm (QAOA) can require exponential time to optimize linear functions when the number of layers is limited, highlighting the need for new algorithms for quantum advantage.

Contribution

The paper provides mathematical evidence that QAOA requires exponential time for linear functions with limited layers and conjectures this holds for all constant layer counts, emphasizing the necessity for novel quantum algorithms.

Findings

QAOA needs exponential time for linear functions with limited layers

Exponential runtime occurs when layers are fewer than the coefficients of the linear function

Quantum advantage in optimization requires new algorithms beyond QAOA

Abstract

QAOA is a hybrid quantum-classical algorithm to solve optimization problems in gate-based quantum computers. It is based on a variational quantum circuit that can be interpreted as a discretization of the annealing process that quantum annealers follow to find a minimum energy state of a given Hamiltonian. This ensures that QAOA must find an optimal solution for any given optimization problem when the number of layers, $p$, used in the variational quantum circuit tends to infinity. In practice, the number of layers is usually bounded by a small number. This is a must in current quantum computers of the NISQ era, due to the depth limit of the circuits they can run to avoid problems with decoherence and noise. In this paper, we show mathematical evidence that QAOA requires exponential time to solve linear functions when the number of layers is less than the number of different coefficients of the linear function $n$. We conjecture that QAOA needs exponential time to find the global optimum of linear functions for any constant value of $p$, and that the runtime is linear only if $p \geq n$. We conclude that we need new quantum algorithms to reach quantum supremacy in quantum optimization.