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

TL;DR

This paper investigates the scalability of the Quantum Approximate Optimization Algorithm (QAOA) for linear objective functions under fixed-depth constraints in gate-based quantum computers. It develops the QAOA state for a linear Ising Hamiltonian and shows that the optimum-measurement probability for replicated coefficient vectors factorizes as $Pr_{opt}(a^k,\gamma,\beta)=(Pr_{opt}(a,\gamma,\beta))^k$, leading to an exponential runtime $T(a^k)=(Pr_{opt}(a,\gamma,\beta))^{-n/m}$ when $p$ is fixed. The authors prove exponential-time lower bounds for $p=1$ and $p=2$ and conjecture the same holds for all fixed $p$, supported by small-$p$ proofs and experiments. The result implies that QAOA may not provide subexponential speedups for general linear optimization at fixed depth, underscoring the need for new quantum optimization algorithms, potentially leveraging Fourier-transform-based ideas, to achieve quantum supremacy in optimization tasks.

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.

Figures (2)

• Figure 1: Variational quantum algorithm.
• Figure 2: The QAOA's execution workflow

Theorems & Definitions (6)

• theorem thmcountertheorem: Exponential runtime of QAOA with constant $p$ for linear functions
• proof
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof